Springer Theses Recognizing Outstanding Ph.D. Research Adam Ross Solomon Cosmology Beyond Einstein Springer Theses Recognizing Outstanding Ph.D. Research Aims and Scope The series "Springer Theses" brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student's supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. 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More information about this series at http://www.springer.com/series/8790 Adam Ross Solomon Cosmology Beyond Einstein Doctoral Thesis accepted by the University of Cambridge, UK Springer Author Dr. Adam Ross Solomon Center for Particle Cosmology University of Pennsylvania Philadelphia USA Supervisor Prof. John D. Barrow Department of Applied Mathematics and Theoretical Physics The Centre for Mathematical Sciences University of Cambridge Cambridge UK ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-319-46620-0 ISBN 978-3-319-46621-7 (eBook) DOI 10.1007/978-3-319-46621-7 Library of Congress Control Number: 2016953643 © Springer International Publishing AG 2017 This work is subject to copyright. 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Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland The effort to understand the universe is one of the very few things which lifts human life a little above the level of farce and gives it some of the grace of tragedy. Steven Weinberg, The First Three Minutes This thesis is dedicated to my parents, Scott and Edna Solomon, without whose love and support I could never have made it this far. Supervisor's Foreword Einstein's general theory of relativity is one of the most impressive human achievements. It superseded Newton's great work of 1687 to provide us with a new theory of gravitation that extended Newton's theory to domains where velocities could approach that of light and gravitational forces were correspondingly strong. In the limit that speeds are slow and gravity is weak, Einstein's theory is well approximated by Newton's. This is where modern physics departs from the Kuhnian story of scientific 'revolutions'. Old theories are not simply replaced by new ones. Rather, they become limiting cases of the new theory which will hold good in extreme situations where the old one cannot remain consistent. Yet, Einstein's theory went further than merely extending the domain of applicability of our theory of gravity. For the first time it provided a collection of differential equations whose solutions, all of them, describe entire universes. For the first time, cosmology became a science. Physicists could try to solve Einstein's equations in simple cases where there was lots of symmetry to find possible descriptions of our entire astronomical universe. These solutions could then be tested against the astronomical evidence and the subject began to resemble other experimental sciences. Although you cannot experiment on the universe—we only have one universe on display—you can predict correlations that should be observed between different properties of the same mathematical universe and look to see if they exist. In this way, cosmology has become a major scientific enterprise. It makes use of a host of new technologies to create light detectors of previously unimagined sensitivity right across the electromagnetic spectrum and has even begun to see direct evidence of gravitational waves. It has joined forces with elementary particle physicists to share insights and constraints on the behaviour of matter at the highest possible energies. And it has fully exploited the massive increase in computational ability that allows us to simulate the behaviour of large and complicated agglomerations of matter to follow the processes that have led to the formation of galaxies in the universe. Einstein's theory of general relativity is a spectacular success and agrees with all the observational evidence to extraordinary accuracy. It would be fair to say that the IX X Supervisor's Foreword agreement between theory and observation is so precise that in certain situations, like the binary pulsar's dynamics, it provides us with the surest and most accurate knowledge that human beings have of anything in their experience. So, why do we want to go 'beyond Einstein' in the words of Adam Solomon's thesis title? There are two main reasons. The first is that Einstein's theory has its limits of reliable applicability, just as Newton's theory does. When the density of matter gets too high, as it does near the apparent beginning of an expanding universe and near the centre of black holes, we expect quantum mechanics to modify the character of gravity in a way that will be described by some new theory of quantum gravity. Perhaps this future theory will modify general relativity so as to remove the 'singularities' that presently signal the beginning and of time at the beginning of the universe and the end of time in the inexorable contraction at the centres of black holes? The second reason to look beyond Einstein is of more recent origin. Just 17 years ago, astronomers first discovered that the expansion of the universe smoothly changed gear from deceleration to acceleration about 4 billion years ago. Why this occurred is a big mystery. Three different lines of astronomical evidence find the cause to be a ubiquitous form of energy in the universe—dubbed 'dark energy'— that is gravitationally repulsive. Physicists knew that quantum vacuum energy can have this repulsive effect because of its negative pressure, but no one expected the effects to be dramatically manifested so late in the universe's expansion history. About 70 % of the mass-energy in the universe seems to be in the form of this dark component. What is this dark energy? Is it just a new type of matter field that we have not identified and logged into the energy budget of the universe? This is one line of inquiry that cosmologists explore. The other is to investigate whether there are extensions of Einstein's theory of gravity which introduce new gravitational effects that act to accelerate the expansion of the universe when it is billions of years old and gravity is weak. These new behaviours of gravity need to be well circumscribed. They must not produce new adverse effects locally and in parts of cosmology where observations concur with Einstein's predictions to high accuracy. Adam Solomon's thesis explores a wide class of extensions to Einstein's theory to see whether they can potentially explain the observed acceleration of the universe and account for the existence of galaxies. These extensions cover theories which include a graviton with a non-zero mass and others, like bigravity, where there are two underlying spacetime metrics instead of one. These theories are mathematically more complicated than Einstein's and contain undesirable possibilities that need to be understood and excluded. Adam's thesis contains an elegant and systematic study of these theories, connecting abstract mathematical studies to astronomical predictions and observational tests of the theories. This analysis discovers new ways to solve the equations describing the growth of inhomogeneities and a facility with the observational data and statistical analysis needed to put them to the test. Adam combines a very wide range of mathematical skills and astrophysical understanding to advance our understanding of what a new theory of gravity that Supervisor's Foreword XI solves the dark energy problem is allowed to look like. The result is a valuable comprehensive study that will lead us a step closer towards the solution of the dark energy problem. Cambridge, UK Prof. John D. Barrow August 2016 Abstract The accelerating expansion of the Universe poses a major challenge to our understanding of fundamental physics. One promising avenue is to modify general relativity and obtain a new description of the gravitational force. Because gravitation dominates the other forces mostly on large scales, cosmological probes provide an ideal testing ground for theories of gravity. In this thesis, we describe two complementary approaches to the problem of testing gravity using cosmology. In the first part, we discuss the cosmological solutions of massive gravity and its generalisation to a bimetric theory. These theories describe a graviton with a small mass, and can potentially explain the late-time acceleration in a technically natural way. I describe these self-accelerating solutions and investigate the cosmological perturbations in depth, beginning with an investigation of their linear stability, followed by the construction of a method for solving these perturbations in the quasistatic limit. This allows the predictions of stable bimetric models to be compared to observations of structure formation. Next, I discuss prospects for theories in which matter "doubly couples" to both metrics, and examine the cosmological expansion history in both massive gravity and bigravity with a specific double coupling which is ghost-free at low energies. In the second and final part, we study the consequences of Lorentz violation during inflation. We consider Einstein-aether theory, in which a vector field spontaneously breaks Lorentz symmetry and couples nonminimally to the metric, and allow the vector to couple in a general way to a scalar field. Specialising to inflation, we discuss the slow-roll solutions in background and at the perturbative level. The system exhibits a severe instability which places constraints on such a vector-scalar coupling to be at least five orders of magnitude stronger than suggested by other bounds. As a result, the contribution of Lorentz violation to the inflationary dynamics can only affect the cosmic microwave background by an unobservably small amount. Xlll Parts of this thesis have been published in the following journal articles: Following the tendency of modern research in theoretical physics, most of the material discussed in this dissertation is the result of research in a collaboration network. In particular, Chaps. 3-7 were based on work done in collaboration with Yashar Akrami, Luca Amendola, Jonas Enander, Tomi Koivisto, Frank Konnig, Edvard Mortsell, and Mariele Motta, published in Refs. [1-5] while Chap. 8 is the result of work done in collaboration with John Barrow, published as Ref. [6]. I have made major contributions to the above, in terms of both results and writing. References 1. Y. Akrami, T.S. Koivisto, and A.R. Solomon, The nature of spacetime in bigravity: two metrics or none?, Gen.Rel.Grav. 47 (2014) 1838, [arXiv:1404. 0006]. 2. A.R. Solomon, Y. Akrami, and T.S. Koivisto, Linear growth of structure in massive bigravity, JCAP 1410 (2014) 066, [arXiv: 1404.4061 ]. 3. F. Konnig, Y. Akrami, L. Amendola, M. Motta, and A.R. Solomon, Stable and unstable cosmological models in bimetric massive gravity, Phys.Rev. D90 (2014) 124014, [arXiv: 1407.4331]. 4. J. Enander, A.R. Solomon, Y. Akrami, and E. Mortsell, Cosmic expansion histories in massive bigravity with symmetric matter coupling, JCAP 1501 (2015) 006, [arXiv: 1409.2860]. 5. A.R. Solomon, J. Enander, Y. Akrami, T.S. Koivisto, F. Konnig, and E. Mortsell, Cosmological viability of massive gravity with generalized matter coupling, arXiv: 1409.8300. 6. A.R. Solomon and J.D. Barrow, Inflationary Instabilities of Einstein-Aether Cosmology, Phys.Rev. D89 (2014) 024001, [arXiv: 1309.4778]. XV Acknowledgements First and foremost, it is my pleasure to thank my supervisor, John Barrow, for his consistent support throughout my Ph.D. I am also deeply indebted to the many collaborators with whom I embarked on the work contained in this thesis, including Jonas Enander, Frank Konnig, Edvard Mortsell, and Mariele Motta, for their insights and enlightening discussions. I am extremely grateful to Luca Amendola and Tomi Koivisto for, in addition to their collaboration, their generous hospitality in Heidelberg and Stockholm, stays which have been memorable both for their productivity and for their great fun. My very special thanks, finally, go to Yashar Akrami, for being a constant collaborator and close friend. The work discussed in this thesis has benefitted from conversations with many people, including Tessa Baker, Phil Bull, Claudia de Rham, Pedro Ferreira, Fawad Hassan, Macarena Lagos, Johannes Noller, Angnis Schmidt-May, Sergey Sibiryakov, and Andrew Tolley. Within DAMTP I have had the privilege of engaging in one of the most active cosmological communities I have known. Particular thanks belong to Peter Adshead, Mustafa Amin, Neil Barnaby, Daniel Baumann, Camille Bonvin, Anne-Christine Davis, Eugene Lim, Raquel Ribeiro, Paul Shellard, and Yi Wang for fostering such a dynamic intellectual atmosphere. DAMTP has also provided respite from work when times were tough. I am grateful to the many friends I have made in the department, who are too numerous to list, for their seemingly inexhaustible appetite for any event where coffee or beer could be found. Special thanks must go to the DAMTP Culture Club for their company practically any time a concert, opera, Shakespearean play, or other staging of the finer things graced Cambridge. It was my great fortune to come through the Ph.D. at the same time as Valentin Assassi and Laurence Perreault-Levasseur, who each are both brilliant cosmologists and great friends. I am extraordinarily grateful to have had an academic "big brother" in Jeremy Sakstein, who has been a tireless friend both scientifically and personally, always as willing to lend an ear, his knowledge, or career advice as to run off to Scotland in a motorhome to tour whisky xvii xviii Acknowledgements distilleries. Despite my technically not being in her group, Amanda Stagg has been a constant source of administrative and moral support, except when it came to switching offices. Finally, I am grateful to the many friends outside the department, from Wolfson College, Sidney Sussex College, and wherever else I somehow managed to run into such incredible people, who have made my years in Cambridge some of the most fun and exhilarating of my life. Finally, all of my love and thanks go to my entire family. Once again there are far too many people to name, but being family I can assume every one of them knows how grateful I am to them personally. This section would not be complete, however, without special thanks to my aunt and uncle, (Unk) Dane and (Tia) Reyna Solomon, for their preternatural willingness to open their hearts and their home to me. And last but not least, words cannot begin to express my gratitude to my family at home, Mom, Dad, Arielle, Jess, Oscar, and even Bubba. During the course of my Ph.D., I have been supported by the David Gledhill Research Studentship, Sidney Sussex College, University of Cambridge; and by the Isaac Newton Fund and Studentships, University of Cambridge. flange var! Contents 1 Introduction.............................................. 1 1.1 Conventions.......................................... 5 1.2 General Relativity...................................... 6 1.3 The Cosmological Standard Model......................... 8 1.4 Linear Perturbations Around FLRW........................ 12 1.5 Inflation............................................. 14 References................................................ 19 2 Gravity Beyond General Relativity........................... 21 2.1 Massive Gravity and Bigravity............................ 21 2.1.1 Building the Massive Graviton...................... 22 2.1.2 Ghost-Free Massive Gravity........................ 28 2.1.3 Cosmological Solutions in Massive Bigravity........... 34 2.2 Einstein-Aether Theory.................................. 41 2.2.1 Pure Aether Theory............................... 42 2.2.2 Coupling to a Scalar Inflation....................... 44 2.2.3 Einstein-Aether Cosmology......................... 45 References................................................ 47 Part I A Massive Graviton 3 Cosmological Stability of Massive Bigravity.................... 55 3.1 Linear Cosmological Perturbations......................... 56 3.1.1 Linearised Field Equations......................... 57 3.1.2 Counting the Degrees of Freedom.................... 60 3.1.3 Gauge Choice and Reducing the Einstein Equations...... 61 3.2 Stability Analysis...................................... 62 3.3 Summary of Results.................................... 68 References................................................ 69 xix xx Contents 4 Linear Structure Growth in Massive Bigravity.................. 71 4.1 Perturbations in the Subhorizon Limit...................... 72 4.2 Structure Growth and Cosmological Observables.............. 74 4.2.1 Modified Gravity Parameters........................ 74 4.2.2 Numerical Solutions.............................. 76 4.3 Summary of Results.................................... 98 References................................................ 100 5 The Geometry of Doubly-Coupled Bigravity.................... 103 5.1 The Lack of a Physical Metric............................ 104 5.2 Light Propagation and the Problem of Observables............ 107 5.3 Point Particles and Non-Riemannian Geometry............... 109 5.4 Summary of Results.................................... 113 References................................................ 114 6 Cosmological Implications of Doubly-Coupled Massive Bigravity................................................ 117 6.1 Doubly-Coupled Bigravity............................... 118 6.2 Cosmological Equations and Their Solutions................. 120 6.2.1 Algebraic Branch of the Bianchi Constraint............ 122 6.2.2 Dynamical Branch of the Bianchi Constraint........... 123 6.3 Comparison to Data: Minimal Models...................... 125 6.4 Special Parameter Cases................................. 128 6.4.1 Partially-Massless Gravity.......................... 128 6.4.2 Vacuum Energy and the Question of Self-Acceleration. . . . 129 6.4.3 Maximally-Symmetric Bigravity..................... 131 6.5 Summary of Results.................................... 131 References................................................ 132 7 Cosmological Implications of Doubly-Coupled Massive Gravity .... 135 7.1 Cosmological Backgrounds............................... 136 7.2 Do Dynamical Solutions Exist?........................... 139 7.3 Einstein Frame Versus Jordan Frame....................... 140 7.4 Massive Cosmologies with a Scalar Field.................... 141 7.5 Adding a Perfect Fluid.................................. 143 7.6 Mixed Matter Couplings................................. 147 7.7 Summary of Results.................................... 149 References................................................ 151 Part II Lorentz Violation 8 Lorentz Violation During Inflation........................... 155 8.1 Stability Constraint in Flat Space.......................... 157 8.2 Cosmological Perturbation Theory......................... 162 8.2.1 Perturbation Variables............................. 162 8.2.2 Linearised Equations of Motion..................... 163 Contents xxi 8.3 Spin-1 Cosmological Perturbations......................... 164 8.3.1 Slow-Roll Limit................................. 167 8.3.2 Full Solution for the Vector Modes................... 169 8.3.3 Tachyonic Instability.............................. 170 8.3.4 What Values Do We Expect for A?.................. 173 8.4 Spin-0 Cosmological Perturbations: Instability and Observability...................................... 177 8.4.1 The Spin-0 Equations of Motion..................... 178 8.4.2 The Instability Returns............................ 179 8.4.3 The Small-Coupling Limit......................... 180 8.4.4 The Large-Coupling Limit: The O Evolution Equation. ... 181 8.4.5 The Large-Coupling Limit: CMB Observables.......... 184 8.5 Case Study: Quadratic Potential........................... 186 8.5.1 Slow-Roll Inflation: An Example.................... 186 8.5.2 The Instability Explored........................... 189 8.6 Summary of Results.................................... 192 References................................................ 194 9 Discussion and Conclusions................................. 197 9.1 Problems Addressed in This Thesis........................ 197 9.2 Summary of Original Results............................. 199 9.2.1 Massive Gravity and Bigravity...................... 199 9.2.2 Lorentz-Violating Gravity.......................... 200 9.3 Outlook............................................. 201 References................................................ 205 Appendix A: Deriving the Bimetric Perturbation Equations.......... 207 Appendix B: Explicit Solutions for the Modified Gravity Parameters...................................... 213 Appendix C: Transformation Properties of the Doubly-Coupled Bimetric Action................................... 215 Appendix D: Einstein-Aether Cosmological Perturbation Equations in Real Space.................................... 221 Curriculum Vitae............................................ 225 Chapter 1 Introduction / am always surprised when a young man tells me he wants to work at cosmology; I think of cosmology as something that happens to one, not something one can choose. Sir William McCrea, Presidential Address, Royal Astronomical Society One of the driving aims of modern cosmology is to turn the Universe into a laboratory. By studying cosmic history at both early and late times, we have access to a range of energy scales far exceeding that which we can probe on Earth. It falls to us only to construct the experimental tools for gathering data and the theoretical tools for connecting them to fundamental physics. The most obvious application of this principle is to the study of gravitation. Gravity is by far the weakest of the fundamental forces, yet on sufficiently large distance scales it is essentially the only relevant player; we can understand the motion of the planets or the expansion of the Universe to impressive precision without knowing the details of the electromagnetic, strong, or weak nuclear forces.1 As a result, we expect the history and fate of our Universe to be intimately intertwined with the correct description of gravity. For nearly a century, the consensus best theory has been Einstein's remarkably simple and elegant theory of general relativity [1, 2]. This consensus is not without reason: practically all experiments and observations have lent increasing support to this theory, from classical weak-field observations such as the precession of Mercury's perihelion and the bending of starlight around the Sun, to the loss of orbital energy to gravitational waves in binary pulsar systems, observations remarkable both for their precision and for their origin in the strongest gravitational fields we have ever tested [3]. Modulo the fact that we need, as input, to know which matter gravitates, and that the quantum field theories describing these forces are essential to understanding precisely which matter we have. © Springer International Publishing AG 2017 1 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7_1 2 1 Introduction Nevertheless, there are reasons to anticipate new gravitational physics beyond general relativity. In the ultraviolet (UV), i.e., at short distances and high energies, it is well known that general relativity is nonrenormalisable and hence cannot be extended to a quantum theory [4]. It must be replaced at such scales by a UV-complete theory which possesses better quantum behaviour. The focus of this thesis is on the infrared (IR), i.e., long distances and low energies. While general relativity is a theoretically-consistent IR theory, the discovery in 1998 that the expansion of the Universe is accelerating presents a problem for gravitation at the longest distances [5, 6]. The simplest explanation mathematically for this acceleration is a cosmolog-ical constant, which is simply a number that we can introduce into general relativity without destroying any of its attractive classical features. However, from a quantum-mechanical point of view, the cosmological constant is highly unsatisfactory. The vacuum energy of matter is expected to gravitate, and it would mimic a cosmological constant; however, the value it would generate is as much as 10120 times larger than the value we infer from observations [7-9]. Therefore, the "bare" cosmological constant which appears as a free parameter in general relativity would need to somehow know about this vacuum energy, and cancel it out almost but not quite exactly. Such a miraculous cancellation has no known explanation. Alternatively, one could imagine that the vacuum energy is somehow either rendered smaller than we expect, or does not gravitate—and theories which achieve this behaviour are known [8, 9]—but we would then most likely need a separate mechanism to explain what drives the current small but nonzero acceleration. For these reasons, it behoves us to consider the possibility that general relativity may not be the final description of gravity on large scales. To put the problem in historical context, we may consider the story of two planets: Uranus and Mercury. In the first half of the nineteenth century, astronomers had mapped out the orbit of Uranus, then the farthest-known planet, to heroic precision. They found anomalies in the observed orbit when compared to the predictions made by Newtonian gravity, then the best understanding of gravitation available. Newton's theory had not yet been tested at distances larger than the orbit of Uranus: it was, for all intents and purposes, the boundary of the known universe. A natural explanation was therefore that Newtonian gravity simply broke down at such unimaginably large distances, to be replaced by a different theory. In 1846, French astronomer Urbain Le Verrier put forth an alternative proposal: that there was a new planet beyond Uranus' orbit, whose gravitational influence led to the observed discrepancies. Le Verrier predicted the location of this hitherto-unseen planet, and within weeks the planet Neptune was unveiled. Buoyed by his success, Le Verrier turned his sights to another planet whose orbit did not quite agree with Newtonian calculations: Mercury, the closest to the Sun. As is now famous, the perihelion of Mercury's orbit precessed at a slightly faster rate than was predicted. Le Verrier postulated another new planet, Vulcan, within Mercury's orbit. However, the hypothesised planet was never found, and in the early parts of the twentieth century, Einstein demonstrated that general relativity accounted precisely for the perihelion precession. In the case of Mercury, it was a modification to the laws of gravity, rather than a new planet, which provided the solution. 1 Introduction 3 We find ourselves in a similar position today. Our best theory of gravity, general relativity, combined with the matter we believe is dominant, mostly cold dark matter, predict a decelerating expansion, yet we observe something different. One possibility is that there is new matter we have not accounted for, such as a light, slowly-rolling scalar field. However, we must also consider that the theory of gravity we are using is itself in need of a tune-up. The project of modifying gravity leads immediately to two defining questions: what does a good theory of modified gravity look like, and how can we test such theories against general relativity? This thesis aims to address both questions, although any answers we find necessarily comprise only a small slice of a deep field of research. Einstein's theory is a paragon of elegance. It is practically inevitable that this is lost when generalising to a larger theory. Indeed, it is not easy to even define elegance once we leave the cosy confines of Einstein gravity. Consider, as an example, two equivalent definitions of general relativity, each of which can be used to justify the claim that GR is the simplest possible theory of gravity. First we can say that general relativity is the theory whose Lagrangian, known as the Einstein-Hilbert term, is the simplest diffeomorphism-invariant Lagrangian that can be constructed out of the metric tensor and its derivatives.2 Alternatively, we could look at general relativity as being the unique Lorentz-invariant theory of a massless spin-2 field, or graviton [4, 10-13]. These serve equally well to tell us why general relativity is so lovely, but they diverge once we move to more general theories. Consider, for example, modifying the Lagrangian (1.1) by promoting the Ricci scalar R to a general function f(R), This is the defining feature of f(R) gravity, a popular theory of modified gravity [14-16]. One can certainly make the argument that this is mathematically one of the simplest possible generalisations of general relativity. However, when considered in terms of its fundamental degrees of freedom, we find a theory in which a spin-0 or scalar field interacts in a highly nonminimal way with the graviton [17]. Alternatively, one can consider massive gravity, in which the massless graviton of general relativity is given a nonzero mass. While this has a simple interpretation in the particle picture, its mathematical construction is so nontrivial that over seven decades were required to finally find the right answer. The resulting action, given in Eq. (2.21), is certainly not something one would have thought to construct had it not been for the guiding particle picture. There are additional, more practical concerns when building a new theory of gravity. General relativity agrees beautifully with tests of gravity terrestrially and in the solar system, and it is not difficult for modified gravity to break that agreement. 2For an introduction to the Lagrangian formulation of general relativity, see Ref. [2]. (1.1) (1.2) 4 1 Introduction While this may be surprising if we are modifying general relativity with terms that should only be important at the largest distance scales, it is not difficult to see that this problem is fairly generic. Any extension of general relativity involves adding new degrees of freedom (even massive gravity has three extra degrees of freedom), and in the absence of a symmetry forbidding such couplings, these will generally couple to matter, leading to gravitational-strength fifth forces. Such extra forces are highly constrained by solar-system experiments. Almost all viable theories of modified gravity therefore possess screening mechanisms, in which the fifth force is large cosmologically but is made unobservably small in dense environments. The details of these screening mechanisms are beyond the scope of this thesis, and we refer the reader to the reviews [18, 19]. In parallel with these concerns, we must ask how to experimentally distinguish modified gravity from general relativity. One approach is to use precision tests in the laboratory [20-27]. Another is to study the effect of modified gravity on astrophys-ical objects such as stars and galaxies [28-31]. In this thesis we will be concerned with cosmological probes of modified gravity. Because screening mechanisms force these modifications to hide locally (with some exceptions), it is natural to look to cosmology, where the new physics is most relevant. Cosmological tests broadly fall into three categories: background, linear, and nonlinear. Background tests are typically geometrical in nature, and try to distinguish the expansion history of a new theory of gravity from the general relativistic prediction. Considering small perturbations around the background, we obtain predictions for structure formation at linear scales. Finally, on small scales where structure is sufficiently dense, nonlinear theory is required to make predictions, typically using N-body computer simulations. This thesis is concerned with the construction of theoretically-sensible modified gravity theories and their cosmological tests at the level of the background expansion and linear perturbations. In the first part, we focus on massive gravity and its extension to a bimetric theory, or massive bigravity, containing two dynamical metrics interacting with each other. In particular, we derive the cosmological perturbation equations for the case where matter couples to one of the metrics, and study the stability of linear perturbations by deriving a system of two coupled second-order evolution equations describing all perturbation growth and examining their eigen-frequencies. Doing this, we obtain conditions for the linear cosmological stability of massive bigravity, and identify a particular bimetric model which is stable at all times. We next move on to the question of observability, constructing a general framework for calculating structure formation in the quasistatic, subhorizon regime, and then applying this to the stable model. After this, we tackle the question of matter couplings in massive gravity and bigravity, investigating a pair of theories in which matter is coupled to both metrics. In the first, matter couples minimally to both metrics. We show that there is not a single effective metric describing the geometry that matter sees, and so there is a problem in defining observables. In the second theory, matter does couple to an effective metric. We first study it in the context of bigravity, deriving its cosmological background evolution equations, comparing some of the simplest models to data, and examining in depth some particularly interesting parameter choices. We next exam- 1 Introduction 5 ine the cosmological implications of massive gravity with such a matter coupling. Massive gravity normally possesses a no-go theorem forbidding flat cosmological solutions, but coupling matter to both metrics has been shown to overcome this. We examine this theory in detail, finding several stumbling blocks to observationally testing the new massive cosmologies. The remainder of this thesis examines the question of Lorentz violation in the gravitational sector. We focus on Einstein-aether theory, a vector-tensor model which spontaneously breaks Lorentz invariance. We study the coupling between the vector field, or "aether," and a scalar field driving a period of slow-roll inflation. We find that such a coupling can lead to instabilities which destroy homogeneity and isotropy during inflation. Demanding the absence of these instabilities places a constraint on the size of such a coupling so that it must be at least 5 orders of magnitude smaller than the previous best constraints. The thesis is organised as follows. In the rest of this chapter, we present background material, discussing the essential ingredients of general relativity and modern cosmology which will be important to understanding what follows. In Chap. 2 we give a detailed description of the modified gravity theories discussed in this thesis, specifically massive gravity, massive bigravity, and Einstein-aether theory, focusing on their defining features and their cosmological solutions. In Chaps. 3 and 4 we examine the cosmological perturbation theory of massive bigravity with matter coupled to one of the metrics. In Chap. 3 we study the stability of perturbations, identifying a particular bimetric model which is stable at all times, while in Chap. 4 we turn to linear structure formation in the quasistatic limit and look for observational signatures of bigravity. In Chaps. 5-7 we examine generalisations of massive gravity and bigravity in which matter couples to both metrics. Chapter 5 focuses on the thorny problem of finding observables in one such theory. In Chap. 6 we examine the background cosmologies of a doubly-coupled bimetric theory, and do the same for massive gravity in Chap. 7. Finally, in Chap. 8 we study the consequences of coupling a slowly-rolling inflaton to a gravitational vector field, or aether, deriving the strongest bounds to date on such a coupling. We conclude in Chap. 9 with a summary of the problems we have addressed and the work discussed, as well as an outlook on the coming years for modified gravity. 1.1 Conventions Throughout this thesis we will use a mostly-positive (—h ++) metric signature. We will denote the flat-space or Minkowski metric by r\iliV. Greek indices jx, v,... = (0, 1, 2, 3) represent spacetime indices, while Latin indices i, j,... = (1, 2, 3) are used for spatial indices. Latin indices starting from a, b, c,... are also used for field-space and local Lorentz indices. Partial derivatives are denoted by 3 and covariant derivatives by V. Commas and semicolons in indices will occasionally be used to represent partial and covariant derivatives, respectively, i.e., 0iAt = 3M0 and = VM0. Symmetrisation and antisymmetrisation are denoted by 6 1 Introduction Table 1.1 Abbreviations used throughout this thesis Abbreviation Expression BAO Bary on-acoustic oscillations CDM Cold dark matter CMB Cosmic microwave background FLRW Friedmann-Lemaitre-Robertson-Walker GR General relativity SNe Supernovae VEV Vacuum expectation value S(fiv) = ~ [Sfiv + Svfi) , ^[/xv] = 2 V^MV ^vfi) , (1-3) and similarly for higher-rank tensors. In lieu of the gravitational constant G we will frequently use the Planck mass, M\x = 1 /8ttG. Cosmic time is denoted by t and its Hubble rate is H, while we use x for conformal time with the Hubble rate Jfl?. For brevity we will sometimes use abbreviations for common terms, listed in Table 1.1. 1.2 General Relativity This thesis deals with modified gravity. Consequently it behoves us to briefly overview the theory of gravity we will be modifying: Einstein's general relativity. The theory is defined by the Einstein-Hilbert action, (1.4) where R = g^v R/1V is the Ricci scalar, with g/iv and R/1V the metric tensor and Ricci tensor, respectively. Allowing for general matter, represented symbolically by fields with Lagrangians Jzfm determined by particle physics, the total action of general relativity is S = SEH+ J d4x^g~J?m (g, O,-). (1.5) Varying the action S with respect to g^ we obtain the gravitational field equation, the Einstein equation, R^v - \Rg^ = 87rGrMV, (1.6) 1.2 General Relativity 7 where the stress-energy tensor of matter is defined by rMV = ——-W-JLt—i. (i.7) It is often convenient to define the Einstein tensor, 1 = Rfjiv — -Rgfj,v, (1.8) which is conserved as a consequence of the Bianchi identity, VMG^v=0. (1.9) Note that we are raising and lowering indices with the metric tensor, gliv. The Bianchi identity is a geometric identity, i.e., it holds independently of the gravitational field equations. The stress-energy tensor is also conserved, vMr\ = o. (l.io) This is both required by particle physics and follows from the Einstein equation and the Bianchi identity, which is a good consistency check. A consequence of stress-energy conservation is that particles move on geodesies of the metric, gliv, x11 + r^xaxp = o, (l.ii) where x^iX) is the position 4-vector of a test particle parametrised with respect to a parameter X, an overdot denotes the derivative with respect to X, and = \gliV(gav,p + gpv,a ~ gap,v) are the Christoffel symbols. Einstein's equation relates the curvature of spacetime to the distribution of matter. Freely-falling particles then follow geodesies of the metric. The combination of the Einstein and geodesic equations leads to what we call the gravitational force. John Wheeler's description of gravity's nature is perhaps the most eloquent: "Spacetime tells matter how to move; matter tells spacetime how to curve" [32]. As discussed above, it seems clear to the eye that Eq. (1.4) is the simplest action one can construct for the gravitational sector, if one restricts oneself to scalar curvature invariants. Indeed, the simplicity of general relativity can be phrased in two equivalent ways. Lovelock's theorem states that Einstein's equation is the only gravitational field equation which is constructed solely from the metric, is no more than second order in derivatives,3 is local, and is derived from an action [34]. Alternatively, as alluded to previously, the same field equations are the unique nonlinear equations of motion for The requirement that higher derivatives not appear in the equations of motion comes from demanding that the theory not run afoul of Ostrogradsky's theorem, which states that most higher-derivative theories are hopelessly unstable [33]. 8 1 Introduction a massless spin-2 particle [4, 10-13]. There is but one extension to the gravitational action presented which neither violates Lovelock's theorem nor introduces any extra degrees of freedom: a cosmological constant, A, which enters in the action as S=^- I d4xV=g(R-2A)+ I d4x^g-j?m (g, f (t)dt / N it). Two common choices for the time coordinate are cosmic time, Nit) = 1, and conformal time, N(t) = ait). Cosmic time is more physical, as it corresponds to the time measured by observers comoving with the cosmic expansion (such as, for example, us). Conformal time is often computationally useful, especially since in those coordinates the metric is conformally related to Minkowski space if k = 0, gliv = a{t)2rjlJiV\ consequently, photons move on flat-space geodesies, and their motion in terms of conformal time can be computed without additionally calculating the cosmic expansion. It is worth keeping the lapse in mind because this thesis deals in large part with theories in which the time-time part of the metric (or metrics) cannot so freely be fixed to a desired value. As long as a single metric couples to matter, however (which is the case everywhere except in Chap. 5), the time coordinate defined by dr = N(t)dt, where N(t) is the lapse of the metric to which matter couples, will function as cosmic time when computing observables. The dynamical variable determining the expansion of the Universe is the scale factor, a(t). Its evolution is determined by the Einstein equation. At this point we specialise to cosmic time; the conformal-time equivalents of the equations we present can be easily derived by switching the time coordinate from í to t, where dr = dt/a(t). We use as the matter source a perfect fluid with the stress-energy tensor T^v = (p + p) u»uv + pg^, (1.14) where mm is the fluid 4-velocity, p is the energy density, and p is the pressure. Then, taking the time-time component of the Einstein equation we obtain the Friedmann equation, 8ttG k A az 3 H2 = —^p- — + -, (1.15) where the Hubble rate is defined by a H = -. (1.16) a Conservation of the stress-energy tensor leads to the fluid continuity equation, p + 3H(p + p) = 0. (1.17) Note that, in a Universe with multiple matter species which do not interact, this conservation equation holds both for the total density and pressure, as well as for the density and pressure of each individual component. The spatial part of the Einstein equation—or, equivalently, the trace—will lead to the acceleration equation, - =-47tG(p + 3p) +—. (1.18) a 3 10 1 Introduction This can also be derived using the equations we already have, by taking a derivative of the Friedmann equation and then applying the continuity equation to remove p. The utility of the acceleration equation is therefore limited for the purposes of this thesis. In order to close the system comprising the Friedmann and continuity equations, it is typical to specify an equation of state relating the pressure to the density, p = p(p), for each individual matter species. The simplest and most commonly-used equation of state is p = wp, (1-19) where w is a constant. Examples of perfect fluids obeying such an equation of state include pressureless matter, or "dust," with w = 0, radiation, with w = 1/3, and vacuum energy or a cosmological constant, with w = — 1. Indeed, we only need these three fluids to model the Universe back to about a second after the big bang, so we will focus on them. Let us briefly discuss some simple properties of the cosmological solutions to this system of equations. Because observations are consistent with a flat Universe, i.e., ic = 0 [35], we will neglect the spatial curvature from here out. With w = const., the continuity equation is solved by p = p0a-3(1+^, (1.20) where po is a constant corresponding to the density when a = 1 (usually taken to be the present day). This leads to the following behaviours for the relevant cosmic fluids: p ~ a~4 radiation (1-21) p ~ a~3 dust (1-22) p = const. vacuum energy, cosmological constant. (1-23) Plugging these into the Friedmann equation, we obtain the following expansion rates during the various cosmic eras: a(t) ~ t1/2 radiation-dominated era (1-24) a(t) ~ t2^3 matter-dominated era (1-25) ait) ~ eHt A-dominated era. (1-26) In general, a universe dominated by a w = const. ^ — 1 perfect fluid will evolve as a(t) ~ t2/3(1+w). Notice that, as time goes on, the densities of radiation and matter (and any fluid with w > — 1) will decay, while that of a cosmological constant stays the same (which is sensible, since it has a constant contribution to the Friedmann equation). Therefore, if A > 0, there is necessarily a time after which the cosmological constant 1.3 The Cosmological Standard Model 11 dominates the Friedmann equation, with H ~ const, and an exponential expansion. This makes quantitative the claim from above that a cosmological constant leads to late-time cosmic acceleration. Observations show that such a late-time acceleration is happening in our own Universe, and if it is caused by a perfect fluid then its equation of state is consistent with w = — 1 [35]. Following this, our criterion for self-acceleration in a theory of modified gravity will generally be that H tends to a constant at late times. Finally, we note that it is common to define a density parameter, Qt, for each matter species, 87T Gpi fi« = -t-, (1-27) 3H2 where the subscript i indexes each matter species. In particular, we will define Qm, Qb, Qc, QY, and QA for all matter (specifically dust), baryons, cold dark matter, radiation, and a putative dark energy, respectively. In terms of the density parameter, the Friedmann equation can be written in the simple and general form as long as we define appropriate density parameters for the curvature and cosmological-constant terms. It is also common to parametrise the present-day density of each species in terms of the density parameter evaluated at the present, denoted by Qifi. Broadly speaking, observations suggest í2mo ~ 0.3 and í2A,o ~ 0-7, while all other contributions are negligibly small or vanishing [35]. The fact that £2A,o is nonzero tells us that in order to match observations using general relativity, we need to introduce a "dark energy" component, of which the simplest example is a cosmological constant. The precise best-fit cosmological parameters from the Planck satellite are presented in Table 1.2. We have progressively constructed the cosmological standard model, or A-cold dark matter (ACDM). Its main ingredients are radiation (which is important mostly at early times), baryons and cold dark matter comprising pressureless dust, and a small cosmological constant. The gravitational theory is general relativity. It is the aim of this thesis to explore alternatives in which the cosmological constant is removed at the expense of introducing a different gravitational theory. Table 1.2 The Planck best-fit cosmological parameters, taken from Ref. [35]. Here H0 = lOO/i km/s/Mpc Qbßh2 0.022 0.12 ^(lO^5) 0.68 Ho 68.14 km/s/Mpc 12 1 Introduction 1.4 Linear Perturbations Around FLRW The FLRW metric was constructed to be consistent with spatial homogeneity and isotropy. The Universe is, of course, not really homogeneous and isotropic: in various places it contains stars, planets, galaxies, people, and Cambridge. The FLRW approximation holds on scales of hundreds of megaparsecs and higher, and breaks down at smaller distances. At slightly smaller distance scales, spacetime is well described by linear perturbations to FLRW. That is, taking gliv to be an FLRW background metric, we consider g/iv = g/iv +Sg/lv, (1.29) where 8gliv <$C 1 is a small perturbation, add a similar small piece to the matter sector, and calculate the Einstein equations to first order in 8gliv. This proves to be a powerful tool for testing gravity: using probes of structure to test gravity at the linear level complements and can even be more constraining than studies of the expansion history which operate at the background level. Let us be more explicit. We will work in conformal time (N = a) and write the perturbed metric as gllvdxlidxv = a2 {- (1 + E) dx2 + IdiFdtdx' + [(1 + A) Stj + 3,3,5] dx'dxj} . (1.30) We define the perturbed stress-energy tensor by r°0 = -p(l + 5), Ti0 = -(p + P)vi, T°t = {p + P) (vf + diF), Tj = {P + SP) 8* j + H'j, S1',- =0, K ■ ) where vl = dxl /dt and barred quantities refer to background values. Let us specialise to pressureless dust (P = 8 P = 5ľ7 =0). Because of the coordinate independence of general relativity, not all of the perturbation variables represent genuine degrees of freedom: as we have things currently set up, it is possible for some of the perturbations to be nonzero while the spacetime is still purely FLRW, only written in funny coordinates. This could lead to unphysical modes propagating through the equations of motion. To remove this problem, we choose a coordinate system, or fix a gauge. We will work in conformal Newtonian gauge, in which F = B = 0. We will also decompose each variable into Fourier modes and suppress the mode index; the only effect of this for our purposes is that we can write 5iJ 3,3,0 = — £20 where O represents any of the perturbations. We are left with four perturbation variables, A, E, 8, and 6 = 3,V. There is only one dynamical degree of freedom among these; any three of the variables can be written in terms of the fourth, which in turn obeys a second-order evolution equation. We will choose to use 8 as our independent degree of freedom. The removal of the other three variables in order to find the evolution equation for 8 alone proceeds as follows. By taking the off-diagonal part of the space-space 1.4 Linear Perturbations Around FLRW 13 Einstein equation, we can find that A = — E. The potential E is related to the density perturbation, 8, by the time-time Einstein equation, 2 - 3je (E + J^E) +k2E = -^-8. (1.32) V ' M2 Most of the modes which we can access observationally are within the horizon, k > M'. To simplify the analysis we can focus on these modes by taking the subhorizon limits, k2<$> » 20, and the quasistatic limit, 20 ~ 4> ~ O, where again O represents any of the perturbation variables. The quasistatic assumption holds if the timescale for the growth of perturbations is of order the Hubble time. In this limit, Eq. (1.32) takes the simple form 9 a2p k2E =--f 8. (1.33) M2 Recalling that k2 = —V2 is the Fourier-space version of the Laplace operator, we recognise this as the Poisson equation; the matter density contrast, 8, sources the gravitational potential, E, in a familiar way. We additionally have the v = 0 and v = i components of the stress-energy conservation equation, 3 • 8 + 6 - -E = 0, v = 0, (1.34) 2 2 9+jee- -k2E = 0, v = i, (1.35) where 9 = 9,V. In the subhorizon and quasistatic limit, these can be combined, along with the Poisson equation, to obtain a closed evolution equation for the density contrast,5 2 a p 8 + Jť8--V<5=0. (1.36) 2M2 The evolution equation for 8 can be integrated to obtain the growth rate of structure. While in general this requires numerical integration, as an illustration we can obtain exact solutions in general relativity during the various cosmic eras. During matter domination, we have p ~ 3MpV3ť2/a2, a ~ r2, and^f = 2/r, soEq. 1.36 becomes .. 2 . 6 8+ -8 - —8 = 0, (1.37) x x 5 Outside the quasistatic limit, this evolution equation will be sourced by E, which in turn obeys its own closed equation. Notice that there is still only one independent degree of freedom. 14 1 Introduction which has, in addition to a decaying mode which we ignore, the growing solution S~T2~a. (1.38) During dark energy domination, p (which is the density of matter) becomes negligibly small, and the only solutions to Eq. (1.36) are a decaying mode and 8 = const. We see that during the dark energy era, matter stops clustering. This makes intuitive sense: as the expansion of the Universe accelerates, it becomes more and more difficult for matter to gravitationally cluster "against" the expansion. A useful parametrisation for comparison to data is based on the growth rate, d log 8 f(a,k) = —2-. (1.39) a log a In the recent past, the growth rate of solutions to Eq. (1.36) is well approximated in terms of the matter density parameter defined above, f(a,k)*Wm, (1.40) where the growth index y has the value y ~ 0.545. The growth index typically deviates from this in theories of modified gravity. 1.5 Inflation To this point we have discussed some of the essential ingredients of modern cosmology, particularly general relativity and its application to background and linearised FLRW spacetimes. We then used this to discuss the expansion history and the growth of structure in the "late Universe," i.e., during the matter- and dark-energy-dominated eras. The standard cosmological model includes, at earlier times, two other important eras: radiation-domination and, before it, inflation. We will skip the radiation era, as it is not directly relevant to this thesis. This leaves inflation. Inflationary cosmology was originally developed as a solution to some of the glaring problems with a big-bang cosmology, which can be summarised as problems of initial conditions: a Universe that was always decelerating (until the recent dark-energy era) requires highly tuned initial conditions to be as flat and uniform as we see it. Not long into the development of inflation, another significant motivation arose: quantum fluctuations during inflation are blown up to sizes larger than the cosmic horizon before they can average out, leaving a spectrum of perturbations which would seed the formation of cosmic structure, in excellent agreement with observations. For a comprehensive review of these motivations and inflationary physics, we point the reader to Ref. [36]. The simplest physical model for inflation, and the one with which we will be concerned in this thesis, is single-field slow-roll inflation. In this model, inflation is 1.5 Inflation 15 driven by a canonical scalar field or inflaton, . Now let us specialise to homogeneous and isotropic cosmology. As argued above, the metric must take the FLRW form. The scalar can only depend on time,

=0, (1.47) where H = a/(aN) is the cosmic-time Hubble parameter. At this point we will specialise, for simplicity, to cosmic time (TV = 1), although when relevant we will present results in terms of conformal time (TV = a) as well. The expressions presented up to this point, keeping ./V general, will be necessary in Chap. 7 when we consider a scalar field in a theory where we cannot freely rescale TV. By taking a derivative of the Friedmann equation and removing terms using the Klein-Gordon equation, we obtain H = —^j-. (1.48) 2M2X This formalises the result we had derived less rigorously earlier: if

50-60, where a, and af are the scale factors at the start and end of inflation, respectively. In the slow-roll limit, both s and r\ are constant at first order and we can integrate to find, to first order in s, = eHt (l - ^fe + 0(e2)^ (1.53) H = H (1 - Hte + Ü{e2)) , (1.54) or, in conformal time, 1 9 a = —^(l + e + ^(e2)), (1.55) Hx je = --(l+e + 0(e2)), (1.56) x where H is the Hubble rate of the de Sitter background that emerges in the limit e —>• 0. Note that in conformal time, x runs from — oo at the big bang to 0 in the far future. In practise, "the far future" actually corresponds to the end of inflation, taken 7We define s with a minus sign so that e > 0: in order to satisfy the null-energy condition, we need H < 0. 18 1 Introduction to be when e, r\ ~ 1 and the slow-roll expansion breaks down. This is assumed to be followed by a period of reheating, in which the scalar field decays into standard-model particles, and the radiation era thus commences. We now, finally, have the tools to understand the microphysics of a scalar field satisfying the slow-roll conditions. Using the expression (1.48) for H and the definition of the slow-roll parameters, we can find V H) (1.57) Therefore, in addition to the aforementioned condition, P1 V ' We have already shown that 4>/H

should be small in such a way that <$C V/Mpi and <$C V/MpV A prototypical inflationary potential is shown in Fig. 1.1. Popular forms for the potential include V ~ (2-2) where indices are raised and lowered with the Minkowski metric, r\iliV, and we have defined the Lichnerowicz operator, <§, by ifvhaß = -l- (D/v - 2dilJLdahav) + dßdvh - n^iUh - dadßhaß)) , (2.3) where h = rj^h^ is the trace of hßV. No other kinetic terms are consistent with locality, Lorentz invariance, and gauge invariance under linearised diffeomorphisms, -> + 23(/x|v). (2.4) Indeed, this uniqueness is a necessary (though not sufficient) part of the aforementioned uniqueness of general relativity as the nonlinear theory of a massless spin-2 field. The role of gauge invariance is to ensure that there are no ghosts, i.e., no degrees of freedom with higher derivatives or wrong-sign kinetic terms. Ostrogradsky's theorem 2.1 Massive Gravity and Bigravity 23 tells us that, up to a technical condition, a Lagrangian with higher than second derivatives will lead to a Hamiltonian which is unbounded from below, and thus states with arbitrarily negative energy are allowed (for a thorough, modern derivation, see [19]). If we had included in Eq. (2.2) other terms that can be constructed out of hliv and its first and second derivatives, then the action would no longer be invariant under Eq. (2.4). In that case, we could split h/iv into a transverse piece, hT , and a vector field, x/x, as h^v = h^v + 2d(llXv), (2.5) and any terms not included in the action (2.2) would contain pieces with higher derivatives of Xfi- Therefore we can see the linearised Einstein-Hilbert term as the kinetic term uniquely set by three requirements: locality, Lorentz invariance, and the absence of a ghost. We would like to give hliv a mass—i.e., add a nonderivative interaction term— while maintaining those three requirements. Unfortunately, it is impossible to construct a local interaction term which is consistent with diffeomorphism invariance (2.4). Since this was useful in exorcising ghost modes, we will need to take care to ensure that no ghost is introduced by the mass term. At second order, this is not especially difficult as there are only two possible terms we can consider: /í/xv/í/xv and h2. We can then consider a general quadratic mass term, ^mass = -\m2 {h^v - (1 - a)h2) . (2.6) o This leads to a ghostlike, scalar degree of freedom with mass m2 = ^J2T~rn2. The only way to remove the ghost from this theory, besides setting m = 0, is to set a = 0. The ghost then has infinite mass and is rendered nondynamical. We see the unique ghost-free action for a massive graviton at quadratic order is the Fierz-Pauli action, j^fp = Ah^fihafi _ l_m2 (h/ivhllV _ h2j (2i?) The Stiickelberg "Trick" Before moving on to higher orders in h^, let us take a moment to count and classify the degrees of freedom it contains at linear order. Recall that a massless graviton contains two polarisations. Because we lose diffeomorphism invariance when we give the graviton a mass, it will contain more degrees of freedom. In fact, there are five in total. In principle, a sixth mode can arise, but it is always ghost-like and must therefore be removed from any healthy theory of massive gravity. To separate the degrees of freedom contained in the massive graviton, we use the Stiickelberg "trick." Stiickelberg's idea is based on the observation that a gauge freedom such as diffeomorphism is not a physical property of a theory so much as a redundancy in Namely that the Lagrangian be nondegenerate, i.e., that dL/dq depend on q. 24 2 Gravity Beyond General Relativity description, and that redundancy can always be introduced by bringing in redundant variables. Let us consider splitting up hliv as 2 2 hfj.v -> H--d(flAv) H---d^dvíj). (2.8) m mz Defining the field strength tensor for AIjL analogously to electromagnetism, FI1V = jd[fiAV], as well as = 3M3V0 and the trace notation [A] = rj^A^, the Fierz-Pauli action (2.7) becomes ^fp = -\h^ifvhaP - Vv (nMV - [n]i/MV) - i Vv = - im2 (h^ - h2) - X-m (h^ - hrjn d(llAv). (2.9) This action is invariant under the simultaneous gauge transformations m hllv -> hjlv + 2d(^v), AjX^ AjX- (2.10) for h/lv and Af,^ A^ + d^X, ý^ý-mX (2.11) for AIjL . With these gauge invariances restored, one can find that hliv contains the usual two independent components of a spin-2 degree of freedom, AIjL similarly contains the standard two independent components, and

• 0. Intuitively we would expect this to reduce to general relativity. In this limit, the vector completely decouples from the other two fields, while the scalar remains mixed with the tensor. They can be unmixed by transforming hliv —>• hliv + $>r\iJiV. However, this transformation introduces a coupling between

• 0, nonlinear effects cure this discontinuity: this is the celebrated Vainshtein mechanism which restores general relativity in environments where

(2.1V) ZMpi where Tqv is the stress-energy tensor of our matter source. Diffeomorphism invariance is preserved in the matter sector if stress-energy is conserved, i.e., if 3M Tqv = 0. However, the coupling to hliv necessarily induces a violation of this conservation. For a simple example of this using a scalar field, see Ref. [17]. This problem is fixed by adding nonlinear corrections both to the matter coupling, •^matter,nonlinear = 0 , . ^/xv^o ~l~ _ o ^/xv^a/S^i > (2.18) ZMp\ 2Mpj for some tensor T^va^, and to the gauge symmetry, symbolically written as 1 h^h + d% +-d(h$). (2.19) Mpi While this ensures the conservation of the stress-energy tensor at the linear level, it is broken at the next order, and so we must continue this procedure order by order, ad infinitum. For a massless spin-2 field, the end result of this procedure is well-known: it is general relativity. The fully nonlinear gauge symmetry is diffeomorphism invariance, and as long as the matter action is invariant under this symmetry, the stress-energy tensor is covariantly conserved, V/xr/xv = 0. The linear action (2.2) must be promoted to something which is also consistent with this symmetry, and there is one answer: the Einstein-Hilbert action (1.4). 2.1 Massive Gravity and Bigravity 27 If we wish to extend this procedure to a massive graviton, i.e., nonlinearly complete the Fierz-Pauli mass term, then, as discussed in Sect. 2.1.1, the Boulware-Deser ghost looms as a pitfall. Demanding that this ghost be absent will severely restrict the allowed potentials to a very specific and special set of functions of M. However, the gauge invariance can still be restored quite easily by a nonlinear version of the Stiickelberg trick. Because this elucidates several of the properties of massive gravity, we will review it here. When constructing a nonlinear candidate theory of massive gravity in Sect. 2.1.1, we employed the matrix M = g~lrj, where g~l is the inverse of the dynamical metric and r\ is the Minkowski metric. The appearance of a second, fixed metric in addition to the dynamical one is new to massive gravity. Indeed, it is necessary to have such a second metric, or reference metric, in order to give the graviton a mass. A mass term is a nonderivative interaction,3 and the only nonderivative scalars or scalar densities we can construct out of gliv alone are ix g = 4 and detg. The first possibility is trivial, while it was shown in Ref. [8] that functions of the metric determinant can only consistently lead to a cosmological constant as well. Therefore we need a reference metric in order to construct a massive graviton.4 Note that, while we have so far taken the reference metric to be Minkowski, in principle we could extend the theory to a more general reference metric, fliv. Consequently, even once we have specified the interaction potential there are many different massive gravity theories, one for each reference metric. Alternatively, /MV can be viewed as a "constant tensor" which must be specified by hand. Physically, the reference metric corresponds to the background around which linear fluctuations acquire the Fierz-Pauli form [15, 34]. This is why we have naturally discussed theories with a Minkowski reference metric: we began by considering fluctuations around that metric, and so it remains when extending to the nonlinear theory. Note that fliv = r\iiv is a natural choice, as the theory then possesses a Poincare-invariant preferred metric, allowing us to define mass and spin regardless of the solutions of the theory.5 The nonlinear Stiickelberg trick is simply to introduce into the reference metric four Stiickelberg fields, //xv = fabd„4>advb. (2.20) Note that here Latin indices are in field space, not spacetime; in particular, each of the adv(pb is trivial. 2.1 Massive Gravity and Bigravity 29 M;2fliv, pn M:jln, (2.31) where M* = Mf /Mg. Therefore the /-metric Planck mass is a redundant parameter. We will generally perform this rescaling implicitly in later chapters, although for now we will leave both Planck masses in to help to elucidate some of the physical features of the theory. In terms of free parameters bigravity is simpler than massive gravity: we have traded a constant matrix (/MV) for a constant number {Mf) which is not even physically relevant. We thus need to specify fewer theory ingredients to test its solutions.7 7Note however that the /-metric cosmological constant, /?4, is physically relevant in bigravity but not in massive gravity, as it is independent of and hence only contributes to the equation of motion for fflv. 2.1 Massive Gravity and Bigravity 31 On the other hand, it is less simple from the more theoretical point of view that it contains more degrees of freedom. The mass spectrum of bigravity contains two spin-2 fields, one massive and one massless. We can see this at the linear level by expanding each metric around the same background, gliv, as 1 _ _ 1 Mg Mf For simplicity we assume the "minimal model" introduced in Ref. [13], given by A) = 3, /31 = -1, 02=O, #,=0, 04 = 1- (2.33) The quadratic Lagrangian is given by [43] =^HR,linear = ~-h^£^v na/3 ~ ^^^v1^ where we have defined Meff = M~ + M^ . Indices are raised and lowered with the background metric. We notice the usual Einstein-Hilbert terms for each of the two metric perturbations, as well as two Fierz-Pauli terms with some additional mixing between and lMV. This can be easily diagonalised by performing the change of variables 1 _ 1 1 1 _ 1 1 Meff Mf Mg Meff Mf Mg The resultant unmixed Lagrangian, ^HRJmear = -\u^£fvUap - £fv Vap - ^m2 (v^V - V2) , (2.36) contains a Fierz-Pauli term for vliv and no interaction term for uliv. Therefore in the linearised theory uliv corresponds to a massless graviton and vliv to a ghost-free massive one with mass m. We can see that in the limit where one Planck mass is much larger than the other, the metric with the larger Planck mass corresponds mostly to the massless graviton: if Mf 2> Mg, then (u/iv, v/iv) -> (1/1V, h/iv), and similarly if Mg 2> Mf, then (u/iv, v/iv) -> (h/iv, —1/1V). This formalises the notion, which is intuitive from Eq. (2.30), that we can recover dRGT massive gravity by taking one of the Planck masses to infinity. In that case, the massless mode corresponds entirely to the metric with the infinite Planck mass, its dynamics freeze out so that it becomes fixed, and the massless mode decouples from the theory, leaving us with what we 32 2 Gravity Beyond General Relativity expect for massive gravity.8 Note, finally, that the notion of mass is only really well-defined around Minkowski space, as it follows from Poincaré invariance. More generally we can identify modes with a Fierz-Pauli term as massive, by analogy to the Minkowski case. As shown above, one can identify massive and massless linear fluctuations in this way around equal backgrounds for a special parameter choice, and indeed this can be done for general parameters as long as gliv and fliv are conformally related [40], but for general backgrounds there is no unambiguous splitting of the massive and massless modes in bigravity. An interesting and useful property of Hassan-Rosen bigravity is that while gliv and fliv do not appear symmetrically in the action (2.30), it nevertheless does treat does metrics on equal footing, ignoring the matter coupling. In particular, the mass term has the property 4 4 V=8~Y.P"e" {^f) = T^Z^-^ (VT31?) , (2.37) n=0 n=0 which follows from the identity *J—gen {^j1 g~lf^j = ^/—fe^-n {^Jf~xg^ [43]. This can easily be seen by formulating the en polynomials in terms of the eigenvalues of X as in Eq. (2.26). The result follows from using basic properties of the determinant and the fact that, because yj1 g~x f and yjf~xg are inverses of each other, their eigenvalues are inverses as well. As a consequence, the entire Hassan-Rosen action in vacuum is invariant under the exchange of the two metrics up to parameter redefinitions, £/xv //xv, MgoMf, fa-n- (2.38) The fact that the matter coupling breaks this duality by coupling matter to only one metric will motivate the search for "double couplings" in later chapters. By varying the action (2.30) with respect to the g and / metrics we obtain the generalised Einstein equations for massive bigravity [13], 3 j n=0 S 2 3 G>v(/) + Z fo-nfv*{n)v (/Pi) = 0, (2.40) n=0 where GI1V is the Einstein tensor computed for a given metric. The interaction matrices F(„) (X) are defined as 8 See, however, [45] for some caveats on taking the massive-gravity limit of bigravity 2.1 Massive Gravity and Bigravity 33 r(o)(X) y(1)(X) = X- - I[X], r(2)(X) = X2 - X[X] + h ([X]2 - [X2]), ^(3) (X) = X3 -X2[X] + ix([X]2-[X2]) 1 - -I 6 ([X]3-3[X][X2] + 2[X3]). (2.41) Notice that they satisfy the relation [40] n Y(n)(X) = ^(-iyX"-^-(X). (2.42) i=0 The tensors g^iY^ are symmetric and so do not need to be explicitly symmetrised [40], although this fact has gone unnoticed in much of the literature. Finally, TI1V is the stress-energy tensor defined with respect to the matter metric, g, 2 8U-detgž%) T„v =-- —-s__mf_ (243) V-det^ Sg^ It is not difficult to check that when TI1V = 0, the Einstein equations are symmetric under the interchanges (2.38). General covariance of the matter sector implies conservation of the stress-energy tensor as in general relativity, V^rMV = 0. (2.44) Furthermore, by combining the Bianchi identities for the g and / metrics with the field Eqs. (2.39) and (2.40), we obtain the following two Bianchi constraints on the mass terms: KY Z ßn8^Y(n)v (VáŤV) = 0, (2.45) n=0 2 3 after using Eq. (2.44). Only one of Eqs. (2.45) and (2.46) is independent: a linear combination of the two divergences can be formed which vanishes as an identity, i.e., regardless of whether gliv and fliv satisfy the correct equations of motion [46], so either of the Bianchi constraints implies the other. 34 2 Gravity Beyond General Relativity Field Equations for Massive Gravity Note that we can also easily obtain the equations of motion for dRGT massive gravity from the bimetric equations: the Einstein equation is simply Eq. (2.39) with fliv fixed to the desired reference metric, and matter is conserved with respect to g/iv as usual. This quick "derivation" should be taken purely as a heuristic—i.e., if we had started off with the dRGT action (2.21) and varied with respect to gliv, we would clearly obtain Eq.(2.47) regardless of whether fliv is dynamical—and not as the outcome of a limiting procedure. Indeed, because fliv lacks dynamics there is no analogue of its Einstein equation (2.40), and including that equation would lead to extra constraints. It turns out that massive gravity can be obtained as a limit of bigravity, but the process is more subtle than simply taking My —^ oo (which freezes out the massless mode and equates it with fliv, as discussed above) [45, 47]. Alternatively, the dRGT action can be obtained from the bigravity action by taking Mf -> 0, but to obtain massive gravity we must throw away the fliv Einstein equation (2.40) by hand. If we leave it in then it is determined algebraically in terms of gliv? Plugging this into the mass term we simply obtain a cosmological constant; thus this is the general-relativity limit of massive gravity.10 This agrees with the linear analysis above, where we found that in the limit Mf -> 0, the fluctuations of gliv become massless. 2.1.3 Cosmological Solutions in Massive Bigravity In this subsection we review the homogeneous and isotropic cosmology of massive bigravity. We will follow the framework derived in Refs. [48, 49], and use, with some generalisations, the notation and approach summarised in Ref. [50]. As discussed above, we will rescale fliv and fin so that the two Planck masses are equal, Mf = Mg. Cosmological Equations of Motion We assume that, at the background level, the Universe can be described by Friedmann-Lemaitre-Robertson-Walker (FLRW) metrics for both gliv and fliv. Specialising to spatially-flat metrics, we have yThis is because we are effectively taking the dRGT action and varying with respect to , treating it like a Lagrange multiplier. Hence fflv cannot be picked freely in this case but is rather constrained in terms of g/lv. 10We thank Fawad Hassan for helpful discussions on these points. (2.47) 2.1 Massive Gravity and Bigravity 35 dsi = -N(tfdt2 + a(tfdx2, (2.48) o dsl = -Xitfdt1 + Y(t)2dx2, (2.49) where a(t) and Y(t) are the spatial scale factors for g/iv and f/iv, respectively, and N(t) and X(t) axe their lapses. In the rest of this thesis we will leave the time dependences of these functions implicit. We will find it useful to define the ratios of the lapses and scale factors, X Y X ee - jee-. (2.50) N a Notice that these quantities are coordinate-independent: while we can freely choose either lapse or rescale either scale factor, their ratio is fixed. This is because bigravity is still invariant under general coordinate transformations as long as the same transformation is acted on each metric.11 With these choices of metrics, the generalised Einstein equations (2.39) and (2.40), assuming a perfect-fluid source with density p = —T°q, give rise to two Friedmann equations, 3H2 = ^-p + m2N2 (fr + 3 fry + 3/32y2 + fry3), (2.51) 8 3K2 = m2X2 (fry-3 + 3fry~2 + 3fry-1 + fr) , (2.52) where we have defined the Hubble rates as12 H = a/a, K = Y/Y, (2.53) and overdots denote time derivatives. We will specialise in this thesis to pressureless dust, which obeys p + 3Hp = 0. (2.54) The Bianchi constraint—either Eq. (2.45) or Eq. (2.46)—yields m2a2P (Xa - NY) = 0, (2.55) where we have defined P ee fr + 2fry + fry2. (2.56) 11 In group-theoretic terms, there are two diffeomorphism groups, one for each metric, and bigravity breaks the symmetry under each of them but maintains the symmetry under their diagonal subgroup. This is obvious from the fact that the mass term only depends on the metrics in the combination 5 Jav- 12In order to present the cosmological solutions for general lapses, we will define the g-metric Hubble rate differently here than in the rest of this thesis; in particular, H is not necessarily the cosmic-time Hubble rate. 36 2 Gravity Beyond General Relativity The Bianchi constraint has two branches of solutions: Algebraic branch: P = 0, Y Dynamical branch: á The algebraic branch is satisfied if ft + 2ft _y + ft J2 = 0, which seems to be nongeneric as it requires tuned initial conditions. Because the solutions on this branch have y = const., the mass term in Eq. (2.51) clearly reduces to a cosmological constant. Thus the algebraic branch, at the background level, is equivalent to ACDM [48, 51]. At the level of linear perturbations, evidence has been found for several modes being strongly coupled [52]. Consequently we will focus our attention on the dynamical branch. In this case, the Bianchi constraint implies that the ratio of the lapses, x, can be written in terms of other background quantities as The Friedmann equations, (2.51) and (2.52), and the Bianchi identity (2.57) can be combined to find a purely algebraic, quartic evolution equation for y, ft/ + (3ft - ft) y3 + 3 (ft - ft)y2 + ( -f^ + ft - 3ft h-ft = 0. (2.58) The g-metric Friedmann equation (2.51) and quartic equation (2.58) completely determine the expansion history of the Universe. As in standard cosmology, we see that the cosmic expansion is governed by a Friedmann equation. It is sourced by a mass term that depends on y, the evolution of which is in turn determined by the quartic equation. It will be useful to simplify the dynamics by expressing all background quantities solely in terms of y (t) and then solving for y (a). We can rearrange Eq. (2.58) to solve for p(y), Ky (2.57) P = -ft/ + (ft - 3ft)/ + 3(ft - fa)y + 3ft - ft + fry -l (2.59) m We can then substitute this into the Friedmann equation to find H(y), 3H2 = m2N2 (fry2 + 3fty + 3ft + fry-1) . (2.60) By taking a derivative of the quartic equation (2.58) and using the fluid conservation Eq.(2.54) and our solution (2.59) for pij) we can find an evolution equation for yifl), 2.1 Massive Gravity and Bigravity 37 dlny = J_= fry4 + (3fe - f34)y3 + 3(/3i - [33)y2 + (/30 - 3fe)y - ft dlna Hy 3fay4 + 2(3ft - ft)y3 + 3 (ft - ft)y2 + ft (2.61) Using the definition of y to find y, we can easily write K(y), K = H + -. (2.62) y We can now write any background quantity in terms of y alone, except for y and a themselves, and further we have two avenues for determining yia): integrating Eq. (2.61), or using the quartic equation (2.58) with p = poa~3. These expressions will be crucial throughout this thesis since they reduce the problem of finding any parameter—background or perturbation—to solving for y(z), where z = 1/a — 1 is the redshift.13 We note briefly that there has been some discussion in the literature over how to correctly take square roots in bigravity. There exist cosmological solutions in which det V' g~l f becomes zero at a finite point in time (and only at that time), and so it is important to determine whether to choose square roots to always be positive, per the usual mathematical definition, or to change sign on either side of the point where det j g~l f = 0. This was discussed in some detail in Ref. [53] (see also Ref. [54]), where continuity of the vielbein corresponding to jg~l f demanded that the square root not be positive definite. We will take a similar stance here, and make the only choice that renders the action differentiable at all times, i.e., such that the derivative of jg~l f with respect to gjiv and fjiv is continuous everywhere. In particular, for the FLRW backgrounds we are dealing with in this section, this choice implies that sj— det / = XY3. This is important because, as we will see in Chap. 3, it turns out that in the only cosmology with linearly-stable perturbations, the / metric bounces, so X = KY/H changes sign during cosmic evolution. With our square-root convention, the square roots will change sign as well, rather than develop cusps. Note that sufficiently small perturbations around the background will not lead to a different sign of this square root. Properties of Bimetric Cosmologies We can understand the qualitative behaviour of bimetric cosmologies by taking the early- and late-time limits, p -> oo and p -> 0, respectively. We will use heuristic arguments to motivate results which were determined more rigorously in Ref. [55] and can also be seen from a statistical comparison to observations of the expansion history [49]. At early times, the quartic equation (2.58) is solved either by y -> 0 or y —>• oo. The former solution is quite easy to see: the quartic equation is of the form ... + py = 0, where ... contains only positive powers of y, so y —>• 0 will clearly be a solution. These are called finite-branch solutions. The solutions with y —>• oo Note that while we have expressed all background quantities in terms of y only, perturbations will in general depend on both y and a. 38 2 Gravity Beyond General Relativity at early times, or infinite-branch solutions, occur when one of the higher powers of y in the quartic equation scales at just the right rate to cancel out the py term. These solutions are rather less common; in order to enforce Qm -> 1 and H2 > 0 at early times, viable infinite-branch solutions require /?2 = ft = 0 and ft > 0 [55]. To see this, notice that Qm = N2 p / (3M2H2) can be written in the limit y -> oo, using Eqs. (2.59) and (2.60), as ft 0k ft Qm = l- —y + 3^ -3 — . (2.63) ftJ 01 ft The condition ft > 0 (rather than just ft ^ 0) arises from demanding, per Eq. (2.60), that H2 be positive at all times. On either branch, at late times y will always flow to a constant, yc, given by the quartic equation with p = 0, ftyc4 + Oft - ft) yl + 3 (ft - ft) y2 + (ft - 3ft) yc - ft = 0. (2.64) Moreover, by taking a derivative of the quartic equation we see that y ^ 0 unless either p = 0or_y = 0. Therefore y evolves monotonically throughout cosmic history, flowing either from 0 up to yc or from oo down to yc.14 As long as yc > 0, the mass term asymptotes in the future to a cosmological constant. Hence bimetric cosmologies generally possess late-time acceleration with H2 ~ ^(m2ft). This is the case even if the g-metric cosmological constant, ft, is turned off, hence these theories self-accelerate. Because y cannot be constant in these models,15 the effective dark energy is dynamical. In particular, we do not have w = — 1 except at the asymptotic future. Crucially, the parameters and the potential structure leading to the accelerated expansion are thought to be stable under quantum corrections [56], in stark contrast to a cosmological constant, which would need to be fine-tuned against the energy of the vacuum [57-59].16 Thus we find that bigravity is an excellent candidate for technically-natural self-acceleration. Comparisons to background data—specifically the cosmic microwave background, baryon acoustic oscillations, and type la supernovae—show that these cosmological models can agree well with observations [48, 49, 55]. Before ending this subsection, let us consider a worked example: the model with only ft nonzero. Because theoretical viability conditions require this term to be nonzero (ignoring the exact ACDM case with ft = ft = 0), it is the simplest nontrivial one-parameter model which will lead to sensible cosmologies [55]. It has Note that these are not necessarily the same yc, as Eq. (2.64) can have multiple roots. 15Unless it is either 0 at all times, which is trivial, or the special case f}\ = /?3 = 0, in which case the Friedmann equation can be rewritten, with the help of the quartic equation, as a ACDM Friedmann equation with a rescaled gravitational constant [48]. 16If matter couples to gllv then matter loops will still contribute to m2f3o as usual. It is the rest of the dRGT potential which is stable under quantum corrections. Consequently we focus on self-accelerating models and assume—as is common in the literature—that some unknown mechanism removes the dangerous cosmological constant. 2.1 Massive Gravity and Bigravity 39 been shown in Refs. [49, 55] that this model provides a self-accelerating evolution which agrees with background cosmological observations and, as it possesses the same number of free parameters as the standard ACDM model, is a viable alternative to it. Indeed, it may be more viable than ACDM if the graviton mass turns out to be stable to quantum corrections, as mentioned above. The graviton mass in this case is given by +J~fi~\m. Note that in order to give rise to acceleration at the present era, the graviton mass typically should be comparable to the present-day Hubble rate, frm2 ~ H2. In this simple case, the evolution equation (2.61) for y(a), dlny din a can be integrated exactly to find (2.65) (2.66) Assuming y > 0 forces us to select the positive branch. Using the Friedmann and quartic equations, we can set the value of C using initial conditions, m2ßi H2 C = -—^ + 3-^, (2.67) where Hq is the cosmic-time Hubble rate today. Equivalently, we can use the quartic equation to solve for y and express the Friedmann equation as a modified expression fortf(p), iV2 H2 = 6M2 (p + Jl2m4Mp2 + p2) . (2.68) In either formulation, the late-time approach to a A-like behaviour is evident. The minimal /Ji-only model is also distinctive for having a phantom equation of state, w(z) ~ -1.22j$jg - 0.64+°,.o4z/(l + z) at small redshifts. Moreover, w is related in a simple way to the matter density parameter [55]. This provides a concrete and testable prediction of the model that can be verified by future large-scale structure experiments, such as Euclid [60, 61], intensity mappings of neutral hydrogen [62, 63], and combinations of structure and cosmic microwave background measurements [64]. The model has also been proven in [65] to satisfy an important stability bound at all times, avoiding the Higuchi ghost which plagues theories of a massive graviton on expanding backgrounds [66]. It is, however, worth noting that its linear cosmological perturbations are unstable until z ~ 0.5, as shown in Chap. 3 of this thesis. We emphasise that this instability does not rule out the /3i-only model, but rather impedes our ability to use linear perturbation theory to describe perturbations at all times. This raises the interesting question of how to make predictions for structure formation during the unstable period, a question which is beyond the scope of this thesis. 40 2 Gravity Beyond General Relativity The aforementioned studies have largely been restricted to the background cosmology of the theory. As the natural next step, in Chaps. 3 and 4 we will extend the predictions of massive bigravity to the perturbative level, and study how consistent the models are with the observed growth of structures in the Universe. A No-Go Theorem for Massive Cosmology As discussed in the previous section, we can easily obtain the equations of motion for massive gravity from those of bigravity; the g-metric equation is the same, and we lose the /-metric equation. Note that we also still have the Bianchi constraint (2.55). This fact will turn out to be crucial. Let us assume that the reference metric is Minkowski, fliv = r\iliV. Under the assumption of homogeneity and isotropy for gliv in unitary gauge, the Friedmann equation is simply equation (2.51) with y = a-1. This alone would define perfectly acceptable cosmologies. However, the Bianchi constraint (2.55) causes trouble. In the bigravity language we now have X = Y = 1, so the constraint becomes m2a2Pd = 0. (2.69) Because Y = 0, we no longer have an interesting dynamical branch: it merely suggests a = 0. Unfortunately, the algebraic branch, P = 0, also only has a = const, as a solution; in bigravity it would fix y = Y/a while allowing a and Y to change, but in massive gravity, it is a which is fixed. Therefore a is generally fixed to be constant, and this system has no dynamical solutions. This is the famous no-go theorem on cosmological solutions in dRGT massive gravity, and it is present for both flat and closed universes [67]. If we instead consider open universes or different reference metrics, such as FLRW or de Sitter, then FLRW solutions do exist, but they are unstable to the aforementioned Higuchi ghost and other linear and nonlinear instabilities [68-73]. The search for a viable cosmology with a massive graviton which avoids these conclusions has involved two routes. One is to extend dRGT by adding extra degrees of freedom. As discussed above, these problems are cured when the second metric is given dynamics. Other extensions of massive gravity, such as quasidilaton [74], varying-mass [67, 75], nonlocal [29, 76, 77], and Lorentz-violating [78, 79] massive gravity, also seem to possess improved cosmological behaviour. The other approach is to give up on homogeneity and isotropy entirely. While FLRW solutions are mathematically simple, the Universe could in principle have anisotropics which have such low amplitude, are so much larger than our horizon, or both, that we cannot easily observe them. Remarkably, these cosmologies are much better behaved in massive gravity than is the standard FLRW case [67]. The general scenario of an FLRW metric with inhomogeneous Sttickelberg fields has been derived in Refs. [80, 81]. This includes, but is not limited to, the case in which the reference metric is still Minkowski space, but only has the canonical form r\iliV = diag(—1, 1, 1, 1) in coordinates where gliv is not of the FLRW form [82]. The inhomogeneous and anisotropic solutions are reviewed thoroughly in Ref. [17]. See Ref. [83] for a review of cosmology in massive gravity and some of its extensions. 2.2 Einstein-Aether Theory 41 2.2 Einstein-Aether Theory In the final section of this chapter, we explore a different route to modifying gravity: allowing Lorentz symmetry to be violated. This is not a step taken lightly; Lorentz invariance is a cornerstone of modern physics. The two theories which have been separately successful at predicting nearly all experimental and observational data to date, general relativity to explain the structure of spacetime and gravity and the standard model of particle physics to describe particles and nongravitational forces in the language of quantum field theory, both contain Lorentz symmetry as a crucial underlying tenet. What do we gain from exploring the breakdown of this fundamental symmetry? Given its foundational significance, the consequences of violating Lorentz invariance deserve to be fully explored and tested. Indeed, while experimental bounds strongly constrain possible Lorentz-violating extensions of the standard model [84], Lorentz violation confined to other areas of physics—such as the gravitational, dark, or inflationary sectors—is somewhat less constrained, provided that its effects are not communicated to the matter sector in a way that would violate the standard-model experimental bounds. Moreover, it is known that general relativity and the standard model should break down around the Planck scale and be replaced by a new, quantum theory of gravity. If Lorentz symmetry proves not to be fundamental at such high energies—for instance, because spacetime itself is discretised at very small scales— this may communicate Lorentz-violating effects to gravity at lower energies, which could potentially be testable. The study of possible consequences of its violation, and the extent to which they can be seen at energies probed by experiment and observation, may therefore help us to constrain theories with such behaviour at extremely high energies. A pertinent recent example is Hořava-Lifschitz gravity, a potential ultraviolet completion of general relativity which achieves its remarkable results by explicitly treating space and time differently at higher energies [85]. The consistent nonpro-jectable extension [86-88] of Hořava-Lifschitz gravity is closely related to the model we will explore. Moreover, since we will be dealing with Lorentz violation in the gravitational sector, through a vector-tensor theory of gravity, the usual motivations for modifying gravity apply to this kind of Lorentz violation. Indeed, there are interesting models of cosmic acceleration, based on the low-energy limit of Hořava-Lifschitz gravity, in which the effective cosmological constant is technically natural [89, 90]. Generalised Lorentz-violating vector-tensor models have also been considered as candidates for both dark matter and dark energy [91, 92]. Lorentz violation need not have such dramatic, high-energy origins. Indeed, many theories with fundamental Lorentz violation may face fine-tuning problems in order to avoid low-energy Lorentz-violating effects that are several orders of magnitude greater than existing experimental constraints [93]. However, even a theory which possesses Lorentz invariance at high energies could spontaneously break it at low energies, and with safer experimental consequences. 42 2 Gravity Beyond General Relativity Spontaneous violation of Lorentz invariance will generally result when a field that transforms nontrivially under the Lorentz group acquires a vacuum expectation value (VEV). A simple example is that of a vector field whose VEV is nonvanishing everywhere. As mentioned above, in order to avoid the experimental constraints such a vector field should not be coupled to the standard model fields, but in order to not be completely innocuous we will ask it to couple to gravity. Moreover, to model Lorentz violation in gravity without abandoning the successes of general relativity—in particular, without giving up general covariance—the (spontaneously) Lorentz-violating field must be a spacetime vector and must be dynamical.17 A particularly simple, yet quite general, example of a model with these features is Einstein-aether theory (ae-theory) [94, 95]. It adds to general relativity a dynamical, constant-length timelike vector field, called the aether and denoted by ua, which spontaneously breaks Lorentz invariance by picking out a preferred frame at each point in spacetime while maintaining local rotational symmetry, thus breaking only the boost sector of Lorentz symmetry [95, 96]. The constant-length constraint plays two crucial roles. The first is phenomenological: it ensures that the aether picks a globally-nonzero VEV and so guarantees that Lorentz symmetry is in fact broken. The other role is to ensure that the theory is not sick: if the length is not fixed and the kinetic term is not gauge-invariant18 then the length-stretching mode has a wrong-sign kinetic term and hence is ghostlike [97]. Note that ae-theory is the most general effective field theory in which the rotation group is unbroken [98], and hence it can be seen as the low-energy limit of any theory which violates boosts but maintains rotational symmetry. 2.2.1 Pure Aether Theory Einstein-aether theory (which we will often refer to as "pure" Einstein-aether theory or ae-theory) is a theory of the spacetime metric gliv and a vector field (the "aether") ui1. The action is [95, 99] S = I d Xs/—g 1 R - KI1VpvV^ďVyU0 + X (u^u^ +m2) (2.70) 16ttG where we have defined K"vpa = clg^vgpa + c2^X + c3i^v - (V^X W), where i> = 2V[/Xuv] is the field strength tensor, and so is already included in the ae-theory action [94]. In what follows we will follow much of the literature on aether cosmology (e.g., Refs. [99, 100]) and ignore the quartic self-interaction term by setting ca, = 0. It is generally assumed that (standard-model) matter fields couple to the metric only. Any coupling to the aether would lead to Lorentz violation in the matter sector by inducing different maximum propagation speeds for different fields, an effect which is strongly constrained by experiments [84]. These problematic standard-model couplings may, however, be forbidden by a supersymmetric extension of ae-theory [101]. The work on ae-theory which we detail in Chap. 8 will be interested in exploring and constraining Lorentz violation in the gravitational sector and in a single non-standard-model scalar, hence we will not need to worry about such a coupling. The gravitational constant G that appears in Eq. (2.70) is to be distinguished from the gravitational constants which appear in the Newtonian limit and in the Friedmann equations, both of which are modified by the presence of the aether [99]. The Newtonian gravitational constant, G#, and cosmological gravitational constant, Gc, are related to the bare constant G by G 1 + 8ttG5' G (2.72) Gc = 1 + 87rGa' (2.73) where 8 = —c\m2, a = (ci3 + 3c2)m . (2.74) (2.75) We have introduced the notation en = c\ + C3, etc., which we will use throughout. 44 2 Gravity Beyond General Relativity 2.2.2 Coupling to a Scalar Inflation We now introduce to the theory a canonical scalar field

) (2/76) Let us pause to motivate the generality of this model. Our aim in Chap. 8 will be to constrain couplings between a Lorentz-violating field and a scalar, in particular a canonical, slowly-rolling scalar inflaton, in as general a way as possible. As mentioned above, Einstein-aether theory is the unique Lorentz-violating effective field theory in which rotational invariance is maintained [98],19 so any theory which spontaneously violates Lorentz symmetry without breaking rotational invariance will be described by the vector-tensor sector of our model at low energies. As for the scalar sector, the main restriction is that we have assumed a canonical kinetic term. While there are certainly coupling terms between the aether and the scalar which do not fall under the form V(6, + v - ev^ gflv + (upVpV0)(gllv + m-2Ulluv), (2.81) where JzfM = K^v paVljLupVvua is the Einstein-aether Lagrangian. Finally, the infla-ton obeys the usual Klein-Gordon equation, □0 = V0. (2.82) Notice that while this equation has the standard form, it couples the scalar to the aether since generally we will have = V^(0,(/)). Note that the equations of motion for the pure ae-theory follow simply by setting 0 = Oand V(0,4>) = 0. 2.2.3 Einstein-Aether Cosmology In this section we examine the evolution of FLRW cosmological solutions in Einstein-aether theory. Consider a flat FLRW background geometry evolving in conformal time, t, ds2 = a2(x)(-dx2 + dx2). (2.83) 46 2 Gravity Beyond General Relativity In pure ae-theory, we take the 0-0 and trace Einstein equations to obtain the Friedmann equations, ,„i Sic Gr o Jť2 = —^a2pm, (2.84) At: G c , M" =--j^a2pm(í + 3w), (2.85) where Jíľ = a'/a = dlna/dr is the conformal time Hubble parameter. These are exactly the Friedmann equations of general relativity except that, as discussed above, the bare cosmological constant, G, is renormalised, °c l + 87rGa' (2'86) with a = (ci +3c2+c3)m2. The aether does not change the cosmological dynamics at all, but just modifies the gravitational constant. This arises because in a homogeneous and isotropic background the Einstein-aether terms for the vector field only contribute stress-energy that tracks the dominant matter fluid, so the associated energy density is proportional to H2 [99].20 The aether does contribute dynamical stress-energy once we couple it to a scalar. In the theory (2.76) the Friedmann equations are 2 _ 8jrGc ^2 (v -eve + pm+l-'2a (2.88) For completeness we have included a matter component, but in the rest of this section and in Chap. 8 we will assume that 4> is gravitationally dominant and ignore any pm. The scalar field obeys the usual cosmological Klein-Gordon equation, 4>" + 2Jf?4>' + a2V

- B = E, f Ee (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) as well as Ag + Eg H UFg 3Bg\ 2Fg 1 A. N.\ 7 {— ~ If) + A^ " T2 {Bg ~ N*') ■ (3J5) N The linearised Einstein equations for the / metric are • 0-0: 3JL(kef-af) + ^ % + K (24v - ^)] - —2 4 (3AA + v2AB) = o. (3-16) Y2 \ XY X2 J\ 2M2 y3 V / • O-i: — di(Af-kef) + mz P la M2 y2 x + y X di (yFg - xFf) = 0. (3.17) • i-i: l x2 (2k X X . 1 1 2K + 3KA - 2 — K)Ef + KEf-af- 3KAy + —af + m2 1 M2 xy2 (32 + 32) Df l-pae +q(aA+^ (d) + 92) AB^j = 0, (3.18) Off-diagonal i-j: l-didjDf + m2 Q 2M2 xy2 J dld;AB = 0, (3.19) 3.1 Linear Cosmological Perturbations 59 where K = Y/Y is the /-metric Hubble parameter and we have again denned D f — Af + Ef K (4Ff 3Bf\ 2Ff 1 /.. X. \ + y \— - ITJ + y¥ ~ F \Bf ~ xBf) (3.20) Finally the fluid conservation equation, V^rMv = 0, can be split into time and space parts, neither of which is changed from general relativity, • Energy conservation (v = 0): VflT»o = -{p + 3Hp) (1 + 8) - p 8 +9 + Iäb + rV2Še (3.21) • Momentum conservation (v = i): 1 Vjr",- = (p+ 4Hp) (Vi + dtFg) + p (vi + dtFg) + -pdtEg. (3.22) We have assumed for simplicity that the fluid comprises only pressureless dust. These are in agreement with the results found elsewhere in the literature using various choices of gauge-invariant variables [4, 8, 9]. It is worth mentioning that the g-metric i-i equation, (3.6), is identically zero in GR after taking into account the 0-i, off-diagonal i-j, and momentum conservation equations and hence gives no information; in massive (bi)gravity, however, it is crucial, and is manifestly only important when m ^ 0. In a gauge with Fg = Ff = 0 it has the simple form [P (xEf-yEg)+2yQAA]=0. (3.23) Performing the same steps on the /-metric i-i equation, we arrive again atEq. (3.23). Hence both i-i equations carry the same information. We see there is an extra algebraic constraint hidden in the system of perturbation equations; this is closely related to the nontrivial constraint which eliminates the Boulware-Deser ghost,3 and will become important shortly when discussing the degrees-of-freedom counting in bigravity. Herein we will decompose the perturbations into Fourier modes without writing mode subscripts: every variable will implicitly refer to the Fourier mode of that variable with wavenumber k. 3We thank Shinji Mukhoyama for discussions on this point. 60 3 Cosmological Stability of Massive Bigravity 3.1.2 Counting the Degrees of Freedom While we have ten equations for ten variables, there are only two independent degrees of freedom.4 These can be seen as corresponding, for example, to the scalar modes of the two gravitons or to the matter perturbation and the scalar mode of the massive graviton. To understand the degrees-of-freedom counting, we will start with the simpler waters of general relativity, using the language we have employed for bigravity and following the spirit of the discussion in Ref. [10]. The time-time and time-space perturbations, Eg and Fg, as well as the velocity perturbation, 6, are auxiliary in that they appear in the second-order action without derivatives.5 Therefore their equations of motion are algebraic constraints which relate them to other perturbation variables, and they can be removed from the system trivially. This leaves us with three dynamical variables, Ag, Bg, and 8, two of which can be gauge fixed, or set to a desired value (such as zero) by a coordinate transformation. Hence at linear order general relativity only has one dynamical (scalar) degree of freedom propagating on FLRW backgrounds. In inflation, for instance, this is often taken to be the comoving curvature perturbation, £. This is relatively straightforward to extend to massive bigravity; we point the reader to Refs. [10, 11] for in-depth discussions. Few complications are introduced because the only new components of the theory are an Einstein-Hilbert term for /MV, which has exactly the same derivative structure as in general relativity, and a mass term, which has no derivatives. Therefore we can see immediately that five of the perturbations—Eg, E f, Fg, F f, and 6—are nondynamical and can be integrated out in terms of the dynamical variables and their derivatives. As discussed in Sect. 2.1.3, the coordinate invariance in massive bigravity is effectively the same as in general relativity, as long as we view the gauge transformations as acting on the coordinates, rather than on the fields. This can be seen by the fact that the Einstein-Hilbert terms are clearly invariant under separate diffeomorphisms for gliv and fliv, and the mass term is invariant as long as g~l f is, which is the case if we act the same coordinate transformation on each of them. We can therefore gauge fix two of the dynamical variables. Because we have started with ten perturbation variables, five of which were auxiliary and two of which can be gauge fixed, we are now left with three dynamical variables. However, they are not all independent. After the auxiliary variables are integrated out, one of the initially-dynamical variables becomes auxiliary, i.e., its derivatives drop out of the second-order action, and it can itself be integrated out. This leaves us, as promised, with two independent, dynamical degrees of freedom. It is therefore possible to reduce the ten linearised Einstein equations to a much simpler system of two coupled second-order differential equations. As we will see, The discussion in this section is indebted to useful conversations with Macarena Lagos and Pedro Ferreira. 5 Specifically, they appear without time derivatives. Recall that we are working in Fourier space where spatial derivatives effectively amount to multiplicative factors of ik. 3.1 Linear Cosmological Perturbations 61 with the right choice of gauge this process is fairly simple. This will allow us, in Sect. 3.2, to check whether the solutions to that system are stable. 3.1.3 Gauge Choice and Reducing the Einstein Equations In Ref. [10] a method for identifying the gauge-invariant degrees of freedom was presented in which Noether identities are used to identify "good" gauges, i.e., gauges in which the equations of motions for the gauge-fixed variables are contained in the remaining equations of motion. This methodology was applied to massive bigravity in Ref. [11]. While the method was developed with a focus on deriving the second-order action for the perturbations, rather than starting with the equations of motion as we do here, we will find that this method of picking a gauge will be convenient. Many common gauges choose to fix auxiliary variables, but this makes the job of reducing the perturbation equations to the minimal number of degrees of freedom difficult. By contrast, in the Noether-identity method only dynamical variables are fixed. Specifically, one chooses to eliminate those variables whose equations of motion are redundant, i.e., are contained within the equations of motion for fields which we leave in, so that no physical information is lost. Because the equations of motion for the perturbation variables are the same as the Einstein equations we are using, such a gauge choice works well for our purposes. The end result is that we should choose to eliminate one of {Ag, Af} and one of {Bg, Bf, x) [11], where X = k~28 + (3/2)k~2Ag — (l/2)Bg, which characterises the fluid flow, is the basic scalar dynamical degree of freedom for a perturbed fluid [12]. This uses up all of the available gauge freedom. In this chapter we will work in a gauge in which A f = x = 0- This has the advantage of treating the two metrics symmetrically: the remaining independent, dynamical fields are Bg and Bf, with Ag having become auxiliary in the process. Our goal is to derive the reduced system of equations of motion for Bg and Bf, as well as expressions relating all of the rest of the perturbation variables to these. Because the resultant equations are extraordinarily lengthy, we will not present them but will simply summarise the steps. Five equations—the 0-0 and 0-i Einstein equations for each of the metrics and the energy-conservation equation—correspond directly to the equations of motion for the five auxiliary variables. In these equations the auxiliary variables only appear linearly and without derivatives. Therefore we can easily "integrate them out" by solving the system of those five equations to obtain {Eg, Ef, Fg, Ff,6) in terms of the remaining degrees of freedom, {Ag, Bg, Bf}, and their derivatives. We are left with Ag, Bg, and Bf, and their equations of motion are the g-metric i-i, g-metric i-j, and /-metric i-j equations, respectively. As discussed above, the i-i equations effectively become constraints after manipulation with the other equations of motion. We demonstrated this in a gauge where Fg = Ff = 0; while mathematically simple, this gauge is not very helpful as it only eliminates auxiliary variables. In the more convenient gauge we are now using there is an equivalent 62 3 Cosmological Stability of Massive Bigravity statement: after integrating out the five auxiliary variables, all derivatives of Ag vanish from the g-metric i-i equation, so that it can be used to solve algebraically for Ag in terms of Bg, Bf, and their first derivatives. This is the result, mentioned above, that after integrating out the auxiliary variables, one of the dynamical variables becomes auxiliary. We note that Ag only depends on Bg and Bf up to first derivatives. This fact is crucial because Ag (though not Ag) appears in the remaining equations of motion. If Ag depended on second derivatives, then higher derivatives would appear upon integrating it out, and we would be in danger of a ghost instability. Indeed, as mentioned above, the fact that Ag loses its dynamics is nothing other than the Boulware-Deser ghost being rendered nondynamical by the specific potential structure of massive bigravity [13]. 3.2 Stability Analysis Having reduced the system of linearised Einstein and conservation equations using the steps outlined in Sect. 3.1.3, we can write our original ten equations as just two coupled second-order differential equations. Defining the vector (3.24) this system takes the simple form x + Ax + Bx = 0, (3.25) where A and B are matrices with extremely unwieldy forms which depend only on background quantities and on k. Equivalently we can write Eq. (3.25) in the same form in terms of N = In a (not to be confused with the g-metric lapse, N), x" + Ax' + Bx = 0, (3.26) where we use primes to denote derivatives with respect to N. We will choose this formulation to simplify the analysis and better understand its physical consequences. We are now in a good position to analytically probe the stability of linear cosmological perturbations in bigravity. If we neglect the dependence of A and B and treat them as constants, then Eq. (3.26) is clearly solved by a linear superposition of exponentials x = ^xieiMiN. (3.27) i We refer to as eigenfrequencies because they can be determined by solving for the eigenvalues of ia>i A + B. While A and B are not truly constant, they do in fact vary slowly enough for the WKB approximation to hold, in which case Eq. (3.27) 3.2 Stability Analysis 63 is the correct first-order solution for x. The criterion for the WKB approximation to hold is \co'/co2\ <$C 1; this will be satisfied in the cases in which we are interested. The criterion for stability is that all eigenfrequencies be real. This is necessary to obtain purely oscillating solutions for the perturbation variables; if any eigenfre-quency had an imaginary piece, then there would be exponentially growing modes. The eigenfrequencies we obtain from Eq. (3.26) are inordinately complicated.6 To simplify the analysis and focus on subhorizon scales, which account for most of the modes we can observe, we will take the limit k ^> J$?, where, as elsewhere in this thesis, we denote the conformal-time Hubble rate by J$?. As discussed in Sect. 2.1.3, the only theory with one nonzero parameter which allows for viable cosmological background expansion is the fi\ model, providing an excellent fit to background data including Type la supernovae, the cosmic microwave background, and baryon-acoustic oscillations [2, 3]. However, the analysis in this chapter shows that it suffers from instabilities throughout most of cosmic history. In the limit of large k/Jf? we find the eigenfrequencies for this model are k y-l + 12y2 + 9y4 'JF l+3y2 The condition for stability, i.e., for the object inside the square root to be positive, is v>yi(V5-2)^ 0.28. (3.29) This suggests that there is an instability problem at early times; recall from Sect. 2.1.3 that the fi\ model has only finite-branch solutions, meaning that y evolves monoton-ically from 0 at early times to, at late times, a positive constant. Therefore the fi\ model always suffers from an instability at early times, with a turnover from unstable to stable occurring when y ~ 0.28. This instability is quite dangerous. Consider scales of k ~ lOO^f, which is a typical mode size for structure observations. Those modes would then grow, assuming Im(o)) ~ as roughly e100N, which is far too rapid for linear theory to be applicable for more than a fraction of an e-fold. During what time period is this instability present? If y reached 0.28 at sufficiently early times, one might expect that the presence of radiation or other new fields, which we have ignored in favour of dust, could ensure that modes are stable. However, the time at which the instability turns off is generically close to the time at which the expansion begins to accelerate, so this is clearly a modern problem. We can find the exact region of instability recalling that, in this model, we can solve for y(a) exactly, c.f. Eq. (2.66), y(a) = (-C + 7l2a6 + C2) . (3.30) We recognise that the number of unused synonyms for "these equations are very long" is growing short as this chapter progresses. 64 3 Cosmological Stability of Massive Bigravity where C = -ft + J-, (3.31) where B\ = m2ft /Hq and Hq is the cosmic-time Hubble rate today. The best-fit value for ft using a combination of SNe, CMB, and BAO data is ft = 1.448 ± 0.0168 [2, 14]. For ft = 1.448 exactly, oo^ switches from imaginary to real at N = —0.49, corresponding to a relatively recent redshift, z = 0.63435. This number is fairly sensitive to the choice of datasets. The CMB and BAO data are taken from observations which assume general relativity in their analysis. Restricting the analysis to supernovae alone, the best fit is ft = 1.3527 ± 0.0497. In this case, the instability ends at N = —0.38 or z = 0.47. At any epoch before this, the perturbation equations are unstable for large k. This behaviour invalidates linear perturbation theory on subhorizon scales and may rule out the model if the instability is not cured at higher orders. This is not necessarily out of the realm of possibility. As discussed earlier, massive bigravity possesses the Vainshtein mechanism, in which nonlinear effects suppress the helicity-0 mode of the massive graviton in dense environments, thereby recovering general relativity [15, 16]. It may be the case that such a mechanism will also impose general-relativistic behaviour on nonlinear cosmological perturbations. Now let us move on to more general models. As we mentioned in Sect. 2.1.3, the other one-parameter models are not viable in the background,7 i.e., none of them has a matter-dominated epoch in the asymptotic past and produces a positive Hubble rate [3].8 Nevertheless it is worthwhile to calculate the eigenfreqencies in these cases in order to study the asymptotic behaviour of the viable multiple-parameter models. For simplicity, from now on we refer to a model in which, e.g., only ft and ft are nonzero as the ft ft model, and so on. At early times, every viable, finite-branch, multiple-parameter model is approximately described by the single-parameter model with the lowest-order interaction. For instance, the ft ft, ftft, and ftftft models all reduce to ft, the ft ft model reduces to ft, and so on. Similarly, in the early Universe, the viable, infinite-branch models reduce to single-parameter models with the highest-order interaction. This is clear from the structures of the terms in the Friedmann equation and the P and Q parameters introduced above. It is only through these terms that the ft parameters enter the perturbation equations. Therefore, in order to determine the early-time stability, we need only look at the eigenfrequencies of single-parameter models. In addition to copl presented in Eq. (3.28) above, we have 7With the exception of the /?o model, which is simply ACDM. 8We frequently discuss the viability of various models in this section; all such results were derived inRef. [3]. 3.2 Stability Analysis 65 k 1 ^2 = ±^V (3.32) k y-3 + 8y2-/ ^ V3(l-y2) (3.33) fc 1 «/34 = ± (3.34) We see that the ft and ft models are stable at all times, while the ft model suffers from an early-time instability just like the ft model. We can now extend these results to the rest of the bigravity parameter space by using the single-parameter models to test the early-time stability. Since much of the power of massive bigravity lies in its potential to address the dark energy problem in a technically-natural way, let us first consider models without an explicit g-metric cosmological constant, i.e., ft = 0. On the finite branch, all such models with ft ^0 reduce, at early times, to the ft model. As we have seen, this possesses an imaginary sound speed for large k, cf. Eq. (3.28), and is therefore unstable in the early Universe. Hence the finite-branch ft ft ft ft model and all its subsets with ft 7^ 0 are all plagued by instabilities. This is particularly significant because all of these models otherwise have viable background evolutions [3]. This leaves the ft ft ft model; this is stable on the finite branch as long as ft 7^ 0, but its background is not viable. We conclude that there are no models with ft = 0 which live on a finite branch, have a viable background evolution, and predict stable linear perturbations at all times. This conclusion has two obvious loopholes: we can either include a cosmological constant, ft,or turn to an infinite-branch model. We first consider including a nonzero cosmological constant, bearing in mind this may not be as interesting theoretically as the models which self-accelerate. Adding a cosmological constant can change the stability properties, although it turns out not to do so in the finite-branch models with viable backgrounds. In the ft ft model, the eigenfrequencies, are unaffected by ft at early times and therefore still imply exponential mode growth in the asymptotic past. This result extends (at early times) to the rest of the bigravity parameter space with ft, ft 7^ 0. No other finite-branch models yield viable backgrounds. In conclusion, all of the solutions on a finite branch, for any combination of parameters, are either unviable (in the background) or linearly unstable in the past. Let us now turn to the infinite-branch models. There are two candidates with viable background histories. The first is the the ftftft model. The reality of a>p2 and cop4, cf. Eq. (3.32), suggests that this model is linearly stable. At the background level, this case is something of an exception as it is the only bimetric cosmology with exact ACDM evolution: the structure of the Friedmann equation and quartic equation conspire to allow the dynamics to be rewritten with a modified gravitational cop0p, = ± k V-l+2(ft/ft)y + 12y2 + V Jf? l+3y2 (3.35) 66 3 Cosmological Stability of Massive Bigravity constant and an effective cosmological constant [1], ,„2 A P , 2ftft~9ft2 3H =--t + m-• (3.36) ft-3ft M2 ft-3ft The quartic equation for y can be solved to find 2 ^ + A)-3ft y = —--• (3.37) ft-3ft This implies that all solutions to this model live on the infinite branch. In order for y to be real at all times, we are required to have ft — 3ft > 0 and ft — 3ft > 0. Unfortunately, for these reasons we cannot have a viable self-accelerating solution; if ft were set to zero (or were much smaller than ft and ft), then the effective cosmological constant would be negative. The modified gravitational constant would also be negative if ft were positive. From a cosmological point of view, these models are therefore not altogether interesting. Finally there is a small class of viable and interesting models which have stable cosmological evolution: the self-accelerating ft ft model and its generalisation to include ft.9 Here, y evolves from infinity in the past and asymptotes to a finite de Sitter value in the future. For these ftft ft models we perform a similar eigenfrequency analysis and obtain k y~l + 2 (fa/fa) y + I2y2 + (9 + 2 fa fa/ft) y4 -2 (fa/fa) [4y^ + 3y5 - (fa/fay6] *AA=±Jf l + 3y2-2(fa/fa)yi ' (3.38) Restricting ourselves to the self-accelerating models (i.e., ft = 0), we obtain k + I2y2 + 9y4-2 (ft/ft) [4y3 + 3y5 - (ft/ft)/] ^ = * 1 + 3y2 - 2 (ft/ft) y3 • (339) Notice that at early times, i.e., for large y, the eigenvalues (3.38) and (3.39) reduce to the expression (3.34) for cop4. This frequency is real, and therefore the ftft model, as well as its generalisation to include a cosmological constant, is stable on the infinite branch at early times. Interestingly, the eigenfrequencies for this particular model can also be written as y We do not have the freedom to include nonzero /?2 or ft; in either case the background evolution would not be viable [3]. We can see this from the expressions (2.59) and (2.60) for p(y) and H2(y). If ft were nonzero, then Qm = p/(3MgH2) would diverge as y at early times. Setting ft=o, we find Qm —> 1 — 3ft /ft as y —> oo. If we demand a matter-dominated history, then ft must at the very least be small compared to ft. 3.2 Stability Analysis 67 Fig. 3.1 Plot of the function y'{y) for the /61/64 model for yg4 = 2/?i. For both the finite q and infinite branches, the final state is the de Sitter point. The arrows show the ^ -1 direction of movement of r -3 —1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—1—|—: /finite brancn^^^S. /\ de Sitter point > ■ ^S'^^^^^^n^n^^ra nch ■ _1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y k I y" k I 1 dy' *>A,AA = ^yj-Jy = ±^"3^ (3-40) Therefore, the condition for the stability of this model in the infinite branch, where y' < 0, is simply y" > 0. One might wonder whether this expression for 00 is general or model specific. While it does not hold for the fa and fa models, c.f. Eqs. (3.32) and (3.33), it is valid for all of the submodels of fa fa fa, including the single-parameter models presented in Eqs. (3.28) and (3.34). We can see from this, for example, that the finite-branch (y' > 0) fa model is unstable at early times because initially y" is positive. In Fig. 3.1 we show schematically the evolution of the fa fa model on the finite and infinite branches. The stability condition on either branch is y"/y' = dy'/dy < 0. For the parameters plotted, fa = 2fa, one can see graphically that this condition is met, and hence the model is stable, only at late times on the finite branch but for all times on the infinite branch. Our remaining task is to extend this to other parameters. Let us now prove that the infinite-branch fa fa model is stable in the subhorizon limit at all times as long as the background expansion is viable, which restricts us to the parameter range 0 < fa < 2fa [3]. From Eq. (3.39) we can see that the subhorizon perturbations are clearly stable if and only if - 1 + I2y2 + 9y4-2 (fa/fa) [4y3 + 3y5 - (fa/fa)y6] > 0 (3.41) At early times, y -> 00, this is dominated by a manifestly positive term. Indeed we have already seen that the eigenfrequencies match those in the fa model (3.34) which are purely real. At later times, Eq. (3.41) is satisfied for all y > 1 as long as we restrict to the viable parameter range, 0 < fa/fa < 2. We can therefore rephrase the question of stability as a question about the background evolution: do the infinite-branch models in this region of the parameter space always have y > 1? The answer is yes. Recall from Sect. 2.1.3 that, on the infinite branch, y evolves monotonically from y = 00 to y = yc, where yc is defined by Eq. (2.64), 68 3 Cosmological Stability of Massive Bigravity fay3c - 3 fay2 + fa = 0. (3.42) Because the evolution of y is monotonic, y > 1 at all times if yc > 1. Moreover, because y = yc corresponds, through the quartic Eq. (2.58), to p = 0, we are only interested in the largest real root of Eq. (3.42). For the largest allowed value of fa, fa = 2fa exactly, we find yc = 1. We must then ask whether for 0 < fa < 2fa, yc remains greater than 1. Writing p = yc — 1, using Descartes' rule of signs, and restricting ourselves to 0 < fa < 2fa, we can see that p has one positive root, i.e., there is always exactly one solution with yc > 1 in that parameter range. Therefore, in all infinite-branch solutions with 0 < fa < 2fa, y evolves to some yc > 1 in the asymptotic future. We conclude that all of the infinite-branch fa fa cosmologies which are viable at the background level are also linearly stable at all times in the subhorizon limit, providing a clear example of a bimetric cosmology which is a viable competitor to ACDM. This stability has been confirmed and extended to the superhorizon limit in a complementary analysis in Ref. [11]. As a side remark, we note that in this model the asymptotic past corresponds to the limit y -> oo and y' -> —\y, i.e., y -> a~3^2. This implies that Y ~ a-1/2, i.e., the second metric initially collapses while "our" metric expands. On the approach to the final de Sitter stage, y approaches a constant yc, so the scale factors a and Y both expand exponentially. This infinite-branch model therefore contains a bouncing cosmology for the / metric. This bounce has an unusual consequence. Recall from Eq. (2.57) that, after imposing the Bianchi identity, we have /oo = —Y21M'2. Therefore, when y bounces, /oo becomes zero: at that one point, the lapse function of the / metric vanishes.10 Nevertheless, this does not render the solution unphysical, for the following reasons. First, the / metric does not couple to matter and so, unlike the g metric, it does not have a geometric interpretation. A singularity in fliv therefore does not necessarily imply a singularity in observable quantities. In fact, we find no singularity in any of our background or perturbed variables. Second, although the Riemann tensor for the / metric is singular when /oo = 0, the Lagrangian density s/—fR(f) remains finite and nonzero at all times, so the equations of motion can be derived at any points in time. 3.3 Summary of Results In this chapter, we introduced the tools for perturbation theory in massive bigravity and used them to test the stability of the theory. We began by presenting the cosmological perturbation equations; these are derived in Appendix A. We went on to detail the way in which the physical degrees of freedom are counted and described how to pick a good gauge and integrate out nondynamical 10Moreover, the square root of this, Y/Jff, appears in the mass terms. We choose branches of the square root such that this quantity starts off negative at early times and then becomes positive. 3.3 Summary of Results 69 variables. By doing so we reduced the ten linearised Einstein and fluid conservation equations to a system of two coupled, second-order differential equations. These describe the evolution of the two independent, dynamical degrees of freedom present at linear order around FLRW solutions in massive bigravity. We then identified the stable and unstable models by employing a WKB approximation and calculating the eigenfrequencies of the perturbation equations. This analysis revealed that many models with viable background cosmologies exhibit an instability on small scales until fairly recently in cosmic history. However, we also found a class of viable models which are stable at all times. These are defined by giving nonzero, positive values to the interaction parameters j5\ and #4, setting f52 = /?3 = 0, and choosing solutions in which the ratio y = Y/a of the two scale factors decreases from infinity to a finite late-time value. A cosmological constant can be added without spoiling the stability, although it is not necessary; the theory is able to self-accelerate. On the surface, these results would seem to place in jeopardy a large swath of bigravity's parameter space, such as the "minimal" ^i-only model which is the only single-parameter model that is viable at the background level [3]. It is important to emphasise that the existence of such an instability does not automatically rule these models out. It merely impedes our ability to use linear theory on deep subhorizon scales (recall that the instability is problematic specifically for large k). Models that are not linearly stable can still be realistic if only the gravitational potentials become nonlinear, or even if the matter fluctuations also become nonlinear but in such a way that their properties do not contradict observations. The theory can be saved if, for instance, the instability is softened or vanishes entirely when nonlinear effects are taken into account. We might even expect such behaviour: bigravity models exhibit a Vainshtein mechanism [15, 16] which restores general relativity in environments where the new degrees of freedom are highly nonlinear. Consequently two very important questions remain: can these unstable models still accurately describe the real Universe, and if so, how can we perform calculations for structure formation? Until these questions are answered, the infinite-branch model seems to be the most promising target at the moment for studying massive bigravity. In the next chapter, we will calculate its predictions for structure formation, confront them with data, and discuss the potential of near-future probes like EUCLID to test this model against ACDM. References 1. M. von Strauss, A. Schmidt-May, J. Enander, E. Mortsell, S. Hassan, Cosmological Solutions in Bimetric Gravity and their Observational Tests. JCAP 1203, 042 (2012). arXiv:lll 1.1655 2. Y. Akrami, T.S. Koivisto, M. Sandstad, Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality. JHEP 1303, 099 (2013). arXiv: 1209.0457 3. F. Konnig, A. Patil, L. Amendola, Viable cosmological solutions in massive bimetric gravity. JCAP 1403, 029 (2014). arXiv: 1312.3208 70 3 Cosmological Stability of Massive Bigravity 4. D. Comelli, M. Crisostomi, L. Pilo, Perturbations in massive gravity cosmology. JHEP 1206, 085 (2012). arXiv: 1202.1986 5. A. De Felice, A.E. Gumrukcuoglu, S. Mukohyama, N. Tanahashi, T. Tanaka, Viable cosmology in bimetric theory JCAP 1406, 037 (2014). arXiv: 1404.0008 6. D. Comelli, M. Crisostomi, L. Pilo, FRW Cosmological perturbations in massive bigravity. Phys. Rev. D90(8), 084003 (2014). arXiv: 1403.5679 7. M. Fasiello, A.J. Tolley, Cosmological stability bound in massive gravity and bigravity. JCAP 1312, 002 (2013). arXiv:1308.1647 8. M. Berg, I. Buchberger, J. Enander, E. Mortsell, S. Sjors, Growth histories in bimetric massive gravity. JCAP 1212, 021 (2012). arXiv: 1206.3496 9. F. Konnig, Y. Akrami, L. Amendola, M. Motta, A.R. Solomon, Stable and unstable cosmological models in bimetric massive gravity. Phys. Rev. D 90, 124014 (2014). arXiv: 1407.4331 10. M. Lagos, M. Bahados, PG. Ferreira, S. Garcia-Saenz, Noether identities and gauge-fixing the action for cosmological perturbations. Phys. Rev. D89(2), 024034 (2014). arXiv:1311.3828 11. M. Lagos, PG. Ferreira, Cosmological perturbations in massive bigravity. JCAP 1412(12), 026 (2014). arXiv: 1410.0207 12. V.F Mukhanov, H. Feldman, R.H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions. Phys. Rep. 215, 203-333 (1992) 13. D. Langlois, S. Mukohyama, R. Namba, A. Naruko, Cosmology in rotation-invariant massive gravity with non-trivial fiducial metric. Class. Quantum Gravity 31, 175003 (2014). arXiv:1405.0358 14. A.R. Solomon, Y. Akrami, T.S. Koivisto, Linear growth of structure in massive bigravity. JCAP 1410, 066 (2014). arXiv: 1404.4061 15. A. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B 39, 393-394 (1972) 16. E. Babichev, C. Deffayet, An introduction to the Vainshtein mechanism. Class. Quantum Gravity 30, 184001 (2013). arXiv: 1304.7240 Chapter 4 Linear Structure Growth in Massive Bigravity The wonder is, not that the field of the stars is so vast, but that man has measured it. Anatole France, The Garden of Epicurus We have, to this point, reviewed the background FLRW solutions of massive bigravity in Chap. 2 and begun to analyse linear perturbations around these solutions in Chap. 3. In addition to introducing the formalism for cosmological perturbation theory in bigravity, the specific aim of the previous chapter was to identify which models are stable at the linear level and which are not. The natural next step is to use perturbation theory to derive observable predictions for the stable models. In this chapter we undertake a study of the cosmological large-scale structure (LSS) in massive bigravity with the aim of understanding the ways in which bigravity deviates from general relativity and its potentially-testable cosmological signatures. This is motivated in particular by anticipation of the forthcoming Euclid mission which is expected to improve the accuracy of the present large-scale structure data by nearly an order of magnitude [1, 2]. At the background level, careful statistical analyses show that several bimetric models can provide as good a fit as the standard ACDM model, including one case, the ft-only model, which has the same number of free parameters as ACDM [3-5].1 This is a blessing and a curse; while it is encouraging that massive bigravity can produce realistic cosmologies in the absence of a cosmological constant, the quality of background observations is not good enough to distinguish these models from ACDM or from each other. In particular, the parameter constraints obtained from the expansion history have strong degeneracies within the theory itself. To efficiently test the theory and distinguish its cosmology from others one needs to move beyond the FLRW metric and study how consistent bimetric models are with the observed growth of cosmic structure. We restrict ourselves to the linear, As discussed in Chap. 3, this model is linearly unstable. However, if the instability is cured at higher orders in perturbation theory before the background solution is spoiled, then at the background level this is a perfectly viable model. © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7_4 71 72 4 Linear Structure Growth in Massive Bigravity subhorizon regime and examine whether there are any deviations from the standard model predictions which may be observable by future LSS experiments. Note that due to the aforementioned instability, some (though not all) bimetric models cannot be treated with linear perturbation theory at all times, as any individual mode would quickly grow large during the period of instability (from early times until some recent redshift). Structure formation must be studied nonlinearly in these cases. In order to demonstrate our methodology, we choose to study every model with a sensible background evolution and one or two free parameters, so some unstable cases will be included. For the most part these should be seen as toy models, useful for illustrative purposes, although our results about quantities which do not depend on initial conditions, such as the anisotropic stress, will be observationally relevant at later times. One specific model which we study, the infinite-branch /Ji #4 model, is stable at all times, and so we will pay extra attention to its comparison to observations. This chapter is organised as follows. Taking the subhorizon and quasistatic limit of the full perturbation equations, we arrive at a convenient closed-form evolution equation in Sect. 4.1 that captures the modifications to the general relativistic growth rate of linear structure, as well as the leading-order scale-dependence which modifies the shape of the spectrum at near-horizon scales. The coefficients in the closed-form equation are given in Appendix B. The results are analysed numerically in Sect. 4.2, where we discuss the general features of the models and confront them with the observational data. We conclude in Sect. 4.3. 4.1 Perturbations in the Subhorizon Limit We define the perturbed metrics in conformal time (iV = a) as ds2 = a2 {-(1 + Eg)dx2 + IdiFgdxdx1 + [(1 + Ag)Sij + didjBg] dxldxj] , (4.1) dsj = -X2(l + Ef)dx2 + IXYdiFfdrdx' + Y2 [(1 + Af)8tj + fydjBf] dx'dxj, (4.2) and immediately specialise to dust (P = 8P = 0) and work in Fourier space. The linearised Einstein and fluid conservation equations have been presented in Sect. 3.1 and are derived in Appendix A. These equations are quite complicated; in order to isolate the physics of interest, namely that of linear structure in the subhorizon regime, we focus on the subhorizon, quasistatic limit of the field equations. This limit is defined by taking k2<3> ^> H2<3> ~ H<£> ~ O for any variable 0 limit yields an identity, as expected. There is a further interesting feature: if we substitute the background equations into the /-metric trace equation, (4.7), we obtain Eq. 4.11 again. Hence one of the two trace equations is redundant and can be discarded, so what looks like a system of six equations is actually a system of five. With these relations, the system of equations presented in this section is closed. 4.2 Structure Growth and Cosmological Observables In this section we study the linearised growth of structure in the quasistatic and subhorizon limit, first solving the field equations to obtain predictions and then comparing to data. Deviations from the predictions of general relativity can be summarised by a few parameters which are observable by large-scale structure surveys such as Euclid [1, 2]. The main aim of this section is to see under what circumstances these parameters are modified by observable amounts in the linear regime by massive bigravity. 4.2.1 Modified Gravity Parameters We will focus on three modified growth parameters, defined in the Euclid Theory Working Group review [2] as / (and its parametrisation y), Q, and r\. They are: Growth rate (f) and index (y): These parameters measure the growth of structure, and are defined by m 2 [P (xEf-yEg)+2yQAA]=0. (4.11) f (a, k) = d log 8 £2y (4.12) d loga 'm' where Qm = a2p/(3M 2 H2) is the usual matter density parameter. This equation holds beyond the subhorizon limit, in a particular gauge; see Appendix A. 4.2 Structure Growth and Cosmological Observables 75 Modification of Newton's Constant (0: The function Q(a,k)5 parametrises modifications to Newton's constant in the Poisson equation, k Q(a,k)p —Ag = '8. (4.13) a M2 Anisotropic Stress (j/): Effective anisotropic stress leads the quantity Ag + Eg to deviate from its GR value of zero, which we can parametrise by the parameter r](a, k), n(a,k) = -^-. (4.14) In GR, these parameters have the values y ~ 0.545 and Q = rj = 1. We have five independent Einstein equations (Eqs. (4.3), (4.5), (4.6), (4.8), (4.11)) for five metric perturbations6 and 8. Crucially, this system is algebraic. There are five equations for six variables, so we can only solve for any five of the perturbations in terms of the sixth. Of the modified growth parameters, Q and rj are ratios of perturbations so are insensitive to how we solve the system. However, to find y we need to solve a differential equation for 8. It is therefore simplest to solve for the perturbations {Agj, Egj, Bf} in terms of 8. Solving the system, we find each perturbation can be written in the form fix, k)8, for some function fit). We do not display the solutions here as they are quite unwieldy, although we do note that in the limit with only ft 7^ 0 studied in Ref. [6], and taking into account differences in notation and gauge, our expressions for the perturbations match theirs. With these solutions for {Agj, Egj, Bf} in hand, we can immediately read off Q and rj. To calculate the growth index, y, we need to solve a conservation equation for the density contrast, 8. The fluid conservation Eqs. (4.9) and (4.10), are unchanged from GR, so as in GR we can manipulate them to find the usual evolution equation for 8 sourced by the gravitational potential, ■ 1 , 8 + H8 + -k2Eg(8) =0. (4.15) At this point we diverge from the usual story. In GR, there is no anisotropic stress and the Poisson equation holds; combining the two, we find k2Eg = — (a2p/M2)8. Both of these facts are changed in massive bigravity, so there is a modified (and rather more complicated) relation between k2Eg and 8. However, since we do have such a relation, 8 still obeys a closed second-order equation which we can solve numerically. 5Not to be confused with the background quantity defined in Eq. (3.9), Q = ft + (x + y) ft +xyft 6 After gauge fixing there are six metric perturbations, but once we substitute the 0-i equations into the trace i—j equations, Ff drops out of our system. In a gauge where Fg = Ff = 0, as was used in Ref. [6], the equivalent statement is that the Bg and Bf parameters are only determined up to their difference, Bf — Bg, which is gauge invariant. 76 4 Linear Structure Growth in Massive Bigravity Finally, we note that the three modified gravity parameters are encapsulated by five time-dependent parameters. The expressions for r\ and Q can be written in the forms /l +k2h4\ " = hilT¥K-J' <4'16) (\ + k2h4\ where the ht are functions of time only and depend on m2fa. We present their explicit forms in Appendix B. The same result has been obtained for Horndeski gravity [7, 8], which is the most general scalar-tensor theory with second-order equations of motion [9]. The similarity is a consequence of the fact that massive bigravity introduces only a single new spin-0 degree of freedom, its equations of motion are second-order, and the new mass scale it introduces (the graviton mass) is comparable to the Hubble scale [6, 10]. Furthermore, the structure growth equation, (4.15), can be written in terms of Q and r\ and hence the hi coefficients as . 1 Q a2p 8 + H8 -- — —V<5 = 0. (4.18) 2 r] Ml The quantity Q/rj, sometimes called Y in the literature [6, 8, 11], represents deviations from Newton's constant in structure growth, and is effectively given in the subhorizon regime by ih\hs)/(Ji2h^). 4.2.2 Numerical Solutions In this section we numerically solve for the background quantities and modified gravity parameters for one- and two-parameter bigravity models.7 We look in particular for potential observable signatures, as the growth data are currently not competitive with background data for these theories, although we expect future LSS experiments such as Euclid [1, 2] to change this. The recipe is straightforward: using Eq. (2.61) we can solve directly for y(z), which is all we need to find solutions for rj(z,k) and Q(z, k) using Eqs. (4.16) and (4.17). Finally these can be used, along withEqs. (2.59) and (2.60),to solve equation (4.18) numerically for5(z, k) and hence for/(z, k). We fit f(z, k) to the parametrisation Q/m in the redshift range 0 < z < 5 unless stated otherwise. 7We focus on these simpler models to illustrate bigravity effects on growth. Current growth data are not able to significantly constrain these models, so we would not gain anything by adding more free parameters. 4.2 Structure Growth and Cosmological Observables 77 The likelihoods for these models were analysed in detail in Ref. [3], using the Union2.1 compilation of Type la supernovae [ ], Wilkinson Microwave Anisotropy Probe (WMAP) seven-year observations of the CMB [13], and baryon acoustic oscillation (BAO) measurements from the galaxy surveys 2dFGRS, 6dFGS, SDSS and WiggleZ [14-16]. We compute likelihoods based on growth data compiled in Ref. [17], including growth histories from the 6dFGS [18], LRG20o, LRG6o [19], BOSS [20], WiggleZ [21], and VIPERS [22] surveys. Both the numerical solutions of background quantities and the likelihood computations are performed as in Ref. [3], where they are described in detail. Following Ref. [3], we will normalise the fa parameters to present-day Hubble rate, Hq, by defining 2 m Bi = -2 A. (4.19) Ho Throughout, we will assume that the g-metric cosmological constant, Bq, vanishes, as we are interested in the solutions which accelerate due to modified-gravity effects. The Minimal Model We begin with the "minimal" model in which only B\ is nonzero. This is the only single-Bi theory which is in agreement with background observations [3]; the other models also have theoretical viability issues [5]. Note, however, that the linear perturbations are unstable at early times until relatively recently, z ~ 0.5, as discussed in Chap. 3. This restricts the real-world applicability of the results presented herein, as the quasistatic approximation we employ will not be viable. Our results will hold for observations within the stable period. Specifically, our results for Q and r\ will certainly hold, while the growth rate, /, may vary if the initial conditions for 8 are significantly changed from what we assume herein. Otherwise this should be seen as an illustrative example. The likelihoods for B\ are plotted in Fig. 4.1 based on supernovae, BAO/CMB, growth data, and all three combined, although the growth likelihood is so wide that it has a negligible effect on the combined likelihood. The point was raised in Ref. [6] that the WMAP analysis is performed assuming a ACDM model and hence may not apply perfectly to these data. We will take an agnostic point of view on this and consider both the best-fit value of B\ from supernovae alone (B\ = 1.3527 ± 0.0497) and from the combination of supernovae and CMB/BAO (#i = 1.448 ± 0.0168).8 The results do not change qualitatively with either choice. The growth rate, /, at k = 0.1 h/Mpc is plotted in the first panel of Fig. 4.2, along with the parametrisation £lYm with best fits y = 0.46 for B\ = 1.35 and y = 0.48 for B\ = 1.45. This is in agreement with the results of Ref. [6], who additionally found that f(z) is fit much more closely by a redshift-dependent parametrisation, / ~ &m (l + y+j) • In the second panel we plot the best-fit value of y as a function of B\. All values of B\ consistent with background observations give a value of y that is far 8These differ slightly from the best-fit B\ = 1.38 ± 0.03 reported by Ref. [5], also based on the Union2.1 supernovae compilation. 78 4 Linear Structure Growth in Massive Bigravity 1.2 0.8 ■a o o E 0.6 CD 0.4 0.2 ^1 1.1 1.2 Growth of Structures SNe la CMB/BAO Combined 1.6 1.7 Bh Fig. 4.1 The likelihood for B\ in the B\ -only model from growth data {red), as well as background likelihoods for comparison. The fits for B\ effectively depend only on the background data; the combined likelihood {black) is not noticeably changed by the addition of the growth data from the GR value (including ACDM and minimally-coupled quintessence models) of y ~ 0.545. While present observations of LSS are unable to easily distinguish this model from ACDM (cf. Fig. 4.1), the Euclid satellite expects to measure y within 0.02 [1, 2] and should easily be able to rule out either the minimal massive bigravity model or GR. Note that there is a caveat, in that we have calculated y by fitting over a redshift range (0 < z < 5) which includes the unstable period of this model's history (z > 0.5) during which linear theory breaks down. As emphasised above, these predictions should only be compared to data during the stable period. Therefore if this model does describe reality, Euclid may measure a different growth rate at higher redshifts; a nonlinear analysis is required to answer this with certainty. We next look at the modified gravity parameters rj{z, k) and Q{z, k). In Figs. 4.3 and 4.4 they are plotted with respect to z, B\, and k, respectively, with the other two quantities fixed. Q deviates from the GR value Q = 1 by ~0.05, while r\ deviates from GR by up to ~0.15. From the first panel of Fig. 4.3 we notice that Q and rj lose their dependence on B\ momentarily around z ~ 2.5. This feature persists to other values of B\ as well. Additionally, we can see from the third panel that Q and rj only depend extremely weakly on k in the linear subhorizon regime. Future structure experiments like Euclid will be able to constrain Q and rj more tightly in a model-independent way because they are effectively scale-independent; in particular, 4.2 Structure Growth and Cosmological Observables 79 k = 0.1 h/Mpc f(z): B1 f(z): B1 Best-fit ay: B1 1 1.35 1.45 1.35 1.45 0.56 k = 0.1 h/Mpc Fig. 4.2 First panel The growth rate, / = dlnS/dlna, for the SNe best-fit parameter, B\ = 1.35 (in black), and for the SNe/BAO/CMB combined best-fit parameter, B\ = 1.45 (in red). The full growth rate (solid line) is plotted alongside the QY parametrisation (dotted line) with best fits y = 0.46 and 0.48 for B\ = 1.35 and 1.45, respectively. Second panel The best-fit value of y as a function of B\. For comparison, the GR prediction (y % 0.545) is plotted as a black horizontal line. The lines correspond to the best-fit values of B\ from different background data sets. This is a prediction of a clear deviation from GR 80 4 Linear Structure Growth in Massive Bigravity k = 0.1 h/Mpc 0.98 0.96 0.94 0.92 0.9 0.88 0.86 Q: Bh = 1.35 ....... Q: Bh = 1.45 - n: Bh = 1.35 ....... n: B1 = 1.45 - k = 0.1h/Mpc 0.98 0.96 0.94 0.92 0.9 0.88 0.86 1.1 1.2 1.3 Bh = 1.35 1.4 Bh = 1.45 1.5 Q: z = 0.5 Q: z= 1.5 n: z = 0.5 n: z= 1.5 1.6 1.7 Fig. 4.3 The modified gravity parameters Q (modification of Newton's constant) and r\ (anisotropic stress) in the Z?i-only model as functions of z and B\. They exhibit O(10_2)-C'(10_1) deviations from the GR prediction, which will be around the range of observability of a Euclid-like mission 4.2 Structure Growth and Cosmological Observables 81 B1 = 1.45 0.98 0.96 0.94 0.92 0.9 0.88 0.86 Q: z= 1.5 Q: z = 0.5 n: z= 1.5 n: z = 0.5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 k [h/Mpc] Fig. 4.4 The modified gravity parameters Q (modification of Newton's constant) and r\ (anisotropic stress) in the Z?i-only model as functions of scale. They depend only very weakly on scale; consequently, more stringent constraints can be placed on them in a model-independent way by future surveys because of scale independence they are expected to be able to measure rj within 10 % [11], which would bring this minimal model to the cusp of observability. Two-Parameter Models At the background level, four models with two nonzero Bt parameters provide good fits to the data: B1B2, B\B3, B\B4, and #2#3 [3]. Even though these models all possess a two-dimensional parameter space, only an effectively one-dimensional subspace matches the background data (cf. Figs. 4.7 and 4.8 of Ref. [3] and 4.4 of Ref. [5]). We will restrict ourselves to those subspaces by fixing one Bt parameter in terms of another, usually B\. We do this by identifying the effective present-day dark energy density, Qef,9 from the Friedmann equation (2.51): £2f = Biyo + B2y2Q + ^B3y30, (4.20) yThis does not need to coincide with the value of £2 a derived in the context of ACDM models. For the B\-only model and hence all the two-parameter finite-branch models, they happen to be similar in value, although this was not a priori guaranteed, while in the infinite-branch B\B4 model, the best-fit value to the background data is Qef = 0.84+q ^ [5]. 82 4 Linear Structure Growth in Massive Bigravity and plugging that into the quartic equation for y (2.58), evaluated at the present day (y = y0) and using + Qef = 1. This procedure fixes one Bt parameter in terms of the other and Qef. The value of Qef can be determined by fitting to the data, as was done in Ref. [3]. However, for finite-branch solutions of the B\Bt models we can also simply take the limit in which only B\ is nonzero, recovering the single-parameter model discussed above; by then using the SNe/CMB/BAO combined best-fit value B1 = 1.448 we find Qf = 0.699. A detailed study of the conditions for a viable background was undertaken in Ref. [5]. There are two results that are particularly relevant for the present study and bear mentioning. First, a viable model requires B\ > 0. Second, the two-parameter models are all (with one exception, which we discuss below) finite-branch models, in which, as discussed in Sect. 2.1.3, y evolves from 0 at z = oo to a finite value yc at late times, which can be determined from Eq. (2.59) by setting p = 0 at y = yc. Consequently, the present-day value of _yo, which is generally simple to calculate, must always be smaller than yc (or above yc in a model with an infinite branch). In the B\ #3 and B\B4 models this will rule out certain regions of the parameter space a priori. There is one two-parameter model which was shown in Ref. [3] to fit the background data well but was ruled out on theoretical grounds in Ref. [5]: the B2B3 model. The theoretical issue is that y becomes negative in finite time going towards the past. This itself may not render the model observationally unviable, as long as any issues occur outside of the redshift range of observations. However, we have solved Eq. (2.61) numerically for y(z), and found that y genetically goes to — oc at finite z, which means that at higher redshifts there is not a sensible background cosmology at all. This problem can be avoided by introducing new physics at those higher redshifts to modify the evolution of y, or by increasing B2 enough that the pole in y occurs at an unobservably high redshift.10 However, these are nonminimal solutions, and so we do not study the #2#3 model. Recall from Chap. 3 that all of the two-parameter models except for the infinite-branch B1B4 model suffer from an early-time instability. Consequently, caution should be used when applying the results for any of the other models in this section to real-world data. We emphasise again that our quasistatic approximation is only valid at low redshifts, and that moreover the growth rate should be recalculated using whatever initial conditions the earlier period ends with. (The predictions for Q and r\ do not depend on solving any differential equation and therefore apply without change.) Modulo this caveat, we present the quasistatic results for the unstable models as proofs of concept, as examples of how to apply our methods. The infinite-branch B\ B4 predictions, presented at the end of this section, can be straightforwardly applied to data. We evaluate all quantities at k = 0.1 /z/Mpc. The modified gravity parameters in all of the two-parameter models we study depend extremely weakly on k, as in the #i-only model. For Z?2 = (5, 50, 500), and #3 chosen to give an effective £2"^ % 0.7 today, the pole occurs at z (1-99, 8.19,27.95). 4.2 Structure Growth and Cosmological Observables 83 As we have already mentioned, in these models the two Bt parameters are highly degenerate at the background level. One of the main goals of this section is to see whether observations of LSS have the potential to break this degeneracy. B1B2: The B1B2 models which fit the background data [3] live in the parameter subspace -B\ + 9Qf - JBA1+9B21Q,f B2 = --rf-, (4.21) 9Qf with Qef ~ 0.7. This line has a slight thickness because we must rely on observations to fit Qef. We can subsequently determine _yo from Eq. (4.20). This model possesses an instability when B2 < 0.11 This is not entirely unexpected: the B2 term is the coefficient of the quadratic interaction, and so a negative B2 might lead to a tachyonic instability. However, the instability of the B\B2 model is somewhat unusual: in the subhorizon limit, Q and r\ develop poles, but they only diverge during a brief period around a fixed redshift, as shown in the first panel of Fig. 4.5, regardless of wavenumber or initial conditions. We can find these poles using the expressions in Appendix B. The exact solutions are unwieldy and not enlightening, but there are three notable features. First, as mentioned, the instability only occurs for B2 < 0. (When B2 > 0, these poles occur when y is negative, which is not physical.) Second, the instability develops at high redshifts, z > 2. The redshift of the latest pole (which can be solved for by taking the limit k/H -> 00) is plotted as a function of B\ in the second panel of Fig. 4.5. As a result, measurements of Q and r\ at z < 2 would generally not see divergent values. However, such measurements would see the main instabilities at much lower redshifts. Finally, the most recent pole occurs at y = 0 for B2 = 0 (B\ = yJ^Q^f ~ 1.45), and approaches y = yo/2 as B2 -> —00 (Bi -> 00). This particular instability is avoided if we restrict ourselves to the range 0 < B\ < 1.448, for which B2 > 0. Some typical results for this region of parameter space are plotted in Figs. 4.6 and 4.7. The first panel of Fig. 4.6 plots f(z) and the best-fit Qm parametrisation for selected values of B\ [with B2 given by Eq. (4.21)], while the second panel shows the best-fit value of y as a function of B\. For smaller values of B\ this parametrisation fits / well, more so than in the 5i-only model discussed in Sect. 4.2.2 (which is the B\ = 1.448 limit of this model). We find that y is always well below the GR value of y ~ 0.545, especially at low B\. In Fig. 4.7 we plot Q and rj, both in terms of B\ at fixed z and in terms of z at fixed B\. In comparison to the 5i-only model (at the far-right edge in the first panel), lowering B\ tends to make these parameters more GR-like, except for r\ evaluated at late times (z ~ 0.5), which dips as low as rj ~ 0.6. Because these quantities are all £-independent in the linear, subhorizon regime, future LSS experiments like A similar singular evolution of linear perturbations in a smooth background has been observed in the cosmology of Gauss-Bonnet gravity [23, 24]. This instability is different from the early-time instabilities discussed in Chap. 3, as those do not arise in the quasistatic limit which we are now taking. 84 4 Linear Structure Growth in Massive Bigravity Fig. 4.5 First panel Q and r\ for a few parameter values in the instability range of the B\B2 model. Generally there are two poles, each of short duration. Note that the redshift at which the pole occurs depends only on B\ and B2 and not on the initial conditions for the perturbations. Second panel The redshift of the most recent pole in Q (the pole in r] occurs nearly simultaneously) in terms of B\ > 1.448. This parameter range corresponds to B2 < 0, cf. Eq. (4.21). The minimum is at (Bu B2,z) = (3.11, -2.51,2.24) 4.2 Structure Growth and Cosmological Observables 85 k = 0.1 h/Mpc f(z): Bh = 0.5 f(z): Bh = 1.0 f(z): B1 = 1.4 Best-fit Cl1: Bh = 0.5 Best-fit Cl7: Bh = 1.0 Best-fit Cl7: B1 = 1.4 k = 0.1 h/Mpc Fig. 4.6 Growth-rate results for the 5i52 model, with Qff = 0.699 and B\ < 1.448. The significant deviation from GR in y should be observable by a Euclid-like experiment. Moreover, it has the potential to break the degeneracy between B\ and B2 when fitting to background observations 86 4 Linear Structure Growth in Massive Bigravity Euclid would be able to measure r\ at these redshifts to within about 10% [11] and thus effectively distinguish between ACDM and significant portions of the parameter space of the B\B2 model, testing the theory and breaking the background-level degeneracy. B1B3: The B1B3 models which are consistent with the background data [3] lie along a one-dimensional parameter space (up to a slight thickness) given by -32fl3 + 81fli£2eAff ± J(85? - 21Qf)2 (1652 + 21Qf) #3 =-—-w-,-—-(4-22) 243 (Qf)2 with ~ 0.7. There is a subtlety here: the physical branch is constructed in a piecewise fashion, taking the + root for B\ < (3/2)3^2^Q^ = 1.536 and the — root otherwise [5]. We solve for y(z) using the initial condition,12 derived from (2.58), no3 yo =--• (4.23) Z#3 As discussed at the beginning of this section, _yo should not be larger than the value of y in the far future, yc. Demanding this, we find a maximum allowed value for #3,13 B3 < 243 ("325i + 8151 + V(165i +27) (85i2"27)2) • (4-24) For Qef = 0.699 this implies we need to restrict ourselves to B\ > 1.055. This sort of bound is to be expected: we know that the 53-only model is a poor fit to the data [3], so we cannot continue to get viable cosmologies the entire way through the B\ -> 0 limit of the B1B3 model. We plot the results for the B1B3 model in Figs. 4.8 and 4.9. These display the tendency, which we will also see in the B\B4 model, that large | B[ | values lead to modified gravity parameters that are closer to GR. For example, y can be as low as y ~ 0.45 for the lowest allowed value of B\, but by B\ ~ 3 it is practically indistinguishable from the GR value, assuming a Euclid-like precision of ~0.02 on y [1]. Again we note that this value of y has been obtained assuming 0 < z < 5, which is not a valid range for observations because of the early-time instability. For lower values of B\, current growth data (see, e.g., Ref. [17]) are not sufficient to significantly constrain the parameter space, but these non-GR values of y and rj should be well within Euclid's window. There is also a positive root, but this is not physical. When #3 < 0, that root yields yo < 0. When B3 > 0, which is only the case for a small range of parameters, then the positive root of yo is greater than the far-future value yc and hence is also not physical. 13Note, per Eq. (4.22), that this is equivalent to simply imposing Q.^ < 1, which must be true since we have chosen a spatially-flat universe a priori. 4.2 Structure Growth and Cosmological Observables 87 k = 0.1h/Mpc 0.9 0.8 0.7 0.6 0.5 Q: z = 0.5 Q: z= 1.5 n:z = 0.5 n: z= 1.5 0.2 0.4 0.6 0.8 1.2 1.4 k = 0.1 h/Mpc 0.95 0.85 0.75 Q: B1 = 0.5 Q: Bh = 1.0 Q: B1 = 1.4 n: Bh = 0.5 n: Bh = 1.0 n:B1 = 1.4 0.65 Fig. 4.7 The modified gravity parameters Q and r) for the B\ B2 model, with £2"^ = 0.699 and B\ < 1.448. As with the growth rate in Fig. 4.6, we notice significant deviations from GR. The anisotropic stress, r)(z ~ 0.5), is a particularly good target for observations 0.4 1-'-'-'-'-'-1 0 2 4 6 8 10 Bi Fig. 4.8 Growth-rate results for the BiB3 model, with ttf = 0.699 and Bx > 1.055. While B\ and Z?3 are degenerate in the background (along a /me given by Eq. 4.22), perturbations clearly can break this degeneracy, with significant deviations at low values of B\ (small IB3I), and GR-like behaviour at large values of B\ (large, negative B3) 4.2 Structure Growth and Cosmological Observables 89 k = 0.1 h/Mpc 0.82 0.9 0.85 0.8 0.75 0 2 4 6 Bi k = 0.1 h/Mpc 8 1 i i i i / / Q: Bh = 1.1 - Q: B1 = 1.45 - Q: B1 =2.5 - Q: B1 = 5 -ti: B1 = 1.1 ....... ti.: = 1.45 ....... ti: B1 =2.5 ....... n: B1 =5 0 1 2 3 z 4 E Fig. 4.9 The modified gravity parameters Q and /j for the B\B3 model, with fi^f = 0.699 and B\ > 1.055. As with the growth rate, we notice deviations from GR for small B\ (small | B31), with parameters approaching their GR values for large Bi (large, negative B3) 90 4 Linear Structure Growth in Massive Bigravity B1B4: This model comprises the lowest-order B\ term in conjunction with a cosmo-logical constant for the / metric, B4.14 Note that B4 does not contribute directly to the Friedmann equation (2.51), but only affects the dynamics through its effect on the evolution of y. The B\B4 model has two viable solutions for y(z): a finite branch with 0 < y < yc, and an infinite branch with yc < y < 00. The infinite-branch model is the only two-parameter bimetric model which is linearly stable at all times, as shown in Chap. 3. Therefore this should be considered the most viable bimetric massive gravity theory to date. In this section we will elucidate its predictions for subhorizon structure formation. As discussed in the beginning of this section, yc is the value of y in the asymptotic future and can be calculated by setting p = 0 in Eq. (2.59). We will consider the two branches separately. For a given Qef, B4 is related to B\ by >eff p2 3QfBf - Bl B4 = — + 1 , (4.25) (flf)3 while yo is given by ßeff yo = ~^- (4.26) Background viability conditions impose B\ > 0 for both branches and B4 > 0 on the infinite branch [5]. The late-time value of y, yc, is determined by B4y3c - 3Biy2c + fli = 0, (4.27) from which we can determine that real, positive solutions for yc only exist if B4 < 2B\. Combined with Eq. (4.25) and the requirements that B\, Q^f > 0, we find two possible regions for B\, as can be seen from the example plotted in the first panel of Fig. 4.10. We can identify each of these two regions with the two solution branches by comparing the solutions of Eq.(4.27) for yc to _yo = Qef /B\. Restricting to positive, real roots of yc one can see (graphically, for example) that the first region, with the smaller values of B\, has _yo > yc for all roots of yc, and hence can only comprise infinite-branch solutions, while _yo < yc in the second region, as is plotted in the second panel of Fig. 4.10. This identification is supported by observational data [5] and also makes intuitive sense. Consider the limit B4 = 0. In the second region this has B\ > 0 and corresponds to the 5i-only model, a finite-branch model, which we discussed in Sect. 4.2.2. In the first region the point B4 = 0 coincides with B\ = 0, which is simply a CDM model with no modification to gravity, in agreement with the fact that there should not be an infinite-branch 5i-only model. In the singly-coupled version of massive bigravity we are studying, matter loops only contribute to the g-metric cosmological constant, Bq. 4.2 Structure Growth and Cosmological Observables 91 0.0 0.5 1.0 1.5 2.0 ft Fig. 4.10 Allowed regions in the B\-B/[ parameter space. First panel Regions in parameter space with Z?4 < B\, which is required for the viability of the background. The orange line fixes B4 as a function of B\ for Qs^ = 0.84, per Eq. (4.25). The blue line is B4 = 2B\. Second panel Plotted are yo (orange line) and the three roots of yc, again with £2Aff = 0.84. Of the two regions with B4 < B\, we can see that the region with smaller B\ has yo > yc and therefore corresponds to the infinite branch, while the region with larger B\ has yo < yc and therefore possesses finite-branch solutions These considerations place constraints on the allowed ranges of ft, as in the ftft model, which depend on the best-fit value of £2^. The ft-only model (ft = 0, ft > 0) is on the finite branch, so that on that branch we can use = 0.699 as we did in the other models. This implies ft > 1.244 for the finite branch. On the infinite branch, SNe observations are best fit by Qef = 0.84 [5]; consequently we 92 4 Linear Structure Growth in Massive Bigravity restrict ourselves to B\ < 0.529 for the infinite branch. Note that the infinite-branch model therefore predicts an unusually low matter density, £2m,o ~ 0.18. We plot the results for the finite branch in Figs. 4.11 and 4.12. Qualitatively, this model predicts subhorizon behaviour quite similar to that of the B\B^ model, discussed above and plotted in Figs. 4.8 and 4.9. Now let us move on to the stable infinite-branch model, the modified gravity parameters of which are plotted in Figs. 4.13 and 4.14. This is the only model we study which does not possess a limit to the minimal 5i-only model, and it predicts significant deviations from ACDM. In this model, rj deviates from 1 by nearly a factor of 2 at all observable epochs and for all allowed values of B\, providing a clear observable signal of modified gravity. This is a significant feature of this model; it has no free parameters which can be tuned to make its predictions arbitrarily close to ACDM and therefore is unambiguously testable. We can understand this behaviour analytically as follows. In the asymptotic past, we can take the limits of our expressions for rj and Q to find Unusually, structure growth does not reduce to that of ACDM even though at early times the graviton mass is very small compared to the Hubble scale. In the future one finds r\ -> 1 if A: is kept finite, but this is somewhat unphysical: for any finite k there will be an epoch of horizon exit in the future after which the subhorizon QS approximation breaks down. We can see both the asymptotic past and asymptotic future behaviours in the second panel of Fig. 4.14, although the late-time approach of rj to unity is not entirely visible. The growth rate and index, f(z) and y, also deviate strongly from the ACDM predictions. Using the £lYm parametrisation, we find that y is even lower than the range ~0.45-0.5 which we found in the 5i-only model and in the other two-parameter models. However, the £lYm parametrisation is an especially bad fit to f(z) in this case; we fit y to / (z) in the redshift range 0 < z < 5 (as in the rest of this chapter) and 0 < z < 1 (which is the redshift range of present observations [17]) in Fig. 4.13, obtaining significantly different results and still never agreeing well with the data. As shown in Fig. 4.15, the confidence region obtained from the growth data is in agreement with type la supernovae (SNe) data (see Ref. [5] for the likelihood from the SCP Union 2.1 Compilation of SNe la data [12]). The growth data alone provide ft = and ft = with a = 9.72 (with 9 degrees of freedom) for the best-fit value and is in agreement with the SNe la likelihood. The likelihood from growth data is, however, a much weaker constraint than the likelihood from background observations. Thus, the combination of both likelihoods, providing 0! = 0.48+J?6 and ft = 0-94-S:5i>is similar to the SNe la result alone. In Fig. 4.16 we compare the growth rate directly to the observational data compiled in Ref. [17], using the best-fit values determined above. The available growth data are unable to distinguish between the infinite-branch model and ACDM. We also find that an alternative parametrisation, 1 lim rj = - and —oo 2 (4.28) n- n- 4.2 Structure Growth and Cosmological Observables 93 1.1 k = 0.1 h/Mpc 0.5 f(z): B-| f(z): B1 = = 1.1 1.45 f(z): B1 = 2.5 f(z): B 1 - J Best-fit ar. B1 = 1.1 Best-fit nY: B1 = 1.45 Best-fit Q1: B1 = 2.5 Best-fit Q1: B-t = 5 _i_ 0.65 k = 0.1 h/Mpc CO CD 0.55 0.45 Fig. 4.11 Growth-rate results for the B\Ba, model on the finite branch, with £2^ = 0.699 and B\ > 1.244. The behaviour is quite similar to that of the B\B3 model, plotted in Figs. 4.8 and 4.9 94 4 Linear Structure Growth in Massive Bigravity k = 0.1 h/Mpc 1 i-1-1- Fig. 4.12 The modified gravity parameters Q and r\ for the B\B\ model on the finite branch. Results are displayed for Qf = 0.699 and Bi > 1.244 4.2 Structure Growth and Cosmological Observables 95 k = 0.1 h/Mpc 0 1 2 3 4 5 z k = 0.1 h/Mpc U.DO 0.6 1 i -1- i- Fit f(z) to Q1 over z=[0,5] — Fitf(z) to Qyoverz=[0,1] — i 0.55 GR 0.5 - H—' H— 1 H—i 0.45 - Bes 0.4 0.35 n ^ - U.O 0.25 - 0.2 0.1 0.2 0.3 0.4 0.5 Bi Fig. 4.13 Growth-rate results for the B1B4 model on the infinite branch, with Q 2ft n ■—- o "n IBB (ft =0.48, #1=0.94) BB fit model ACDM (nm0=0.27) ACDM(nm0=0.18) Fig. 4.16 Growth history for the best-fit infinite-branch ft ft model (solid blue) with ft = 0.48 and ft = 0.94 compared to the result obtained from the best fit (4.29) (solid orange) with yo = 0.47 and a = 0.21 and the ACDM predictions for Qm,o = 0.27 (dotted red) and Qm,o = 0.18 (dotted-dashed green). The latter value for the matter density is the same as is predicted by the infinite-branch model. Note that a vertical shift of each single curve is possible due to the marginalisation over 0$. Here we choose 0$ for each curve individually such that it fits the data best. The growth histories are compared to observed data compiled by Ref. [17] f(z)~Q%[l+aj^-y (4.29) is able to provide a much better fit to / (z) than the usual Qm. The best-fit values for this parametrisation are yo = 0-47 and a = 0.21. 98 4 Linear Structure Growth in Massive Bigravity 4.3 Summary of Results In this chapter we have examined the evolution of cosmological perturbations on subhorizon scales in massive bigravity, describing linear structure formation during the matter-dominated era. We solved the linearised Einstein and conservation equations in the quasistatic, subhorizon limit in Newtonian gauge for gliv. In this limit, we found that the perturbations are described by a system of six algebraic equations for five variables. We obtained a consistent solution to this system relating each of the metric perturbations appearing in the subhorizon Einstein equations to the matter density contrast. This allowed us to derive a modified evolution equation for the density contrast, which differs from its GR counterpart by a varying effective Newton's constant. We also obtained algebraic expressions for the anisotropic stress, rj(z,k), and the parameter measuring modifications to Newton's constant in the Poisson equation, Q(z, k), purely in terms of background quantities. We then solved for the background numerically to obtain these parameters, and finally integrated the structure growth equation to derive the growth rate, f(z,k), and its best-fit parametrisation, / ~ Q.vm. We have studied every subset of the theory which is viable at the background level and contains one or two free parameters, excluding the g-metric cosmological constant as we are interested in self-accelerating theories. Among the single-parameter models, only the case with the lowest-order interaction term, fa ^ 0, is in agreement with the background data. As emphasised in Refs. [3-5], this "minimal" bigravity model is especially appealing because it possesses late-time acceleration and fits the background data well with the same number of free parameters as ACDM. We have found that it predicts modified gravity parameters that differ significantly from GR: Y ~ 0.46-0.48 (in agreement with Ref. [6]), Q ~ 0.94-0.95, and r\ ~ 0.88-0.90. For reference, the ACDM predictions are y ~ 0.545 and Q = r\ = 1. Future large-scale structure experiments such as Euclid [1,2] will easily be able to distinguish this simple model from GR, if we can trust its predictions. However, we have shown in Chap. 3 that this model suffers from an early-time instability. If this can be overcome, or if observations are restricted to late times (z < 0.5), our results demonstrate that by going to the level of linear perturbations, this theory can be probed in the near future by multiple observables which deviate significantly from general relativity. We additionally examined the four two-parameter models which are viable in the background, all of which keep fa > 0 while turning on a second, higher-order interaction term. Two of these models, in which either the cubic interaction, fa, or the /-metric cosmological constant, fa, is nonzero (the latter specifically in the "finite branch," which reduces to the minimal model in the fa -> 0 limit), have similar behaviour to each other. They predict GR-like values for all three modified gravity parameters in the limit where m2fa/HQ is large, becoming indistinguishable from ACDM (given a Euclid-like experiment) for m2fa/HQ > 3. These reduce to the predictions of the minimal model in the limit m2fa/HQ ~ 1.45 (the best-fit value for fa in the minimal model). For lower values of fa (corresponding to positive fa or fa) these models predict even more dramatic deviations from GR: y can dip to 4.3 Summary of Results 99 0.45, and rj at recent times can be as low as ~0.75. Euclid is expected to measure these parameters to within about 0.02 and 0.1, respectively [1, 2, 11], and therefore has the potential to break the degeneracies between ft and ft or ft and ft which is present at the level of background observations. The ft ft model has an instability when ft, the coefficient of the quadratic term, is negative. This instability does not necessarily rule out the theory, but it might signal the breakdown of linear perturbation theory, in which case nonlinear studies are required in order to understand the formation of structure. This is different from the early-time instability discussed in Chap. 3 which does not show up in the subhorizon, quasistatic regime. It is possible that these perturbations at some point become GR-like due to the Vainshtein mechanism. Moreover, because the instability occurs at a characteristic redshift (which depends on the ft parameters), there may be an observable excess of cosmic structure around that redshift. The parameter range of the ft ft model over which this instability is absent, 0 < ft < 1A5Hq/m2 (corresponding to Hq/hi2 > ft > 0), is quite small; near the large ft end the predictions recover those of the minimal ft-only model, while at the small ft end the perturbations can differ quite significantly from GR, with y as low as ~0.35 and rj as low as 0.6. However, the exact ft = 0 limit of this theory is already ruled out by background observations [3], so one should take care when comparing the model to observations in the very low ft region of this parameter space. Finally, we examined the "infinite branch" of the ft ft model, which is the only bimetric model15 that avoids the early-time instabilities uncovered in Chap. 3. This is called an infinite-branch model because the ratio of the / metric scale factor to the g metric scale factor, y, starts at infinity and monotonically decreases to a finite value. In the rest of the models we study, y starts at zero and then increases; consequently, in the ft -> 0 limit this theory reduces to pure CDM, rather than to the ft-only model. The predictions of the infinite-branch ft ft theory deviate strongly from GR. The model predicts a growth rate f(z) which is not well-parametrised by a Qm fit, but has best-fit values of y on the low side (0.3-0.4, depending on the fitting range). The anisotropic stress rj is almost always below 0.7 and can even be as low as 0.5, a factor of two away from the GR prediction. Across its entire parameter space, this model has the most significantly non-GR values of any we study. Its predictions should be well within Euclid's window. 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Koivisto, D.E Mota, Cosmology and astrophysical constraints of gauss-bonnet dark energy. Phys. Lett. B644, 104-108 (2007). arXiv:0606078 Chapter 5 The Geometry of Doubly-Coupled Bigravity O, that way madness lies; let me shun that; No more of that. Lear, King Lear, 3.4 The existence of a consistent bimetric theory raises an intriguing question: which is the physical metric? In Chaps. 3 and 4 we chose to couple only one of the two metric, gllv, directly to matter, while the other dynamical metric, fllv, only interacts with matter fields indirectly through its interactions with g^. It is specifically in this singly-coupled context that the absence of the Boulware-Deser ghost was originally proven [1-3], and extending that analysis to other matter couplings is decidedly nontrivial [4-7]. It is therefore natural to interpret g^ in this case as the usual "metric" of spacetime, while fllv is an extra spin-2 field that is required in order to give mass to the graviton. Since the fields gllv and fllv have metric properties, we have called both gllv and fllv "metrics," even though strictly speaking, the singly-coupled theory could more accurately be called a theory of "gravity coupled to matter and a symmetric 2-tensor" [8]. In this chapter we aim to explore the consequences of coupling matter to both metrics in massive bigravity. As discussed in Sect. 2.1.2, the bimetric action (2.30) in vacuum places both metrics on equal footing: it is invariant under the interchanges (2.38) g/xv Uv, Mg Mf, ft -> ft_„. (5.1) This metric-interchange duality is broken by the addition of matter in the singly-coupled version of bigravity. The structure of the vacuum theory might hint that any fundamental theory which gives rise to massive bigravity does not discriminate between the two metrics. Consequently it is important to explore doubly-coupled bigravity, in which matter couples to both metrics in a way that maintains the interchange duality (5.1). © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7_5 103 104 5 The Geometry of Doubly-Coupled Bigravity We are specifically interested in the possibility of double-coupling schemes in which there is no "effective metric" whose geodesies describe the motion of matter. In these cases, we must introduce new tools for understanding the physical geometry of spacetime. We will focus on one of the most straightforward possibilities for double coupling, in which matter is minimally coupled to each metric. This theory was introduced and its cosmological solutions were studied in Ref. [9]. Subsequent work showed this type of coupling revives the Boulware-Deser ghost at arbitrarily low energy scales and therefore cannot be a fundamental theory of bigravity [4, 5]. Indeed, there is only one known double coupling which avoids the ghost at low energies, and even in this theory it re-emerges at or above the strong-coupling scale [5, 10].1 This coupling is the phenomenologically interesting one and will be investigated in depth in Chaps. 6 and 7. These theories couple matter to a single effective metric built out of gllv and fllv and so avoid the problem of determining the physical spacetime. It is, however, not yet clear whether such theories are immune from other types of pathologies, and it may well be the case that the unknown, healthy doubly-coupled theory of bigravity will not admit an effective-metric formulation at all. It is our goal in this chapter to demonstrate the difficulties such theories would have with regards to defining observables by studying arguably the simplest example of doubly-coupled bigravity without an effective-metric description. In this context, the traditional notion of a "physical metric" may have to be discarded, leaving us faced with entirely new conceptual challenges in interpreting even the observables of the theory. This chapter is organised as follows. In Sect. 5.1 we argue that there is no effective metric to which matter minimally couples, and that such a metric does not even exist for most individual fields. In Sect. 5.2 we explore light propagation in this theory in the geometric optics limit, and discuss the problem of relating cosmological observables to the underlying theory when we can no longer describe photon trajectories as null geodesies in a metric. In Sect. 5.3 we examine the dynamics of point particles, finding that they effectively live in a Finsler spacetime, a geometry which depends nontrivially on the coordinate intervals. Finally, we conclude in Sect. 5.4. 5.1 The Lack of a Physical Metric We consider a doubly-coupled bimetric theory in which the action (2.30) is extended by the addition of a minimal-coupling term between matter fields, and /MV, R = J dAx^R{g) - -J- J d4x^fR(f)+m2M2g J d4xV^5>"e«® + agJ d4x^J?m (g, <&,-) + df J d4x^fj?m (/, <&,-). (5.2) 1 Using the same principles, further candidate double couplings have been constructed in Ref. [11], but it is not yet known which, if any, of these are ghost-free. 5.1 The Lack of a Physical Metric 105 This extends the symmetry (5.1) to the entire action, as long as we also exchange ag and oif, giiv fpv, Mg Mf, fin -> p4_n, ag af. (5.3) The presence of the interaction term is crucial; if one were to couple two pure, noninteracting GR sectors to the same matter, the Bianchi identities would constrain that matter to be entirely nondynamical [9]. Note that ag and a f are not both necessary to fully specify the theory; only their ratio is physical, as can be seen by rescaling the action by a"1. For the purposes of this chapter, we will find it useful to leave both in so as to keep the symmetry between the two metrics explicit. An immediate concern is the violation of the equivalence principle. However, because the Vainshtein mechanism screens massive-gravity effects [12], it is not obvious how stringent the constraints from tests of GR in the solar system would be: the modifications might be hidden from local experiments while showing up at cosmological scales. The cosmology of this doubly-coupled theory has been studied and shown to produce viable late-time accelerating background expansion without an explicit cosmological constant term [9], and with a phenomenology which can be interestingly different from that of the singly-coupled theory [13]. We emphasise again that this model itself possesses the Boulware-Deser ghost and hence we cannot trust its cosmological solutions, but a ghost-free doubly-coupled theory may well have similar properties. Indeed, the cosmological phenomenology of this theory is quite similar to that of the healthier doubly-coupled theory introduced in Ref. [5], as we will show in Chap. 6. We can readily confirm that no physical Riemannian metric exists in the sense that all matter species would minimally couple to it and thus follow its geodesies. Indeed, for some matter fields such a metric does not exist at all. Consider a massive scalar field. Its action is given by (5.4) Let us assume that

- V{4>). (5.6) The kinetic and potential terms, respectively, yield the conditions 106 5 The Geometry of Doubly-Coupled Bigravity V=SSMV + = o, (5.ii) yields the correct massless Klein-Gordon equation [9] (ag^Og + af^fUf) 0 = 0. (5.12) This is straightforward to show using the identity g^T? = —7=3/x(Ay/—gg^p). This is the only example we have found of a field for which we can construct an effective metric. Consider the electromagnetic field AIjL. This is of paramount importance for cosmology, since we make observations by tracking photons. Its action is SA = l-agJ d'x^gg'^g^F^ - l-af J dAx^f fp FllvFafi, (5.13) where FI1V = dljLAv — 3V AIjL is the usual field-strength tensor and does not depend on a metric. If AIjL is minimally coupled to an effective metric, hjjiV, then we can write Eq. (5.13) as sa = -\! d4xV^hh^ahvfiF^Fafi. (5.14) 5.1 The Lack of a Physical Metric 107 This implies that h/lv obeys agV^ggllvgaP + OLfyTlf^f^ = 4^hh^haK (5.15) However, this equation overconstrains hllv. Consider the 00-00, 00-n, and ii-ii components, h(hm)\ (5.16) hh°°hu, (5.17) h(hHf. (5.18) where repeated indices are not summed over. Solving for h00 and/z" using Eqs. (5.16) and (5.17), Eq. (5.18) becomes a constraint on g/lv and f/lv, g00r = /V- (5-i9) Note that we have chosen an arbitrary spatial index, i, in an arbitrary coordinate system; Eq. (5.19) therefore applies to any diagonal spatial component in any coordinates. This equation is not satisfied by general choices of gllv and fllv. An FLRW universe is a simple example where this condition fails to be satisfied. Thus, except in special circumstances, there is no physical metric for the electromagnetic field. Similar arguments should hold for other fields. The massless scalar appears to be a special case because it lacks a potential term to constrain and because it has no indices, so its kinetic term only includes one appearance of the metric. *g^{gHÝ+*fyf=f(fllÝ = ^ 5.2 Light Propagation and the Problem of Observables We have shown that the electromagnetic field is not minimally coupled to any effective metric. This case is of particular physical relevance because we make observations by tracking photons. For cosmological observations, especially, it is crucial to know how light propagates. Even in this simplified case, photons turn out not to travel on null geodesies of any metric. To see this, we will consider the plane-wave approximation for the Maxwell field, Alí = Re[(alí+€blí)ei+/e], (5.20) and take the geometric optics limit in which the wavelength is tiny compared to the characteristic curvature scale, e = X/R <$C 1. This provides a rigorous approach to describing light propagation in curved spacetime. In this ansatz, aM is the leading-order polarisation vector and yjr is the phase. Herein we will drop the real evaluation for compactness. Because this is a "pregeometric" approach, we can utilise it to 108 5 The Geometry of Doubly-Coupled Bigravity tackle the propagation of light rays in bigravity. The stress-energy tensor for the electromagnetic field, which can be derived from the action (5.13), is T\ = ^ (f^F^ - l-8\fafifa^ . (5.21) We must be careful about which quantities depend on a metric and which don't. The field tensor is defined as usual in terms of the electromagnetic 4-potential AM, which is itself defined with lower indices just in terms of the fields, so AM is the same in both metrics. Similarly, because of the symmetries of the Christoffel symbols, can be defined equivalently in terms of covariant or partial derivatives; because of the latter, we see that FMV with lower indices is also independent of the metric. Stress-energy conservation is given by [9] «,V=£vjrj\ + a/y^7v/{r;v = o, (5.22) where and V/f are the covariant derivatives defined with respect to g/lv and f/lv, respectively. To apply this to the electromagnetic field, we first need to know (in terms of gllv, for concreteness) the divergence of the stress-energy tensor. The identity V[a fi1v} = 0 holds independently of the theory of gravity and in either metric, because it relies only on the usual expression for the commutation of covariant derivatives and the symmetries of the Riemann tensor. Using this, we find VMr\ = FvaVMF^. (5.23) Plugging this into Eq. (5.22) and using the fact that it should apply for arbitrary fva (because, as mentioned above, this is independent of the metrics), we find a straightforward generalisation of the Maxwell equations, «*V=iVjF^ + afyTf^Ff = 0, (5.24) where g and / subscripts on FMV tell us which metric is being used to raise indices. We have yet to use our gauge freedom. We will choose to work in a Lorenz gauge, where V^yP = 0. Since this cannot be simultaneously satisfied in both metrics, we will choose to apply this gauge with respect to gllv. As we will see shortly, this choice does not make a difference at leading order in the geometric optics approximation. After specialising to this gauge and commuting some covariant derivatives, the Maxwell equation reduces to «gV=g {g'^A, - R^A,) + afJ~f (rnfA, - V? Vjf AM - R'fA,) = 0. (5.25) Plugging the ansatz (5.20) into Eq. (5.25) and keeping only the leading-order terms in e—i.e., those obtained by acting the covariant derivatives on the exponential term twice—we obtain 5.2 Light Propagation and the Problem of Observables 109 aM (ag^g-g^gaß +afJ~frvr? - otfyTf f1™fvß) kakß = 0, (5.26) where kfJi = d^ifr is the wavevector. Note that in the singly-coupled limit, this gives us the standard result that ka is null in gMV. As discussed above, we cannot use this to define a metric, h^, in which ka is lightlike. This creates problems when applying the standard methods of relativistic cosmology to compare observable quantities to the underlying theory. In a bimetric cosmology, as we have seen, there are two scale factors and two Hubble rates. When matter couples to both metrics, neither of these quantities plays the role that they play in general relativity. Had we been able to identify an effective metric from Eq. (5.26), then the scale factor of that metric would have been the geometrical quantity that entered the expression for the redshift, and its Hubble rate (computed using the effective metric's lapse) would be the "physical" Hubble rate. The next step, relating the theoretical redshift to the shift in wavelengths observed by a telescope, would involve understanding the proper time of a massive observer, which we tackle in the next section. However, Eq. (5.26) defies the usual, simple categorisation. While we can, in principle, use this to compute light propagation, this approach does not shed light on the identification of a physical scale factor to compare to observations. 5.3 Point Particles and Non-Riemannian Geometry The situation we have described in bimetric theories is radically different from the extensively-studied nonminimally coupled theories where the behaviour of matter can be described in terms of a single metric. In the context of scalar-tensor theories, for example, it is well known that there are conformally-equivalent descriptions of the theory where either the gravity sector is general relativity whilst matter has a nonminimal coupling (the Einstein frame), or matter is minimally coupled whilst the gravity sector is modified (the Jordan frame). All physical predictions are completely independent of the frame in which they are calculated after properly taking into account the rescaling of units in the Einstein frame [14]. One can generalise to nonuniversal couplings, allowing different Jordan frame metrics for different matter species, or to couplings to multiple fields. These bring about new technical but not fundamental difficulties. However, the doubly-coupled bimetric theories we are studying do not admit a Jordan frame at all for most types of matter. They possess mathematically two metrics but physically none, and to understand them we need to step beyond the confines of metric geometry. For concreteness, let us look at the simplest possible type of matter: a point particle of mass m. Its action is given by (5.27) 110 5 The Geometry of Doubly-Coupled Bigravity where overdots denote derivatives with respect to a parameter X along the particle's trajectory, x^iX). Varying with respect to X, we obtain the "geodesic" equation [9] /duae s \ dsf (du% f u \ a^ \ir8 + r»v*)+afUdfg + r»vf) = a (5-28) where ug = dx^/dsg is the four-velocity properly normalised with respect to gllv, such that g^viigUg = 1, and u1^ is denned analogously for the fllv geometry. In denning ug and u^ we have introduced the line elements for the two metrics, ds2 = gllvdxlldxv and ds2 = fllvdxlldxv. Is Eq. (5.28) the geodesic equation for a Riemannian metric? In other words, can the motion of point particles in this bimetric theory be described as geodesic motion of an effective metric? We can gain insight on this question by writing Eq. (5.27) in the form Spp = —imag J dsg — imag J dsf. (5.29) To see that this is equivalent to Eq. (5.27), note that we can write dXy/-gllvxi1xv = idsgyj g^u'gU^ = idsg, (5.30) where we have used the fact that (by definition) g^u^u? = 1. Similar logic holds for fllv. The form (5.29) is less useful calculationally, particularly for deriving the geodesic equation (5.28), but it opens up a helpful rephrasing of the question of an effective metric: we want to find a line element, ds, for which Spp = —im J ds. This would imply ds = agdsg + oifdsf. (5.31) Squaring this and plugging back in the definitions of dsg and dsf, we find that the cross-term introduces a non-Riemannian piece, ds = ( 0 and so is technically only denned for massive particles. In general relativity, the geodesic equation does hold for massless particles. This can be seen from the fact that m drops out of the geodesic equation, but to show it rigorously, it is common to introduce a Lagrange multiplier, often called the einbein. The same logic carries over to our bimetric theory uninterrupted. Let us write the action (5.27) in terms of a parameter X and introduce the einbein, e(X), as MS 2 2 + mze(X) (5.43) For m/0, varying this with respect to e we find 1 ds e = - — . (5.44) m dX Plugging this into the action (5.43), we obtain the original action, Eq. (5.27). But we can now extend the treatment to m = 0. In this case, varying with respect to e yields ds2 = 0. (5.45) Then, varying with respect to xa, we find the same geodesic equation as for the massive point particles. In other words, we have found that massless point particles travel on null geodesies of We may want to use a different form than (5.28) for the geodesic equation when dealing with massless point particles, since in general dsg and dsf may vanish for a massless particle.3 We can write the geodesic equation (for a massive or massless point particle) in terms of the quasimetric as 3This will be the case in particular if gßV and fßV are conformally related, as then a lightlike path in one metric is also lightlike in the other. 5.3 Point Particles and Non-Riemannian Geometry 113 xvxa = 0. (5.46) Note that we do not write this in terms of Christoffel symbols because we do not have to; if we had, we would need to calculate the inverse quasimetric, which is both difficult and unnecessary. This result is straightforward to extend to theories with N interacting metrics gl/lv, corresponding to a massless graviton with a tower of N — 1 massive gravitons [20, 21]. In this case, the line element is defined by which is clearly Finslerian. 5.4 Summary of Results We have examined an example of a bimetric theory in which, due to their minimal coupling to both metrics, matter fields do not feel a universal physical metric. We have found that when coupling matter to multiple metrics, the massless scalar might be unique in minimally coupling to a Riemannian effective metric. The massive scalar, Maxwell field, and point particle all provide counter-examples. We examined in detail light propagation in the geometric optics limit and showed that there is a distinct problem in relating observations to the underlying theory. We can make progress by generalising the line element beyond a Riemannian form. In particular, we showed that point particles follow geodesies of a Finsler spacetime, which is nonmetric. This geometry that emerges for a pointlike observer depends quite nontrivially upon, in addition to the two metric structures, the observer's own four-velocity through a disformal coupling. These considerations in this chapter may force us to rethink the geometric nature of spacetime, even in a metric theory of gravity. Consider the fundamental question of how to relate bigravity to observations, such as cosmological measurements. The textbook methods lean heavily on the existence of a "Jordan-frame" metric to which matter is minimally coupled. Here, however, such a metric does not exist universally, and may not exist at all for certain species of matter. How, then, should one calculate the redshift and the luminosity distance of a cosmological source in terms of the underlying FLRW geometries? Even the proper time along a timelike path is no longer trivial, as we cannot use the assumption dx = —ds. Indeed, because of the different effective metrics (or lack thereof), the notion of proper time is likely no longer even unique, depending instead on which matter fields an observer uses to construct her clock. (5.47) 114 5 The Geometry of Doubly-Coupled Bigravity Perhaps the best approach to solving physical problems in bimetric spacetimes without an effective metric is to go back to "primitive," pre-geometric constructions. In the absence of a single spacetime on which to formulate physics, we may need to simply consider particle motion coupled to two (or more) spin-2 fields in a way that only looks geometric because it is the nature of the spin-2 particle to invoke geometry [22, 23]. Paradoxically, once we have doubled geometry, we lose the ability to use its familiar methods. This is a call to go back to the basics, and rediscover the justification for results which we have taken for granted for the better part of the last century. References 1. S. Hassan, R.A. Rosen, Resolving the ghost problem in non-linear massive gravity. Phys. Rev. Lett. 108, 041101 (2012). arXiv: 1106.3344 2. S. Hassan, R.A. Rosen, A. Schmidt-May, Ghost-free massive gravity with a general reference metric. J. High Energ. Phys. 1202, 026 (2012). arXiv: 1109.3230 3. S. Hassan, R.A. Rosen, Confirmation of the secondary constraint and absence of ghost in massive gravity and bimetric gravity. J. High Energ. Phys. 1204, 123 (2012). arXiv: 1111.2070 4. Y. Yamashita, A. De Felice, T. Tanaka, Appearance of Boulware-Deser ghost in bigravity with doubly coupled matter. Int. J. Mod. Phys. D23, 3003 (2014). arXiv: 1408.0487 5. C. de Rham, L. Heisenberg, R.H. Ribeiro, On couplings to matter in massive (bi-)gravity. Class. Quant. Grav. 32, 035022 (2015). arXiv: 1408.1678 6. S. Hassan, M. Kocic, A. Schmidt-May, Absence of ghost in a new bimetric-matter coupling. arXiv: 1409.1909 7. C. de Rham, L. Heisenberg, R.H. Ribeiro, Ghosts and matter couplings in massive gravity, bigravity and multigravity. Phys. Rev. D90(12), 124042 (2014). arXiv: 1409.3834 8. M. Berg, I. Buchberger, J. Enander, E. Mortsell, S. Sjors, Growth histories in bimetric massive gravity. J. Cosmol. Astropart. Phys. 1212, 021 (2012). arXiv: 1206.3496 9. Y. Akrami, T.S. Koivisto, D.F Mota, M. Sandstad, Bimetric gravity doubly coupled to matter: theory and cosmological implications. J. Cosmol. Astropart. Phys. 1310, 046 (2013). arXiv: 1306.0004 10. J. Noller, S. Melville, The coupling to matter in massive, bi- and multi-gravity. J. Cosmol. Astropart. Phys. 1501, 003 (2014). arXiv: 1408.5131 11. L. Heisenberg, Quantum corrections in massive bigravity and new effective composite metrics. arXiv: 1410.4239 12. A. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B39, 393-394 (1972) 13. Y. Akrami, T.S. Koivisto, M. Sandstad, Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality. J. High Energ. Phys. 1303, 099 (2013). arXiv: 1209.0457 14. C. Brans, R. Dicke, Mach's principle and a relativistic theory of gravitation. Phys. Rev. 124, 925-935 (1961) 15. E. Cartan, Les Espaces de Finsler (Herman, Paris, 1934) 16. J.D. Bekenstein, The Relation between physical and gravitational geometry. Phys. Rev. D48, 3641-3647 (1993). arXiv:gr-qc/9211017 17. C. Pfeifer, The Finsler spacetime framework: backgrounds for physics beyond metric geometry. Ph.D. thesis, University of Hamburg (2013) 18. T.S. Koivisto, D.F. Mota, M. Zumalacarregui, Screening modifications of gravity through dis-formally coupled fields. Phys. Rev. Lett. 109, 241102 (2012). arXiv:1205.3167 References 115 19. M. Zumalacarregui, T.S. Koivisto, D.F. Mota, DBI Galileons in the Einstein frame: local gravity and cosmology. Phys. Rev. D87, 083010 (2013). arXiv:1210.8016 20. K. Hinterbichler, R.A. Rosen, Interacting spin-2 fields. J. High Energ. Phys. 1207, 047 (2012). arXiv:1203.5783 21. N. Tamanini, E.N. Saridakis, T.S. Koivisto, The cosmology of interacting spin-2 fields. J. Cosmol. Astropart. Phys. 1402, 015 (2014). arXiv: 1307.5984 22. S.N. Gupta, Gravitation and electromagnetism. Phys. Rev. 96, 1683-1685 (1954) 23. S. Weinberg, Photons and gravitons in perturbation theory: derivation of Maxwell's and Einstein's equations. Phys. Rev. 138, B988-B1002 (1965) Chapter 6 Cosmological Implications of Doubly-Coupled Massive Bigravity The universe is full of magical things patiently waiting for our wits to grow sharper. Eden Phillpotts, A Shadow Passes So far we have studied the cosmological solutions of massive bigravity in Chaps. 3 and 4 with matter coupled only to one metric, and discussed some of the theoretical issues with extending to a bimetric matter coupling in Chap. 5. As emphasised in the introduction of Chap. 5, the singly-coupled theory spoils the metric interchange symmetry present in vacuum; the kinetic and mass terms treat the metrics on equal footing, but this is broken when one couples matter to only one metric. It is therefore compelling to investigate other types of matter coupling that extend this metric-interchange symmetry to the entire theory. Moreover, as demonstrated in Chap. 3, cosmological background viability and linear stability rule out all but a small handful of the parameter space of the singly-coupled theory. By extending the matter coupling, we may be able to open up the space of observationally-allowed bimetric theories. The most significant obstacle to the construction of such a theory is that almost all attempts to couple matter to both metrics, such as the double minimal coupling discussed in Chap. 5, reintroduce the Boulware-Deser ghost (cf. Sect. 2.1.1) at arbitrarily low energies [1-3]. One of the papers demonstrating this, Ref. [3], also proposed a double coupling which is significantly better-behaved with respect to the Boulware-Deser ghost. While a ghost does appear in this theory, it appears at a scale at least as high as the strong coupling scale and possibly parametrically larger, in which case it is outside the domain of the validity of the effective theory. While this may present a problem for highly anisotropic solutions, the absence of the ghost around FLRW solutions was demonstrated explicitly [3]. The status of the ghost in this specific coupling has also been investigated in Refs. [4-6]. In this theory, matter couples minimally to an effective metric constructed out of the two metrics appearing in the gravitational sector of the theory, regardless of whether the second metric is dynamical. This would alleviate the problem of con- © Springer International Publishing AG 2017 117 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7_6 118 6 Cosmological Implications of Doubly-Coupled Massive Bigravity structing physical observables discussed in Chap. 5, as matter would move on geodesies of the effective metric. This proposal has been derived using complementary methods and extended to a multi-metric framework in Ref. [7], while the cosmology of this new coupling has been investigated in the dRGT context in Ref. [8] and will be discussed in Chap. 7. In this chapter, we study the background cosmology of massive bigravity when matter couples to the effective metric proposed in Ref. [3]. We show that the background expansion can asymptotically approach ACDM at both early and late times, and for certain parameter values is identical to ACDM always. At the background level, this type of coupling is therefore consistent with observations. In a future study, we will investigate whether this holds true for cosmological perturbations. This chapter is organised as follows. In Sect. 6.1 we present the effective metric and the symmetries that are present in the action. In Sect. 6.2 we derive the cosmological equations of motion and discuss their main features. A parameter scan of the minimal models, where only one of the interaction terms is nonvanishing, is performed in Sect. 6.3. In Sect. 6.4 we discuss some special parameter choices. We conclude in Sect. 6.5. 6.1 Doubly-Coupled Bigravity In this chapter we will extend the bigravity action (2.30) to a doubly-coupled version with an effective metric, g^,1 given by M2 r A M2f r A ,_ W = —f J d4x^R(g) d4x^fR(f) 4 +m2M2gJ d4x^g~Y^l3nen(X) + J d4x^g~Cm (geff, <&,-). (6.1) n=0 2 The effective metric, first introduced in Ref. [3], is defined by gfv = «V + WgvcXv + P2Uv, x\ = (VirWv. (6.2) As shown in Appendix C, the effective metric is symmetric under the interchange gfiv ffiv and a -o- /3. This makes the entire action symmetric under the transformations giiv Uv, Mg o Mf, fin 04_n, a fi. (6.3) We will denote the effective metric with "eff" written as a superscript or subscript interchangeably. 2In Ref. [3] the effective metric is given in an explicitly symmetric form, but this is not needed since §iia^av = gva^aii, as first shown in Ref. [1]; see also Appendix C. 6.1 Doubly-Coupled Bigravity 119 There is thus a duality between the two metrics present in the action which is spoiled when matter couples to only one of the metrics (taken by setting either a = 0 or 0=0). The effective metric has the convenient property that its determinant is in fact in the form of the ghost-free interaction potential in Eq. (6.1). In particular, the determinant can be written as [3] = V^det (a + BX). (6.4) The right-hand side is a deformed determinant, and it appears naturally when constructing a ghost-free potential [9]. Indeed, this deformed determinant is nothing other than a subset of the ghost-free dRGT potential with specific choices for ft, 4 det (a + BX) = ^ aA~nBnen (X). (6.5) n=0 Therefore matter loops, which will generate a term of the form ges A„, will by construction not lead to a Boulware-Deser ghost. This simple criterion in fact dooms many other forms of double coupling and is, in part, what motivated Refs. [3, 7] to construct this specific form of g^. The action (6.1) contains two Planck masses (Mg and Mf), five interaction parameters (f3n, of which ft and ft are the cosmological constants for g and /, respectively), and two parameters describing how matter couples to each metric (a and ft. The Planck masses and the coupling parameters a and B only enter observable quantities through their ratios. Moreover, one of those ratios is redundant: as described in Appendix C, the action can be freely rescaled so that either Mf/Mg or B/a is set to unity.3 Therefore the physically-relevant parameters are ft and either Mf /Mg or B/a. In this chapter we will rescale the Planck masses so that there is one effective gravitational coupling strength, Meff. We will also keep a and /3 explicit to make the a -o- j5 symmetry manifest, but the reader should bear in mind that only their ratio matters physically. All observational constraints will be given solely in terms of B/a, from which it is straightforward to take the singly-coupled limit, B/a -> 0. The Einstein equations have been derived in Ref. [10] and can be written in the form 3 ,- (x-1)^* + m2X(-D"^(x-1)^ay(^ = (atx-^r* + fiT*v) , (6.6) M R I- xKaf +m2^ Yji-Wfr-nr^Jllp = ^2 V^f (aT'lv + ^(ll-Tv)a). (6.7) 3 See also Sect. 2.1.2 for the redundancy of the Planck masses in the singly-coupled theory. 120 6 Cosmological Implications of Doubly-Coupled Massive Bigravity The matrices Y and Y depend on g~l f and f~lg, respectively, and are the same as were denned in Eq. (2.41). The Einstein tensors Ggv and G^v have their indices raised with g/iv and f/iv, respectively. Note that the terms with ges can be simplified using Eq. (6.4). The stress-energy tensor T^v is defined with respect to the effective metric g^ as 8 [V=g^f£m (Seff, O)] = l-^g-^T»v8gfv, (6.8) and obeys the usual conservation equation veffr/xv = ^ (6 9) where V^ff is the covariant derivative for g^fv. 6.2 Cosmological Equations and Their Solutions To describe homogeneous and isotropic cosmologies, we specialise to the Friedmann-Lemaitre-Robertson-Walker (FLRW) ansatze for both gliv and fliv, dsl = -N2dt2 + a2dx2, (6.10) o o o dsj = -N2fdt2 +a2fdx2, (6.11) where Ngj and agj are the lapses and scale factors, respectively, of the two metrics. Because both metrics are on equal footing, we have changed the notation slightly from Chaps. 2 to 4 to be more symmetric between the two metrics. As in general relativity, we can freely rescale the time coordinate to fix either Ng or Nf \ however, their ratio is gauge-invariant and will remain unchanged. The effective metric becomes ds2ff = -N2dt2 + a2dx2, (6.12) where the effective lapse and scale factor are related to those of gliv and fliv by N = aNg + r3Nf, (6.13) a = aag+f3af. (6.14) The equations of motion can be derived either directly from Eqs. (6.6) and (6.7), or by plugging the FLRW ansatze into the action and varying with respect to the scale factors and lapses, as was done in Ref. [3]. We have checked that both approaches yield the same result. Defining B0(y) = ß0 + 3ßiy + 3ß2y2 + ß3y3, Bx(y) = ßiy~3 + 3ß2y~2 + 3ß3y~1 + ß4, (6.15) (6.16) 6.2 Cosmological Equations and Their Solutions 121 where, as before, y = —, (6.17) the Friedmann equations for gßV and fßV are ap a3 3H<: = l^^+m ßo' (6-18) Bp a3 3Hi = ^r—+m'Bl. (6.19) Me2ff a3 2 yjfJ u 2 Here the energy density, p, is a function of the effective scale factor, a, and we have denned the g- and /-metric Hubble rates as Hg = Hf = (6.20) Ngag Nfaf Notice that the two Friedmann equations for Hg and Hf map into one another under the interchange ft -> ft_„, a ^> ft and gMV ^> /MV (which sends ^> y ^ y-1, and ft as expected from the properties of the action described in Appendix C. The stress-energy tensor is conserved with respect to the effective metric, so we immediately have p + 3-(p + p) = 0, (6.21) a where the density, p, and pressure, p, are denned in the usual way from the stress-energy tensor. By taking the divergence of either Einstein equation with respect to the associated metric (e.g., taking the g-metric divergence of Eq. (6.6)) and using the Bianchi identity and stress-energy conservation, we obtain the "Bianchi constraint," 2/2 i\ aßa2p m [ß\ag + 2ß2agaf + fta^j (Nfäg - Ngäf) =0. (6.22) In complete analogy with the singly-coupled case discussed in Sect. 2.1.3, which can be obtained by setting a or fi to zero, Eq. (6.22) gives rise to two possible branches of solutions, one algebraic and one dynamical [11-13].4 4In the singly-coupled theory, Eq. (6.22) would be a constraint equation arising from the Bianchi identity and stress-energy conservation. When using the effective coupling, the stress-energy conservation holds with respect to the effective metric, rather than or fflv. This gives rise to the pressure-dependent term in the left bracket. Due to this term, both branches—obtained by setting either bracket to zero—can be regarded as dynamical. We choose to adopt the terminology from the singly-coupled case here, however. 122 6 Cosmological Implications of Doubly-Coupled Massive Bigravity 6.2.1 Algebraic Branch of the Bianchi Constraint As discussed in Sect. 2.1.3, in the singly-coupled case, setting the first bracket of Eq. (6.22) to zero gives an algebraic constraint on y that can be shown to give solutions that are indistinguishable from general relativity at all scales [12]. In the doubly-coupled theory, the presence of the pressure term makes the phenomenology of the algebraic branch richer. In this section, without any ambition to examine all possible solutions, we briefly outline some of the properties of a few specific solutions on the algebraic branch of the Bianchi constraint (6.22). In this branch we have i/o ->\ oiBa2p m1 (fta2 + 2/32agaf + fta2) = (6.23) If the Universe is dominated by dust (p = 0), then as in the singly-coupled theory this is a polynomial equation for y, Pi + 2/32y + foy1 = 0, (6.24) which is solved by a constant y = yc. Notice that when y is constant, the mass terms in the two Friedmann equations become constant, so Hg and Hf are determined by Friedmann equations containing effective cosmological constants.5 Using the fact that a = (a + fiyc) ag = (a/yc + fi) a/, we can show that the observed Hubble rate, H = d/iaN), for a constant y is given by H = Hg(a + P^ =Hf(a^ + p) . (6.25) If the ratio Ng/Nf is constant, the solutions on this branch contain an exact cosmological constant (at least at the background level) given by a combination of the metric interaction terms. Since for a constant y, the two Hubble rates are related by Hf = H*jjj> ^ the bimetric interactions mimic a cosmological constant when Hg/Hf = const. This is only possible if the parameters satisfy ay3Bi = ^Bq. For more general parameter values, we have . . , 3aH2eyl (6.27) 3pH^ + m2 (aBxyi-pBo)' which is dynamical, so these cosmologies are not exactly ACDM. These are not, however, ACDM cosmologies for the effective metric due to the nontrivial coupling to p. 6.2 Cosmological Equations and Their Solutions 123 For nonzero pressure, p ^ 0, we can rewrite the constraint (6.23) as 2 / h ft 2\ <*3p{l + 24y + &y2)p m2l31ll + 2—y + —y2) = -^-=-(6.28) V Pi ft / M2ff We will not attempt to classify the solutions in this more complicated scenario. However, we note that for the special parameter choice P P2 ft = ft-, ft=ft^, (6.29) we obtain 2\ p m2ßi M2 a3/3 (6.30) i.e., we are required to have a constant p, corresponding to a vacuum equation of state, w = — 1. 6.2.2 Dynamical Branch of the Bianchi Constraint As is most often done in singly-coupled bigravity models—see, for example, Refs. [12, 14-17] and Chaps. 2-4—in the remainder of this chapter we will restrict our study to solutions where the second bracket in (6.22) vanishes, as these will turn out to be consistent with observational data. In this branch we have a dynamical constraint on the ratio between Nf and Ng, Nf df da.f N=-=J- (631) This implies the simple relation Hfy = Hg. Furthermore, the physical Hubble rate H, defined as H — ~rr~' (632) Na becomes He yHf H =-8— = y f . (6.33) a + fiy a + fiy Combining the two Friedmann equations, we obtain the equations for H and y, 124 6 Cosmological Implications of Doubly-Coupled Massive Bigravity 9 P / ix m2(B0 + y2B1) 6Me2ff 6(a + fty)2 0 = (a + fty)3 (a - fty"1) + m2 (B0 - y2Bx) . (6.35) Equations (6.34) and (6.35) determine the expansion history completely and are invariant under the combination of ft -> ft_„, a, -o- ft and _y -> _y_1. They have the same structure as in singly-coupled bigravity (cf. Sect. 2.1.3): there is a single Friedmann equation sourced by p and y, while y evolves according to an algebraic equation whose only time dependence comes from p. Notice that due to Eq. (6.35) one can write many different, equivalent forms of the Friedmann equation for H2. It is therefore dangerous to directly identify the factors in front of p in Eq. (6.34) as a time-varying gravitational constant and the term proportional to m2 as a dynamical dark energy component: both of these effects are present, but they cannot be straightforwardly separated from each other. From Eq. (6.35), we see that as p -> oo in the far past, either y -> /3/a or y -> —a/^. One can show that if p ~ a~p then H2 ~ a~2p^ as y -> —a/ft Since this scenario is observationally excluded, we will not consider this limit. Recall from Sect. 2.1.3 that in the singly-coupled theory there are also infinite-branch solutions where y -> oo at early times [17]. Indeed, as we saw in Chap. 3, these infinite-branch solutions are crucial in order to avoid linear instabilities. However, in the doubly-coupled theory, there are no solutions to Eq. (6.35) in which y -> oo as p -> oo. This is because of the new term proportional to afiy3p; none of the terms in Bq — y2B\, which grows at most as _y3, can possibly cancel off this term as p —>• oo. An interesting feature is that in the early Universe the mass term drops away but we are left with a modification to the gravitational constant, H2 v" ' : /'' . (6.36) 3M2ff Since the coefficient in front of p in the Friedmann equation during radiation domination can be probed by big bang nucleosynthesis, this could in principle be used to constrain the parameters of the theory. However, this will only work if the corresponding factor in front of p in local gravity measurements has a different dependence on a and fi. The solar-system predictions for this theory have not, to date, been worked out. In the far future, as p —>• 0, we have two possibilities. The first is that y goes to a constant yc, determined by foyt + - &) y3c +3 (Pi- ft) y2c + (ft - 3ft) yc-fa = 0. (6.37) (a2 + ß2)p These models approach a de Sitter phase at late times (whether they self-accelerate is a subtle question which we address below), with a cosmological constant given by 6.2 Cosmological Equations and Their Solutions 125 m1 [fa + (fa + 3fa) yc + 3(fa+ fa) y2c + (3fa + fa) y3c + fay4} A =---%---—. (6.38) 2yc (a + Pyc)2 The second possibility is that, for some parameter choices, \y\ -> oo such that the leading-order fa term in Eq. (6.35) exactly cancels the leading density term, y4p. It is unclear whether these solutions are viable; in this chapter, we will restrict ourselves to solutions where y is asymptotically constant in the past and future, starting at y = fa a and ending with y = yc. This implies that ag and cif are proportional to one another in both the early and late Universe. As long as y does not exhibit any singular behaviour, the evolution between y = fa a and y = yc is monotonic. This can be seen by taking a time derivative of Eq. (6.35) and setting y = 0. The monotonicity of the evolution of y implies that in the special case where yc = fa a, then we will have y = fa a at all times, and the expansion history is identical to ACDM. This is a new feature of the doubly-coupled theory: in the singly-coupled case, yc becomes zero in the presence of matter, which makes such a case trivially identical to general relativity. A constant y occurs in any model where the fa parameters and 0/a axe chosen to satisfy fa(-) + (3ft - ft) (-) +3 (ft+ (0o - 302) (-) ~ fa = 0, (6.39) which is simply Eq. (6.37) with yc = fa a. An interesting implication of solutions with constant y is that, since Eq. (6.31) implies Nf/Ng = da,f/dag = y, the two metrics are proportional, f/iv = y2g/iv.6 6.3 Comparison to Data: Minimal Models In this section, we compare the background expansion derived above to observations and perform a parameter scan of the minimal models, in which only one of the fa is nonzero. Due to the duality property of the solutions, we only have to look at the fa, fa, and fa cases. We will restrict ourselves to positive fa a; in principle negative values could also be allowed, but we have not yet investigated the physical implications of these values.7 The minimal models admit exact ACDM solutions when fa a = Jo, 1J for the fa, fa, and fa cases, respectively, as is evident from Eq.(6.39). Since we have so far calculated the equations of motion only for homogeneous backgrounds, we will limit this study to purely geometrical tests of the 6It is not difficult to see that there are no cases in which the two metrics are related by a dynamical conformal factor; from Eq. (6.31) any conformal relation means that daf/dag = df/ag, but this implies af /ag = const. 7Note that fi < 0 leads to instabilities in the case of doubly-coupled dRGT massive gravity, in which one of the metrics is nondynamical [8]. 126 6 Cosmological Implications of Doubly-Coupled Massive Bigravity background expansion, including the redshift-luminosity relation of Type la super-novae (SNe) [18], the observed angular scales of cosmic microwave background (CMB) anisotropics [19], and baryon-acoustic oscillations (BAO) [20-22]. Since the latter two depend on the physical size of the sound horizon scale around the time when the CMB photons decoupled from the baryon plasma, we can cancel out this dependence by using only the ratio of the observed angular scales in the CMB and BAO [12, 23]. In this way, we obtain a cosmological probe that is highly insensitive to the physics of the early Universe, and almost exclusively sensitive to the expansion history of the Universe between z ~ 1000 and today. We can calculate the effective equation of state for the background model described in Eqs. (6.34) and (6.35) using 1 d log H2 «>eff = -i-T-rr—• (6-40) 3 d log a Since in this chapter we restrict ourselves to solutions where y approaches constant values in the infinite past and future, for matter-dominated models we are guaranteed to have an effective equation of state where u>eff -> 0 as a -> 0 (ignoring radiation) and u>eff -> —1 as a -> oo, mimicking the asymptotic behaviour of the ACDM model. Except for some special parameter choices which are exactly ACDM (see the discussion above, as well as Sect. 6.4), we expect the model to deviate from the concordance model at all finite times. It is well-known that ACDM is able to provide an excellent fit to background expansion data, so we expect the success of the bimetric model to depend on how close the effective equation of state is to that of ACDM. All solutions that look exactly like ACDM will trivially be able to fit existing background expansion data. Note, however, that this does not mean that these models are equivalent to ACDM, since they may give different predictions for perturbations, i.e., when studying structure formation. In Fig. 6.1, we study the ft model, i.e., when only ft is turned on. Notice, cf. Eq. (6.34), that this model has no nontrivial interactions between the two metrics, so it deviates from ACDM only through the novel matter coupling. In the left panel of Fig. 6.1, we compare the effective equation of state for different values of ft a with that of ACDM. We fix Qm = 0.3, where Qm EE -f^r, (6.41) 3M2ffH2 and the subscript 0 indicates a value today. In the right panel of Fig. 6.1, we plot background constraints on Qm and fta. Note that the value of ft is set by the requirement that we have a flat geometry. Shaded contours show constraints from SNe and CMB/BAO data, respectively, corresponding to a 95 % confidence level for two parameters. Combined constraints are shown with solid lines corresponding to 95 and 99.9% confidence levels for two parameters. As expected, when /3/a -> 0, the effective equation of state coincides with ACDM since this limit corresponds to 6.3 Comparison to Data: Minimal Models 127 Fig. 6.1 Left panel The effective equation of state, u>eff, for the /3o model with 0 < /3/a < 1 (dotted lines) compared to weff of the ACDM model (solid line). When /3/a —> 0, the effective equation of state for the /3o model approaches that of the ACDM model. In all cases, Qm = 0.3. Right panel Confidence contours for Qm and ft/a for the fio model as fitted to SNe, CMB, and B AO data Fig. 6.2 Confidence contours for Q.m and /3/a for the f>\ and f>2 minimal models as fitted to SNe, CMB, and BAO data. In each case, we are able to obtain as good a fit as the concordance ACDM model the singly-coupled case where Pq acts as a cosmological constant. Note also that as P/a is increased, so is the factor multiplying the matter density in the Friedmann equation, and therefore the preferred matter density, Qm, becomes smaller. In Fig. 6.2 we plot background constraints on the Pi and P2 models. Since we know that the values P/a = -j= and P/a = 1 give exact ACDM solutions for the Pi- and /32-only models, respectively, we expect these values to provide good fits to the data. This is indeed the case, as can be seen in the plots. The P2 model is especially interesting in this regard, as P/a = 1 corresponds to the case where the two metrics gliv and fliv give equal contributions to the effective metric (or Mg = Mf when using the equal coupling strength framework described in Appendix C). Notice that the P2 model favours P > 0, as we would expect since the /32-only singly-coupled model is not in agreement with background data [15] and is ruled out by theoretical viability conditions [17]. 128 6 Cosmological Implications of Doubly-Coupled Massive Bigravity One of the attractive features of the double coupling is that it allows sensible cosmological solutions with only one of the ft turned on. For more general combinations of the ft parameters, we expect the data to favour values that cluster around the value of B/a given by solving Eq. (6.39), since this value yields an exact ACDM background expansion. We do not find it meaningful to do such a parameter scan at this moment, since it is only by including other probes, such as spherically symmetric solutions and cosmological perturbations, that we can exclude a larger part of the parameter space. However, in the next section, we discuss a few special cases that may turn out to be of particular interest for further investigations. 6.4 Special Parameter Cases 6.4.1 Partially-Massless Gravity Partial masslessness arises when a new gauge symmetry is present that eliminates the helicity-0 mode of the massive graviton,8 removing two of the problems with massive gravity discussed in Sect. 2.1: the vDVZ discontinuity in the m -> 0 limit of linearised massive gravity [24, 25] and the need for Vainshtein screening to reconcile the theory with solar system tests [26]. This is because both of these aspects of massive gravity are direct results of the fifth force mediated by the helicity-0 mode. Moreover, this new gauge symmetry would both determine the cosmological constant in terms of the graviton mass and protect a small cosmological constant against quantum corrections. Thus it is potentially a solution to both the old and new cosmological -constant problems: why the cosmological constant is not huge, and why it is not exactly zero, respectively. Massive gravity and bigravity contain a candidate partially-massless theory [27, 28], obtained by making the parameter choices Bo = 3ft = ft, ft = ft = 0. (6.42) For more on partially-massless gravity and its connection to massive (bi)gravity, we refer the reader to Ref. [29], as well as Refs. [28, 30] and references therein. In singly-coupled bigravity, the partially-massless parameter choices could only be imposed in vacuum; including matter forces y to be zero, which trivially reduces to general relativity. The nontrivial implications of the partially-massless scenario have been demonstrated for other doubly-coupled bigravity theories (see Ref. [31], though note that the theory discussed therein appears to have a ghost [2, 3]). Here we discuss this class in the context of the present doubly-coupled theory. For the partially-massless parameter choices, Eq. (6.35) implies that y = P/a at all times, and the Friedmann equation becomes So that a partially-massless graviton has four polarisations rather than the five of a massive graviton, hence the name. 6.4 Special Parameter Cases 129 P + 3(a2 + ß2)' m2ß0 (6.43) The cosmology of the candidate partially-massless theory is therefore equivalent to standard ACDM with an effective cosmological constant, m2ft/(a2 + /32), and a rescaled gravitational coupling for matter. Consequently, the background expansion is identical to that of general relativity, albeit with shifted constants. Notice that this is a qualitatively new feature as compared to the singly-coupled theory. Doubly-coupled bigravity with the parameters (6.42) is thus a strong candidate partially-massless theory of gravity. In the context of single-metric (dRGT) massive gravity, with matter coupled only to the dynamical metric, this parameter choice leads to a theory which is not partially massless and in fact suffers from an infinitely strongly-coupled helicity-0 mode [32]. If doubly-coupled bigravity is shown to possess the partially-massless gauge symmetry nonlinearly and around all backgrounds, it should automatically become one of the most interesting available theories of gravity beyond general relativity. 6.4.2 Vacuum Energy and the Question of Self-Acceleration As discussed in Chap. 1, one of the primary motivations for modifying general relativity is the possibility of having self-accelerating solutions, i.e., cosmologies which accelerate at late times even in the absence of a cosmological constant or vacuum energy contribution. In general relativity, as well as in singly-coupled bigravity, these two are degenerate: the vacuum energy and a cosmological constant may have different origins, but they are mathematically indistinguishable. In bigravity with matter coupled to the effective metric, however, this question becomes rather subtle, as the vacuum energy from the matter sector produces more than just the cosmological constant terms for g/iv and f/iv, which are equivalent to ft and ft. We have shown in Sect. 6.1 that quantum corrections to matter coupled to g^l will generate all of the ghost-free bimetric interaction terms. If we take the matter loops to generate a cosmological constant term ■s/—genAv, then we can see from Eqs. (6.4) and (6.5) a pure vacuum-energy contribution can be written in the form of the bigravity interaction potential with parameters Let us assume that the ft parameters take this particular form, i.e., the only metric interactions arise from matter loops. The quartic equation (6.35) can then be solved only if y = ft/a (or p = —M2ff A„), and the Friedmann equation becomes ft = Ava4~nß (6.44) m 130 6 Cosmological Implications of Doubly-Coupled Massive Bigravity Mls 3 Equations (6.44) and (6.45) reduce to the known expression for the ACDM solutions with a cosmological constant proportional to either ft or ft m the singly-coupled limit (where either ji -> 0 or a, -> 0). It is, of course, not surprising that matter loops lead to an accelerating expansion. However, the appearance of the vacuum energy in all the bigravity interaction terms has novel implications. First, because the vacuum energy contributes to all the interaction terms, the mass scale m is not protected against quantum corrections from matter loops [3]. Therefore, any values we obtain for these parameters from comparison of the theory to observations must be highly fine-tuned.9 This is in contrast to singly-coupled bigravity, in which the only parameter that receives contributions from quantum loops is ft (if one couples matter to gjjiV), just as in general relativity where the cosmological constant is unstable in the presence of matter fields. In the singly-coupled theory, both the scale m and the structure of the interaction potential are stable to quantum corrections [33, 34], a very useful fact which is lost once we couple matter to g^fv.10 This is not a problem in the double coupling studied in Chap. 5, as loops would only induce g- and /-metric cosmological constants, ft and ft, although that theory is not ghost-free. Candidate expressions for g^fv where the matter sector would only contribute quantum corrections to ft and ft have been studied in Ref. [35], although it is not yet known whether any of these are free of the Boulware-Deser ghost at low energies. The other implication is that self-accelerating solutions are no longer straightforward to define in this theory. Typically, self-acceleration refers to cosmologies which accelerate at late times even when the vacuum energy is set to zero. Since in general relativity and singly-coupled bigravity, there is a single parameter which is degenerate with the vacuum energy (A in the former and ft or ft in the latter), one can simply set its value to zero and look for other accelerating solutions. In the present doubly-coupled theory, however, all interaction terms are degenerate with the vacuum energy: given an interaction potential, there is no way to unambiguously determine the value of A„. In that respect, we cannot set some of the parameters to zero in order to restrict ourselves to accelerating solutions arising from nonvacuum, massive-gravity interaction terms (unless we set all the parameters to zero, which will give uninteresting solutions). Therefore, from a particle physics point of view this theory lacks, or at the very least cannot unambiguously define, self-accelerating solutions. yIf the case described in Sect. 6.4.1 is truly partially massless, this may be an exception, as there is a new gauge symmetry to protect against quantum corrections. 10Indeed, the fact that a small graviton mass is stable against quantum corrections is one of the main motivations for studying massive (bi)gravity, particularly as a candidate to explain the accelerating Universe. 6.4 Special Parameter Cases 6.4.3 Maximally-Symmetrie Bigravity 131 The parameter choice ßo = ß4, ßi = ßs, « = ß, (6.46) is special in the sense that the duality transformation (6.3) maps solutions to themselves.11 Thus this theory is maximally symmetric between the two metrics: they appear in the theory in completely equal ways. In this case, the quartic equation (6.35) becomes (y2 -1) 4 ßi {y2 + 1) + 3ß2y - ß0y + -^Sr (1 + y)2 mzM, eff = 0. (6.47) As expected, there is an exact ACDM solution given by y = 1. Indeed, the two metrics are completely equal, gliv = fliv, because the Bianchi constraint imposes Nf/Ng = daf/dag = 1. The second-order polynomial for y in brackets gives two solutions which are inverses of one another. This is not surprising, since when giiv fiiv we have y -> y~l. 6.5 Summary of Results In this chapter we have presented the main features of the background expansion for massive bigravity with matter "doubly coupled" to both metrics through an effective metric, given by 8% = «V + 2a^X\ + f32Uv, T\ = (v^rVA- (6.48) This coupling was introduced in Refs. [3, 7], and has been further discussed in Refs. [4-6, 8]. This matter coupling has several advantages: it retains the metric-interchange symmetry in the presence of matter, leads to sensible cosmological solutions, and has a straightforward physical interpretation. The expansion history is described by a Friedmann equation for the effective metric (6.34) and a quartic equation (6.35) which algebraically describes the evolution of y = af/ag, the ratio of the /- and g-metric scale factors. One can always choose the parameters of the theory such that the background expansion is exactly that of ACDM; any parameter choice which leads to y = /3/a in Eq. (6.35) will have this behaviour. For more general parameter values, the background expansion will deviate from ACDM but may still be consistent with observational data. To explore this, we confronted the models with only fa, fa, or fa nonzero with observational data. Vacuum solutions for this model were previously studied in Ref. [30]. 132 6 Cosmological Implications of Doubly-Coupled Massive Bigravity The other single-parameter models—with ft or ft nonzero—are then automatically included in this analysis due to the duality between solutions under gliv -o- fliv, ft -> ft_„, and a, -o- ft as described in Appendix C. A novel feature of the effective coupling studied here is that gliv and fliv can be conformally related to each other at the background level in the presence of matter. In the singly-coupled case, this is only possible in vacuum, where the solutions are de Sitter. A special example of this is the parameter choice leading to a candidate partially-massless theory. This potentially has a novel gauge symmetry which would eliminate the problematic fifth force and protect a small vacuum energy against quantum corrections. In this case the background is identical to ACDM in the presence of matter. This suggests that doubly-coupled bigravity is a promising candidate for a theory of partially-massless gravity. This matter coupling has a problematic feature, namely that loop corrections for any matter coupled to g^l will generate all five dRGT interaction terms. Therefore the structure of the potential and the mass scale m lose their stability against quantum corrections, which had been one of the most impressive features of the singly-coupled theory. We have discussed an important consequence of this: while many solutions to the theory accelerate at late times, it is no longer possible to unambiguously identify solutions that self-accelerate, as the effective cosmological constant at late times can always be identified at least in part with a vacuum energy contribution. We end with a brief comment concerning our expectations for perturbations around these cosmological solutions. We have shown in Chap. 3 that the singly-coupled models are often unstable for small y. One might hope that these doubly-coupled models will have better stability properties: y is always nonzero and can be made to have a large minimum value by tuning /3/a. Moreover, we found in Chap. 3 that the ft-only model did have stable perturbations in the singly-coupled case, but that model is not viable in the background. As we have shown, this model is in excellent agreement with background data if /3/a is not too small, so it may provide another avenue for stable cosmological solutions in massive bigravity. References 1. S. Hassan, A. Schmidt-May, M. von Strauss, On consistent theories of massive spin-2 fields coupled to gravity. JHEP 1305, 086 (2013). arXiv:1208.1515 2. Y. Yamashita, A. De Felice, T. Tanaka, Appearance of Boulware-Deser ghost in bigravity with doubly coupled matter. Int. J. Mod. Phys. D 23, 3003 (2014). arXiv: 1408.0487 3. C. de Rham, L. Heisenberg, R.H. Ribeiro, On couplings to matter in massive (bi-)gravity. Class. Quantum Gravity 32, 035022 (2015). arXiv: 1408.1678 4. S. Hassan, M. Kocic, A. Schmidt-May, Absence of Ghost in a New Bimetric-Matter Coupling. arXiv: 1409.1909 5. C. de Rham, L. Heisenberg, R.H. Ribeiro, Ghosts and matter couplings in massive gravity, bigravity and multigravity. Phys. Rev. D 90(12), 124042 (2014). arXiv: 1409.3834 6. J. Noller, On Consistent Kinetic and Derivative Interactions for Gravitons. arXiv: 1409.7692 7. J. Noller, S. Melville, The coupling to matter in Massive. Bi- and Multi-Gravity. JCAP 1501, 003 (2014). arXiv: 1408.5131 References 133 8. A.E. Gumriikcuoglu, L. Heisenberg, S. Mukohyama, Cosmological perturbations in massive gravity with doubly coupled matter. JCAP 1502(02), 022 (2015). arXiv: 1409.7260 9. S. Hassan, R.A. Rosen, On non-linear actions for massive gravity. JHEP 1107, 009 (2011). arXiv: 1103.6055 10. A. Schmidt-May, Mass eigenstates in bimetric theory with ghost-free matter coupling. JCAP 1501, 039 (2014). arXiv: 1409.3146 11. D. Comelli, M. Crisostomi, F. Nesti, L. Pilo, FRW cosmology in ghost free massive gravity. JHEP 1203, 067 (2012). arXiv:1111.1983 12. M. von Strauss, A. Schmidt-May, J. Enander, E. Mortsell, S. Hassan, Cosmological solutions in bimetric gravity and their observational tests. JCAP 1203, 042 (2012). arXiv:1111.1655 13. M.S. Volkov, Cosmological solutions with massive gravitons in the bigravity theory. JHEP 1201, 035 (2012). arXiv: 1110.6153 14. M. Berg, I. Buchberger, J. Enander, E. Mortsell, S. Sjors, Growth histories in bimetric massive gravity. JCAP 1212, 021 (2012). arXiv: 1206.3496 15. Y. Akrami, T.S. Koivisto, M. Sandstad, Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality. JHEP 1303, 099 (2013). arXiv: 1209.0457 16. Y. Akrami, T.S. Koivisto, M. Sandstad, Cosmological Constraints on Ghost-Free Bigravity: Background Dynamics and Late-Time Acceleration. arXiv: 1302.5268 17. F. Konnig, A. Patil, L. Amendola, Viable cosmological solutions in massive bimetric gravity. JCAP 1403, 029 (2014). arXiv: 1312.3208 18. N. Suzuki, D. Rubin, C. Lidman, G. Aldering, R. Amanullah et al., The nubble space telescope cluster supernova survey: V. improving the dark energy constraints above z > 1 and building an early-type-hosted supernova sample. Astrophys. J. 746, 85 (2012). arXiv: 1105.3470 19. Planck Collaboration: Collaboration, P. Ade et al., Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys. (2014) arXiv: 1303.5076 20. L. Anderson, E. Aubourg, S. Bailey, D. Bizyaev, M. Blanton et al., The clustering of galaxies in the SDSS-III baryon oscillation spectroscopic survey: Baryon acoustic oscillations in the data release 9 spectroscopic galaxy sample. Mon. Not. R. Astron. Soc. 427(4), 3435-3467 (2013). arXiv:1203.6594 21. F. Beutler, C. Blake, M. Colless, D.H. Jones, L. Staveley-Smifh et al., The 6dF galaxy survey: Baryon acoustic oscillations and the local nubble constant. Mon. Not. R. Astron. Soc. 416, 3017-3032 (2011). arXiv: 1106.3366 22. C. Blake, E. Kazin, F. Beutler, T. Davis, D. Parkinson et al., The WiggleZ dark energy survey: mapping the distance-redshift relation with baryon acoustic oscillations. Mon. Not. R. Astron. Soc. 418, 1707-1724 (2011). arXiv: 1108.2635 23. J. Sollerman, E. Mortsell, T. Davis, M. Blomqvist, B. Bassett et al., First-year sloan digital sky survey-II (SDSS-II) supernova results: constraints on non-standard cosmological models. Astrophys. J. 703, 1374-1385 (2009). arXiv:0908.4276 24. H. van Dam, M. Veltman, Massive and massless Yang-Mills and gravitational fields. Nucl. Phys. B22, 397-411 (1970) 25. V. Zakharov, Linearized gravitation theory and the graviton mass. JETP Lett. 12, 312 (1970) 26. A. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B 39, 393-394 (1972) 27. C. de Rham, S. Renaux-Petel, Massive gravity on de Sitter and unique candidate for partially massless gravity. JCAP 1301, 035 (2013). arXiv: 1206.3482 28. S. Hassan, A. Schmidt-May, M. von Strauss, On partially massless bimetric gravity. Phys. Lett. B 726, 834-838 (2013). arXiv: 1208.1797 29. S. Hassan, A. Schmidt-May, M. von Strauss, Higher Derivative Gravity and Conformal Gravity From Bimetric and Partially Massless Bimetric Theory. arXiv: 1303.6940 30. S. Hassan, A. Schmidt-May, M. von Strauss, Particular solutions in bimetric theory and their implications. Int. J. Mod. Phys. D 23, 1443002 (2014). arXiv: 1407.2772 31. Y. Akrami, T.S. Koivisto, D.F Mota, M. Sandstad, Bimetric gravity doubly coupled to matter: theory and cosmological implications. JCAP 1310, 046 (2013). arXiv: 1306.0004 134 6 Cosmological Implications of Doubly-Coupled Massive Bigravity 32. C. de Rham, K. Hinterbichler, R.A. Rosen, A.J. Tolley, Evidence for and obstructions to nonlinear partially massless gravity. Phys. Rev. D 88(2), 024003 (2013). arXiv: 1302.0025 33. C. de Rham, G. Gabadadze, L. Heisenberg, D. Pirtskhalava, Nonrenormalization and naturalness in a class of scalar-tensor theories. Phys. Rev. D 87(8), 085017 (2013). arXiv: 1212.4128 34. C. de Rham, L. Heisenberg, R.H. Ribeiro, Quantum corrections in massive gravity. Phys. Rev. D 88, 084058 (2013). arXiv: 1307.7169 35. L. Heisenberg, Quantum corrections in massive bigravity and new effective composite metrics. arXiv: 1410.4239 Chapter 7 Cosmological Implications of Doubly-Coupled Massive Gravity If the Lord Almighty had consulted me before embarking upon his creation, I should have recommended something simpler. Alfonso X of Castile In Sect. 2.1.3 we described a no-go theorem for cosmological solutions in dRGT massive gravity, i.e., in the theory where the only gravitational degree of freedom is a massive graviton. If the reference metric is taken to be that of Minkowski space, then dynamical flat and closed FLRW solutions do not exist; the Bianchi constraint (2.55) restricts the scale factor to be constant. This can be avoided by either choosing open solutions or changing the reference metric, but the resultant solutions are unstable. Therefore, the search for a viable cosmology with a massive graviton has necessarily involved extending dRGT by adding extra degrees of freedom (as in the bimetric theory which we have studied in Chaps. 3-6) or by breaking the assumptions of homogeneity and isotropy, either in the metric or in the Stiickelberg sector. The double coupling discussed in Chap. 6 has been shown to avoid both of these no-go theorems, opening up the intriguing possibility of obtaining sensible cosmological solutions with only a single massive graviton [1, 2]. In this scenario, matter is coupled to an effective or Jordan-frame metric, gfv = « V + 2aftv.X\ + £ V> (7-D where gliv is the dynamical metric, r\iliV is the Minkowski reference metric, and XMV = (\/g~l if)11 v. The properties of this effective metric were discussed in some detail in Sect. 6.1. However, we remind the reader that the theory with this matter coupling is believed to be ghost-free at least within the effective theory's regime of validity and that the Boulware-Deser ghost is absent about FLRW backgrounds [1, 3, 4]. In this chapter we explore the basic properties of these newly-allowed massive gravity cosmologies. Unusually, the proof in Ref. [1] that the no-go theorem is © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7_7 135 136 7 Cosmological Implications of Doubly-Coupled Massive Gravity avoided turns out to rely crucially on coupling a fundamental field (in this case, a scalar field) to the effective metric. In a standard late-Universe setup where matter is described by a perfect fluid with a constant equation of state (or even more generally when w only depends on the scale factor), this result does not hold, and FLRW solutions are constrained to be nondynamical, just as in standard dRGT. More generally, the pressure of at least one component in the Universe must depend on something besides the scale factor—such as the lapse or the time derivative of the scale factor—for massive gravity cosmologies to be consistent. This is why fields, which have kinetic terms where the lapse appears naturally, are required in order to obtain sensible cosmological solutions. Consequently the standard techniques of late-time cosmology cannot be applied to this theory. While we do not aim to rule out these models, the inability to obtain cosmological solutions with just, e.g., dust or radiation is an unusual feature which makes it difficult to derive precise predictions for cosmology, as the nature of the "extra matter" is not presently known. These solutions exhibit pathologies in the early- and late-time limits if all matter couples to the effective metric, and the scalar field physics would need to be highly contrived to avoid these issues. Moreover, the reliance on extra matter, such as a scalar field, which may well be gravitationally subdominant and high-energy implies a violation of the decoupling principle, in which the low-energy expansion of the Universe should not be overly sensitive to high-energy physics. The rest of this chapter is organised as follows. In Sect. 7.1 we derive and discuss the cosmological evolution equations in this theory. In Sect. 7.2 we elucidate the conditions under which the no-go theorem is violated and dynamical cosmological solutions exist. We discuss in Sect. 7.3 some of the nonintuitive features of the Einstein-frame formulation of the theory, and how these are resolved in a Jordan-frame description. In Sect. 7.4 we study cosmologies containing only a scalar field, and generalise this to include a perfect fluid coupled to the effective metric in Sect. 7.5. In Sect. 7.6 we consider an alternative setup in which the scalar field couples to the effective metric while the perfect fluid couples to the dynamical metric. We conclude in Sect. 7.7. 7.1 Cosmological Backgrounds The Einstein equation with all matter fields coupled to g^l was derived in Ref. [ ] (see also Sect. 6.1) and can be written in the form1 3 n=0 = At det (a + pX) (a(X-1)(/xarv)a + PT'1V), (7.2) Our convention is that indices on the Einstein tensor G^v are raised with g^v. 7.1 Cosmological Backgrounds 137 where the stress-energy tensor is denned as usual with respect to the effective metric, V-Seff 8gfv and the matrices are defined in Eq. (2.41). Let us assume a flat FLRW ansatz for g/iv of the form2 gllvdx'1dxv = -N2(t)dt2 + a2(t)8ijdxidxj, (7.4) and choose unitary gauge for the Stiickelberg fields, rj^ = diag(—1, 1, 1, 1), so the effective metric is given by gfvdxlxdxv = -N2f(t)dt2 + a^iOStjd^dx', (7.5) where the effective lapse and scale factor are related to N and a by Neff = aN + f3, aeff = aa + f3. (7.6) We will define the Hubble rates for gliv and g^l by a dpff H =-, Heff = ——. (7.7) aN ^effAeff Notice that, because of the inclusion of the lapses in these definitions, these quantities correspond to what would be the cosmic-time Hubble rates in general relativity, obtained by setting jV = 1 or iVeff = 1. While we need not include the lapse in the definition of H when working with diffeomorphism-invariant theories like general relativity or massive bigravity, instead choosing to set jV to a convenient value and thereby pick a physically-meaningful time coordinate like cosmic time or conformal time, the lack of diffeomorphism invariance in massive gravity means that neither the lapse nor the time coordinate has any meaning on its own, but will only appear through the combination Ndt. The time component of Eq. (7.2) yields the Friedmann equation, 3H2 = aP 4f M2 a3 Wk+^ + ^ + ^f), (7.8) where p = —g^T00 is the density of the matter source. The spatial component of Eq. (7.2) gives us the acceleration equation, 2Note the differences in notation between this chapter and Chap. 6, such as our use of a for the scale factor of gllv rather than of gffj. 138 7 Cosmological Implications of Doubly-Coupled Massive Gravity 5ti H---1--t---— = m N Mpj Na2 (7.9) where the pressure is defined by p = (l/3)g?ff riJ. Notice that the double coupling leads to a time-dependent coefficient multiplying the density and pressure terms in Eqs. (7.8) and (7.9). The Friedmann equation for the effective Hubble rate, Heff, can be determined from Eq. (7.8) by the relation Na HeS = a--H, (7.10) Wefftfeff which follows from Eq. (7.6). Matter is covariantly conserved with respect to geJfv, VfTllv = 0, (7.11) Li from which we can obtain the usual energy conservation equation written in terms of the effective scale factor, p+ 3— (p + p) = 0. (7.12) «eff As in general relativity, this holds independently for each species of matter as long as we assume that interactions between species are negligible. Finally, we can take the divergence of the Einstein equation (7.2) with respect to gliv and specialise to the FLRW background to find, after imposing stress-energy conservation, the "Bianchi constraint," 9 9 9 • 9 • m MFla P(a)a = a^aeíípa, (7.13) where we have defined 2ft ft P(a) = ft + ^ + ^f. (7.14) This can equivalently be derived using Eqs. (7.8), (7.9) and (7.12). The pressure, p, appearing in Eq. (7.13) is the total pressure of the Universe, or, if different species eff [IV ' couple to different metrics, the total pressure of all matter coupling to g Let us pause to count the number of equations and variables in this system. We have four free functions—the scale factor, the lapse, the density, and the pressure— and four equations—Friedmann, acceleration, conservation, and Bianchi constraint. Of the four equations, only three are independent, much like in general relativity. The remaining freedom is fixed by specifying an equation of state. The acceleration equation can usually be derived from the other three, but unlike in general relativity it is not always redundant: if the Bianchi constraint yields ä = 0, then the acceleration equation does give new information, and in fact is what would be used to determine 7.1 Cosmological Backgrounds 139 the lapse [6]. This situation is similar to general relativistic cosmology, but with one new variable and one new equation: because we have broken diffeomorphism invariance, the lapse cannot be fixed by a coordinate transformation, and furthermore the divergence of the Einstein equations leads to a nontrivial constraint. This is in contrast to general relativity, where the same procedure results in an identity. We emphasise that if all matter couples to the same g^l then the expansion history inferred from observations is given by aeff and Heff, for the simple reason that all observations are observations of matter (including light). In deriving any cosmological observables, the "proper time," dx = Neffdt, will play the same role as the cosmic time coordinate in general relativity. In particular, x corresponds to the time measured by point-particle clocks, while the distance light travels is given by dr = dx/aeffix). Therefore in principle we need only know Heft(aeff) in order to connect to standard background observables. The coordinate time, t, is just the coordinate in which the reference metric, rjjjiV, has the standard Minkowski form, and has no other physical significance. Since gliv and g^v play the exact same roles as the Einstein-frame and Jordan-frame metrics, respectively, in other modified gravity theories, we will use these terms freely. 7.2 Do Dynamical Solutions Exist? In the original, singly-coupled formulation of massive gravity, ji = 0 and so the right-hand side of Eq. (7.13) vanishes, with the result that a is constrained to be constant. This is nothing other than the no-go theorem on flat FLRW solutions in massive gravity. A nondynamical cosmology is, of course, still a solution when a and P are nonzero, in which case the values of a and iV are determined from Eqs. (7.8) and (7.9). The question is now under which circumstances the theory also allows for dynamical a. To begin with, let us assume that p = wp, where w can depend on the effective scale factor but nothing else. Assuming that a /O, Eq. (7.13) becomes m2Mpla2P(a) = a^wa^p, (7.15) and p is a function only of a (or equivalently aeff). To see this, consider Eq. (7.12) in the form d In p —— + 3[l + w(a)] = 0. (7.16) a In a Integrating this will clearly yield p = pia). Unless the left-hand side of Eq. (7.15) has the exact same functional form for a as the right hand side (which is, e.g., the This is because the Bianchi identity and stress-energy conservation are related to the diffeomorphism invariance of the Einstein-Hilbert and matter actions, respectively, but we have now added a mass term which does not obey this gauge symmetry, after fixing the Stiickelberg fields. 140 7 Cosmological Implications of Doubly-Coupled Massive Gravity case when w = —1/3 and ft = ft = 0), this equation is not consistent with a time-varying a. The theory does therefore not give viable cosmologies using the standard equation of state p = wp, where w is constant or depends on the scale factor. This conclusion is avoided if the pressure also depends on the lapse. In this case, Eq. (7.13) becomes a constraint on the lapse, unlocking dynamical solutions.4 The most obvious way to obtain a lapse-dependent pressure is to source the Einstein equations with a fundamental field rather than an effective fluid. This was exploited by Ref. [1] to find dynamical cosmologies with a scalar field coupled to g^v. We discuss this case in more detail below. Therefore, while physical dust-dominated solutions may exist, we must either include additional degrees of freedom or treat the dust in terms of fundamental fields. The standard methods of late-time cosmology cannot be applied to doubly-coupled massive gravity. 7.3 Einstein Frame Versus Jordan Frame Before examining the cosmological solutions when the pressure depends on the lapse, it behoves us to further clarify the somewhat unusual differences between this theory's Einstein and Jordan frames. It turns out that the Friedmann equation in the Einstein frame is completely independent of the matter content of the Universe (up to an integration constant which behaves like pressureless dust): H(a) always has a predetermined form [see Eq. (7.19)]. In the Einstein-frame description, matter components with nonzero pressure affect the cosmological dynamics through the lapse, N. Because the lapse is involved in the transformation from the Einstein frame, H, to the Jordan frame, Heff, cf. Eq. (7.10), the Jordan-frame Friedmann equation (corresponding to the observable Hubble rate) does depend on matter. We proceed to demonstrate this explicitly. Regardless of the functional form of p, and whether or not it depends on the lapse, for a 7^ 0 the pressure is constrained by Eq. (7.13) to have an implicit dependence on a given by m2Mla2P(a) p(a) =-nQ 2 . (7.17) The continuity Eq. (7.12) can then be integrated to obtain C 3m2 M2 /ft . , \ p(a) = ----—3-^ ^a3 + fta2 + fta , (7.18) 4Another possibility is that the pressure depends on a. The dynamics would be determined by Eq. (7.13), while the lapse would be constrained by the Friedmann equation. It is unclear whether these would give rise to Friedmann-like evolution, and we do not discuss this case any further. 7.3 Einstein Frame Versus Jordan Frame 141 where C is a constant of integration that includes any pressureless dust. Inserting this into Eq. (7.8) we find a generic form for the Einstein-frame Friedmann equation, 3H2 = m2 (c0 + 3cxa~l + 3c2a~2 + c3a~3) , (7.19) where we have defined the coefficients a co = 0o - —0i, a c\ = 01 - -02, a c2 = 02 - -03, 0' 0' a 0' aC c3 = 03 + -r-2-. (7.20) Notice that the functional forms of p (a), p (a), and i/2 (a) are completely independent of the energy content of the Universe, except for an integration constant scaling like pressureless matter. It is interesting to note that in the vacuum energy case studied in Sect. 6.4.2 with 0„ = (a/0)0„+i, all of the a coefficients apart from c3 vanish. In other words, if the metric interactions took the form of a cosmological constant for gjfy, then the Einstein-frame Friedmann equation would scale as a-3. 7.4 Massive Cosmologies with a Scalar Field If we include matter whose pressure does not only depend on the scale factor, aeff, then the Bianchi constraint (7.13) may not rule out dynamical cosmological solutions. For a pressure that also depends on the lapse, Eqs. (7.13) and (7.19) determine H and iV, which in turn can be used to derive the Jordan-frame Friedmann equation. Because the lapse enters into the frame transformation (7.10), the Jordan frame can be sensitive to matter even though, as discussed above, the Einstein frame is not. The lapse thus plays an important and novel role in massive gravity compared to general relativity. As discussed above, lapse-dependent pressures are not difficult to obtain: they enter whenever considering a fundamental field with a kinetic term. Consider a universe dominated by a scalar field, /, with the stress-energy tensor T»v = ^ffX^X ~ (^VaXVe"ffX + V(x)j (7-21) where V^f = g1^Vff and V(x) is the potential for the scalar field. The density and pressure associated to / are 142 7 Cosmological Implications of Doubly-Coupled Massive Gravity • 2 -2 Px = ^T + P* = jh-- V(X)- (7"22) The constraint (7.13) now has a new ingredient: the lapse, iVeff, which appears through the scalar field pressure.5 One can then use the Bianchi identity to solve for the lapse and substitute it into the Friedmann equation to obtain an equation for the cosmological dynamics that does not involve the lapse [1]. A simple way to substitute out the lapse is to use the relation, following straightforwardly from Eq. (7.13), ,2 r/M2 n*2 „2 Xz mzM^azP(a) = V(X) +-ri , , (7.23) as the lapse only appears in the Einstein-frame Friedmann equation through / 2/2N2ff. This explains the result, first noticed in Ref. [1], that after solving for the lapse, the Friedmann equation loses its dependence on the kinetic term. Note however that we can also use Eq. (7.23) to solve for the potential, V(x), and write the Einstein-frame Friedmann equation in a form that does not involve the potential. Of course, if we were to additionally use the continuity equation as discussed above, the Einstein-frame Friedmann equation would take the form of Eq. (7.19) which contains neither the kinetic nor the potential term. Using Eqs. (7.17) and (7.18) we can find expressions for the kinetic and potential energies purely in terms of a, in2 M2 a3 K{a) =-L (da"1 + 2c2a"2 + c3a"3) , (7.24) 2««efF in2 M2 a3 Via) =--(2d0 + dia"1 + 2d2a~2 + d3a"3) , (7.25) 2aaeJff where K = x212N2IV the q are defined in Eq. (7.20), and we have further defined a do = -Pi, di = Pi+5^p2, d2 = p2 + 2jP3, d^p,--^. (7.26) 5The q?2 theory studied in Ref. [1] can be obtained by setting fio = —3, f$\ = 3/2, pi = —1/2, and ^3=0 [7]. With this parameter choice, the Bianchi constraint (7.13) reproduces Eq. (5.8) of Ref. [1]. 7.4 Massive Cosmologies with a Scalar Field 143 Note that the terms proportional to C include any possible pressureless matter component coupled to g^. This integration constant will always appear when solving the continuity equation (7.12). The Friedmann equation is given by the generic equation (7.19). That is, we are left with the peculiar situation that the pressure, energy density, and Einstein-frame Friedmann equation are completely insensitive to the form of the scalar field potential. As discussed above, this lack of dependence on the details of the scalar field physics is illusory; the lapse does depend on V(x) and x, cf. Eq. (7.23), and in turn the physical or Jordan-frame expansion history depends on the lapse, cf. Eq. (7.10). Let us briefly remark on a pair of important exceptions. The no-go theorem forbidding dynamical a still applies when there is a scalar field present if either the potential does not depend on the lapse (such as a flat potential) or the field is not rolling. Let us rewrite Eq. (7.12) (which is equivalent to the Klein-Gordon equation) as - h^2- + V(X) +3^-^=0. (7.27) dt \2N2n ) aeff 7Ye2ff If V (X) is independent of 7Veff then x2/N2ff cannot depend on 7Veff and, by extension, neither can/? = x2/2Nln — V(x)-In the specific case of V(x) = const, this is clearly true, and we find x2//V2ff oc a~^, so p = p(a). Similarly, if the field is not rolling, X = 0, then it is clear from Eq. (7.22) that p loses its dependence on the lapse. To conclude this section, when a scalar field is coupled to the effective metric, we avoid the no-go theorem and it is possible to have dynamical a, unless the potential does not depend on the lapse or the field is not rolling. This result agrees with and slightly generalises that presented in Refs. [1, 2]. In a realistic scenario, however, we will have not only a scalar field but also matter components present. We now turn to that scenario. 7.5 Adding a Perfect Fluid We have seen that the no-go theorem on FLRW solutions in dRGT massive gravity continues to hold in the doubly-coupled theory if the only matter coupled to the effective metric is a perfect fluid whose energy density and pressure depend only on the scale factor. This complicates the question of computing dust-dominated or radiation-dominated solutions in massive gravity. One solution would be to treat the dust in terms of fundamental fields. Another would be to add an extra degree of freedom such as a scalar field. Its role is to introduce a lapse-dependent term into the Bianchi constraint (7.13) and thereby avoid the no-go theorem. It is this possibility which we study in this section. In Sect. 7.4 we examined the scalar-only case. Let us now include other matter components, such as dust 144 7 Cosmological Implications of Doubly-Coupled Massive Gravity or radiation, also coupled minimally to g^v. We assume that the density, pm, and pressure, pm, only depend on aeff .6 We can then write the total density and pressure as p = K + V + pm, p = K-V + pm, (7.28) so that P + P ~ (Pm + Pm) K = 2 V = P-P-• oo in Eq. (7.29), we find X2 aeff^oo m2M2 (ßfa - oifa) 2N2 2a30aeff > -7T-r-„-, (7-38) 2 At2 v(x)^»_AmMp, (7 39) We see that the scalar field slows to a halt: V (x) approaches a constant, while dx/dr, where dx = Nesdt is the proper time, approaches zero. Taking the late-time limit of the Friedmann equation (7.36), we obtain 146 7 Cosmological Implications of Doubly-Coupled Massive Gravity DC Aa5ß 25c~2~ aeff. (7.40) Because / approaches a constant Xc at late times, V = (dV/dx)\x=xc contributes a constant to the Friedmann equation. Therefore we find that Heff genetically blows up, which is potentially disastrous behaviour. This implies a violation of the null energy condition. Notice also that there is no guarantee that V = — f3\m2M^Ja3f3 is within the range of V(x), assuming the scalar field potential is not set by gravitational physics. This may lead to further pathologies, as the form of Via) would be inconsistent with large values of aes. As we discuss below, if V goes to 0 then, depending on the speed at which it does so, Heff may be better behaved. At early times, demanding the existence of a sensible radiation era leads to further problems. Assuming radiation couples to g^fv, then pm ~ a~f4 with pm = pm/3. We have, cf. Eq. (7.29), that 2K = p + p — (pm + pm), but, cf. Eq. (7.18), p and p do not have any terms scaling as steeply as a~ff. Therefore, in the presence of radiation, px and px pick up a negative term going as a~f4 to exactly cancel out pm and pm, leading to K < 0 at sufficiently early times. From Eq. (7.34) we see that this would lead to a negative H2ff, and hence to an imaginary Hubble rate. Equivalently, we can take the early-time limit of Eq. (7.36) to show so that again we see (for a real potential) Heff becoming imaginary. How could these conclusions be avoided? We can reproduce sensible behaviour, but only if the potential is extremely contrived. At early times, we would need to arrange the scalar's dynamics so that V -> oo "before" (i.e., at a later aeff then) K crosses zero.7 We would then reach the initial singularity, Heff -> oo, before the kinetic term turns negative.8 Moreover, we would need to tune the parameters of the theory so that K = 0 happens at extremely early times, specifically before radiation domination. At intermediate times, V would need to scale in a particular way to [through Eq. (7.36)] reproduce H2{{ ~ a~ff and H2{{ ~ a~f3 during the radiation-and matter-dominated eras, respectively. Finally, in order to have Heff -> const, at late times, we see from Eq. (7.40) that we would require V' to decay as a~^2. We can construct such a potential going backwards by setting Heff = HAqdm in Eq. (7.36), but there is no reason to expect such an artificial structure to arise from any fundamental theory. Even then we may still get pathological behaviour: Neff diverges if at some point i/effaeff = Ha, cf. Eq. (7.10). 7The other obvious possibility, having dV/daeff reach 0 before K does, is impossible given the forms of K(a) and V(a). 8This proposal has an interesting unexpected advantage: the Universe would begin at finite aes, so a UV completion of gravity might not be needed to describe the Big Bang in the matter sector. «eff^O (7.41) 7.6 Mixed Matter Couplings 147 7.6 Mixed Matter Couplings Before concluding, we briefly discuss a slightly different formulation which avoids some, but not all, of our conclusions. If we consider a scalar field and a perfect fluid, the avoidance of the no-go theorem on FLRW solutions only requires that the scalar field couple to g^jfv. In principle, all other matter could still couple to gliv. In fact, this is the theory that was studied in Ref. [1]. This theory violates the equivalence principle in the scalar sector, but is not a priori excluded, and will turn out to have slightly better cosmological behaviour. Moreover, there is a compelling theoretical reason to consider such couplings: matter loops would only generate a g-metric cosmological constant and would not destabilise the rest of the potential. However, the scalar field's energy would still contribute to the cosmological constant for g^l and hence to all of the interaction terms unless, for example, this was forbidden by some symmetry. A massless scalar would be better behaved in this sense, but as we have shown above, such a scalar field will not avoid the no-go theorem because after integrating the Klein-Gordon equation, the pressure loses its dependence on the lapse. Because the perfect fluid couples to gliv and we derived the Bianchi constraint (7.13) by taking the g-metric divergence of the Einstein equation, the constraint will now only contain px rather than the total pressure, i.e., This is the same constraint as in the scalar-only case discussed in Sect. 7.4, so the scalar's kinetic and potential energies have the same forms, Kia) and V(a), as in Eqs. (7.24) and (7.25). The physical Hubble rate is now H, which after solving for the lapse is determined by the equation9 where the q coefficients are defined in Eq. (7.20). Because the scalar field does not have to respond to matter to maintain a particular form of p(a) and p(a), we no longer have pathological behaviour in the early Universe, where there will be a standard a~4 evolution. Moreover, as was pointed out in Ref. [2], there is late-time acceleration: as pm -> 0, 3H2 -> m2(ft — (a//*) A) > which, if positive, leads to an accelerating expansion. However, these are not always self-accelerating solutions. We will demand two conditions for self-acceleration: that the late-time acceleration not be driven by a cosmological constant, and that it not be driven by V (x), both of which can easily be accomplished without modifying gravity. In other words, we would like the effective cosmological constant at late times to arise predominantly from the massive graviton. 9 9 9 • 9 ■ m MFla P(a)a = aPaeffpxa. (7.42) 9 A^m 9/ _1 _9 _^ \ 3H =—=- + m Co + 3c\a + 3c2a + c^a , (7.43) Using the transformations to the theory in footnote 5, we recover Eq. (5.9) of Ref. [1]. 148 7 Cosmological Implications of Doubly-Coupled Massive Gravity Let us start with the first criterion, the absence of a cosmological constant. Recall from Sect. 2.1.2 that we can write the dRGT interaction potential in terms of elementary symmetric polynomials of the eigenvalues of either X = yj1 g~x f or K = I — X, with the strengths of the interaction terms denoted by the by-now familiar ft in the first case and by an in the latter. What is notable is that «o # Po- the cosmological constant is not the same between these two parametrisations. Terms proportional to yj—g arise from the other interaction terms when transforming from one basis to the other. In bigravity there is a genuine ambiguity as to how one defines the cosmological constant, and throughout this thesis, because we are concerned with cosmological solutions, we have chosen to identify the cosmological constant with the constant term appearing in the Friedmann equation for the physical metric. In massive gravity with a Minkowski reference metric, however, the presence of a Poincare-invariant preferred metric allows for a more concrete definition of the cosmological constant.10 Consider expanding the metric as gMv = nnv + 2/v + hmhv^v. (7.44) This expansion is useful because the metric is quadratic in hliv but is fully nonlinear, i.e., we have not assumed that hliv is small [8]. In this language, the cosmological constant term, proportional to ^—g, can be eliminated by setting V. We can define a cosmological-constant-like piece solely due to the late-time behaviour of V given by A, aV /aeff\3 a„ff^oo m2 (_£E) (3a2_3a3+a4). (7.48) \ a J 3 5 LX 3M2 V a / 3 ^ Then Eq. (7.46) can simply be written in the form 2 2 m m Aeff = — (6a2 - 4a3 + a4) + Ax = —ft + Ax, (7.49) where in the last equality we mention that the residual term is nothing other than m2ft/3, which is simply a consistency check. The modifications to gravity induced by the graviton mass therefore lead to a constant contribution to the Friedmann equations at late times, encapsulated in m2ft/3 (with «o = «1 = 0). In a truly self-accelerating universe, this term should dominate Ax. If it did not, the acceleration would be partly caused by the scalar field, and one could get the same end result in a much simpler way with, e.g., quintessence. For generic values of an and for /3 ~ Oil), both of these contributions are of a similar size and will usually have the same sign. To ensure self-accelerating solutions, one could, for example, tune the coefficients so that 3«2 — 3«3 +«4 = 0 (the scalar field contributes nothing to Aeff) or 3«2 — 3«3 + a4 < 0 (the scalar field contributes negatively to Aeff), or take ft <$C 1 (the scalar field contributes negligibly to Aeff). 7.7 Summary of Results One can extend dRGT massive gravity by allowing matter to couple to an effective metric constructed out of both the dynamical and the reference metrics. The no-go theorem ruling out flat homogeneous and isotropic cosmologies in massive gravity [6] can be overcome when a scalar field is "doubly coupled" in such a way [1, 2]. We have shown that this result is, unusually, dependent on the use of a fundamental field, such as a scalar field in the aforementioned references, as the no-go theorem is only avoided when the pressure of the matter coupled to g^jfv depends on the cosmic lapse function. This lapse dependence is not present for the types of matter usually considered in late-time cosmological setups, such as radiation (p ~ a~^) and dust (p = 0), and therefore a universe containing only such matter will still run afoul of the no-go theorem. While this may not be a strong physical criterion—cosmological matter is still built out of fundamental fields—it presents a sharp practical problem in relating the theory to cosmological observations. Furthermore, if one uses a scalar field to avoid the no-go theorem, it cannot live on a flat potential and must be rolling. The latter consideration would seem to rule out the use of the Higgs field to unlock massive cosmologies, as we expect it to reside in its minimum cosmologically. 150 7 Cosmological Implications of Doubly-Coupled Massive Gravity Overall, in principle one can obtain observationally-sensible cosmologies in doubly-coupled massive gravity, but either a new degree of freedom must be included, such as a scalar field or some other matter source with a nontrivial pressure, or we must treat cosmological matter in terms of their constituent fields. Thus we cannot apply the standard techniques of late-time cosmology to this theory. We have further shown that if dust and radiation are doubly coupled as well— which is necessary if we demand the scalar obey the equivalence principle—then the cosmologies genetically are unable to reproduce a viable radiation-dominated era, and in the far future the Hubble rate diverges, rather than settling to a constant and producing a late-time accelerated expansion. These pathologies can only be avoided if the scalar field potential is highly contrived with tuned theory parameters, or dust and radiation do not doubly couple. In the latter case, there is generically late-time acceleration, but for much of the parameter space, this is in large part driven by the potential of the scalar field. In those cases the modification to general relativity may not be especially well motivated by cosmological concerns. Otherwise, the parameters of the theory need to be tuned to ensure that the theory truly self-accelerates. It seems that dRGT massive gravity only has viable cosmological solutions— i.e., that evade the no-go theorems on existence [6] and stability [9]—if one either includes a scalar field or some other "exotic" matter with a lapse-dependent pressure (or possibly a pressure depending on ä) and couples it to the effective metric proposed in Ref. [1] or goes beyond the perfect-fluid description of matter. Even if one includes a new scalar degree of freedom, significant pathologies arise if normal matter couples to the same effective metric. In all setups, the need for descriptions beyond a simple perfect fluid makes this theory unappealing from an observational standpoint. We end with three small caveats. Notice that we have assumed that in unitary gauge for the Sttickelberg fields, i.e., choosing coordinates such that r\iiv = diag(—1, 1, 1, 1), the metric has the usual FLRW form (7.4). However, that form is arrived at by taking coordinate transformations of a more general homogeneous and isotropic metric, so that assumption may be overly restrictive.12 Equivalently, one could consider a more general, inhomogeneous and/or anisotropic, gauge for the Sttickelberg fields. We also note that if this theory does possess a ghost, even with a mass above the strong coupling scale, solutions to the nonlinear equations of motion could contain the ghost mode and therefore not be physical.13 However, a Hamiltonian analysis showed that the ghost does not appear around FLRW backgrounds [1], suggesting that we have studied the correct cosmological solutions to any underlying ghost-free theory. Finally, as discussed in Chap. 6, if one simply gives dynamics to the reference metric, we end up with a theory of doubly-coupled bigravity which treats the two metrics on completely equal footing and has been shown to produce observationally viable cosmologies. We thank Fawad Hassan for pointing this out to us. We thank Angnis Schmidt-May for discussions on this point. References 151 References 1. C. de Rham, L. Heisenberg, R.H. Ribeiro, On couplings to matter in massive (bi-)gravity. Class. Quantum Gravity 32, 035022 (2015). arXiv:1408.1678 2. A.E. Gumrukcuoglu, L. Heisenberg, S. Mukohyama, Cosmological perturbations in massive gravity with doubly coupled matter. JCAP 1502(02), 022 (2015). arXiv: 1409.7260 3. C. de Rham, L. Heisenberg, R.H. Ribeiro, Ghosts and matter couplings in massive gravity, bigravity and multigravity. Phys. Rev. D90(12), 124042 (2014). arXiv: 1409.3834 4. S. Hassan, M. Kocic, A. Schmidt-May, Absence of ghost in a new bimetric-matter coupling. arXiv: 1409.1909 5. A. Schmidt-May, Mass eigenstates in bimetric theory with ghost-free matter coupling. JCAP 1501, 039 (2014). arXiv: 1409.3146 6. G. D'Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava et al., Massive cosmologies. Phys. Rev. D 84, 124046 (2011). arXiv: 1108.5231 7. S. Hassan, R.A. Rosen, On non-linear actions for massive gravity. JHEP 1107, 009 (2011). arXiv: 1103.6055 8. C. de Rham, Massive gravity. Living Rev. Rel. 17, 7 (2014). arXiv:1401.4173 9. A. De Felice, A.E. Gumrukcuoglu, C. Lin, S. Mukohyama, Nonlinear stability of cosmological solutions in massive gravity. JCAP 1305, 035 (2013). arXiv: 1303.4154 Part II Lorentz Violation Einstein was a giant. His head was in the clouds, but his feet were on the ground. But those of us who are not that tall have to choose! Richard Feynman Chapter 8 Lorentz Violation During Inflation The only thing that really worried me was the ether. There is nothing in the world more helpless and irresponsible and depraved than a man in the depths of an ether binge. And I knew we'd get into that rotten stuff pretty soon. Probably at the next gas station. Hunter S. Thompson, Fear and Loathing in Las Vegas For this final chapter, we move to the early Universe to ask what constraints we can put on the violation of Lorentz invariance during inflation. As discussed in Sect. 2.2, we can use Einstein-aether theory (ae-theory) [1, 2] to model Lorentz violation in the boost sector, i.e., while maintaining rotational invariance on spatial hypersurfaces, at low energies. In ae-theory, Lorentz invariance is spontaneously broken by the presence of a vector field nonminimally coupled to gravity. A Lagrange multiplier enforces the constraint that this vector, sometimes called the aether and denoted here by mm, be timelike and have fixed norm, u^u^ = -m2, (8.1) where m is a free parameter with mass dimension 1. This forces the aether to acquire a nonzero vacuum expectation value (VEV) at every point in spacetime, so at every point the aether picks out a timelike direction and hence defines a preferred reference frame. Note that ae-theory is a vector-tensor theory of gravity and is, at the level of the theory, completely Lorentz invariant. It is the nontrivial constraint which ensures that Lorentz invariance, and specifically boost invariance, is always broken at the level of the solutions. ^E-theory corresponds, for example, to the low-energy limit of Hořava-Lif schitz gravity, a well-studied candidate UV completion of general relativity which breaks the symmetry between time and space coordinates directly at the level of the action [3]. In Sect. 2.2.2 we considered a generalisation of ae-theory in which a scalar field, ) = (0, 0), because if it diverges there the aether-scalar stress-energy tensor (2.81) will be nonzero and we cannot have a Minkowski solution. We will also assume that V(0, 0) is either vanishing or negligibly small; if not, then this contributes a cosmological constant term to the stress-energy tensor, and our background is (anti-)de Sitter rather than flat. Observations constrain such a term, barring a nonlinear screening mechanism, to be very small.1 In flat space the field equations are solved by a constant-field configuration, The scalar field is canonical, coupled minimally to gravity, and not coupled at all to the matter sector, so we would not expect any screening mechanisms to be present in this theory. 158 8 Lorentz Violation During Inflation = (m, 0,0,0), (8.3) X = 0, (8.4) 0 = 0. (8.5) We introduce small perturbations, 8X, 8(dßv^) (8.11) whose variation yields the equations of motion of the perturbed variables. From here we drop the (0, 0) evaluation on the derivatives of the potential (although they remain implicit). The 8X equation of motion is ü% = 0. (8.12) It constrains the timelike component of the aether perturbation to vanish, v° = 0. (8.13) Inserting this result into Eq. (8.11) and splitting vl into spin-0 and spin-1 fields2 as 2The aether perturbation is in a reducible subgroup of SO(3), so by decomposing vl like this we single out the real dynamical degrees of freedom. Note also that, here and throughout this chapter, we will refer to scalar and vector modes of the aether as spin-0 and spin-1, respectively, so as not to confuse them with the scalar field

0 (8.18) from the start. This was derived in pure ae-theory from requiring positivity of the quantum Hamiltonian for both the spin-0 and spin-1 fields [8], and is suggested by the fact that for c\ < 0 the kinetic terms for Sl and A77 in Eqs.(8.16) and (8.17) have the wrong sign. Since this was proven to be true for the spin-1 perturbations in ae-theory and they remain unchanged in this extension of it, this condition on c\ must continue to hold. Finally, we can vary the action with respect to our three perturbation variables— S', Nl, and 88l}dj8 = 0, (8.19) ci 2ci N{ - d2Nl = 0, (8.20) □50 - V>050 - Vg^diS1 = 0. (8.21) In ae-theory, 4> = 0 = V (9, 4>) and both aether equations are simply wave equations with plane wave solutions [8], 160 8 Lorentz Violation During Inflation S'(k) oc e-^kt+ihx^ (822) N'(k) oc ß-^kt+ilx^ (823) with the propagation speeds for the spin-0 and spin-1 perturbations given by 2 = (8.24) c5(1)2 = 1. (8.25) The scalar coupling modifies the ae-theory situation in two ways. First, c\23 is shifted by ^ Vqq evaluated at (0 = 0, 4> = 0) (remember that implicitly we are evaluating all the derivatives of V there, so they are just constants). This is to be expected: the expansion of the potential around (0, 0) includes, at second order, the term \Vqq8Q2 = ±V^(9^')2, which can be absorbed into the c2 term in the (quadratic) Lagrangian by redefining c2 -> c2 + \ Vqq . We will find this same redefinition of c2 appears in the cosmological perturbation theory. The second change from ae-theory is more significant for the dynamics. When Vqq is nonzero—i.e., when the coupling between mm and

8 = 0, (8.26) 2ci Sif> + (k2 + Vqq)84> - Ve4>k2r = 0. (8.27) This system can be diagonalised3 by defining r = r + ——^-j-8^, (8.28) 2ci {k2 + V2^ - co2_) v &4> = &4>+ ^9£*_ V, (8.29) where the o>± are defined by 3We thank the referee of Ref. [12] for this suggestion, which simplifies an earlier version of the calculation while obtaining the same result. 8.1 Stability Constraint in Flat Space 161 M - *2 (i + 0 where the two fields decouple, &>2 goes to k2 + V^, the squared frequency of a 8

2_ goes to c2k2, the equivalent for "V modes. We see that "V and 8

± to be real, so that the solutions to Eqs.(8.31) and (8.32) are plane waves rather than growing and decaying exponentials. Note that co+ is manifestly real, so the 8

) = (0, 0) evaluation which has been implicit. Equation (8.34) constrains the coupling between the aether and the scalar field in terms of the aether kinetic-term free parameters (or, equivalently, its no-coupling propagation speed) and the effective mass of the scalar in flat space. It agrees with the spin-0 stability constraint in Ref. [4] which was derived in a slightly different fashion for a specific form of V(6, 0,6 which is the combined constraint from subluminal propagation and positivity of the Hamiltonian of the spin-0 field in pure se-theory [8]. Technically, the mixed aether-scalar modes which become arbitrarily close to the aether perturbations in the limit Vq§ —> 0. 5 Our notation is different than that used in Ref. [4] and as a result their constraint looks slightly different. They define the aether to be dimensionless and unit norm while we give it a norm m with mass dimensions. To compensate for this, their c, are l6nGm2Ci in our notation. We have checked, translating between the two notations, that our constraint matches theirs. 6Assuming that the scalar field is nontachyonic. Vgl < 2clC?V(k2 + V**), (8.33) VliO, 0)< 2^23^(0,0), (8.34) 162 8 Lorentz Violation During Inflation 8.2 Cosmological Perturbation Theory The goal of this chapter is to explore the impact of the coupling between 4> and wM on small perturbations to a homogeneous and isotropic cosmology. We will be particularly interested in a period of slow-roll inflation driven by j-jrBi, r9. = + 2*) + - 2Jf8) + 8ikB[jM + y'8) + H'T), 8.2 Cosmological Perturbation Theory 163 rjk = jeB'Sjk + V,k8) + * j8i - + HT)k + HTkj - HTjkJ, (838) where primes denote derivatives with respect to x. We do not reproduce the components of the Einstein tensor here; they can be found in the literature [14]. The aether in the background has only u° = —. Imposing the constant-norm constraint, u^u11 = —m, to first order the aether is given by 171 = — ((1 - O), V1), (8.39) a where V1 is the spatial perturbation to the aether. With lowered indices we have M/i = ma(-(l + 4>),V;--5I-). (8.40) Taking the divergence of Eq. (8.39) we can find the linearised expansion, 9 = — [3jr(1 - O) + 3*' + V'j]. (8.41) Finally, the scalar field

= 4> + &4>, (8.42) where ) = v + v08e + v^ + x- [vee8e2 + v^2 + 2ve4,ses] + o(se3), (8.43) where, per Eq. (8.41), the linear piece of the expansion is given by 8e = - (3*' - 3Jf O + V1 A . (8.44) a In deriving the linearised field equations we will need V (e, + 6>(S02), v^e,0) = y0 + v^h + Ve^e + oise2), Veeie, 0) = Vee + Veee&O + + 6>(502), Vg^O, 0) = Vg

&4> + O(80)2. (8.45) The linearised equations of motion in real space are given in Appendix D. However, the symmetries of the FRW background allow us to decompose the perturbations into spin-0, spin-1, and spin-2 components [15]. In particular, because the background variables (including the aether, which points only in the time direction) do not break the SO(3) symmetry on spatial slices, these components conveniently decouple from each other. Hence we perform this decomposition both to isolate the fundamental degrees of freedom and to make close contact with the rest of the literature on cosmological perturbation theory. We decompose the variables as 50 = ^5#F(O), (8.46) k O = ^OfeF(0), (8.47) k V = ^VkY(0\ (8.48) k V' =X X Vk(±m)Yi(±m\ (8.49) k m=0,l Bl =Y, X flfm)r'(±m), (8.50) k m=0,l ht =S X Hnm)YiJ(±m)> (8-51) k m=0,l,2 where F(0\ etc., are eigenmodes of the Laplace-Beltrami operator, 32 + k2 (see Refs. [8, 14] for the forms of these mode functions and some of their useful properties). From here on, we will drop the k subscripts. The spin-0, spin-1, and spin-2 perturbation equations can then be found by plugging these expansions into the real space equations listed in Appendix D. 8.3 Spin-1 Cosmological Perturbations We begin our analysis by focusing on the spin-1 perturbations. The spin-2 perturbations are unmodified by the aether-scalar coupling because V(0,^)]' + i(ci - c3)k2(B^ - 8T (±i) = 2 (^)2ci3 {a-2 [a2(-*v(±1) + //f1')]'} (±1) (8.54) (8.55) where a = (cb + 3c2)m2 is defined in Eq. (2.75). As a consistency check, these expressions reduce to those found in the literature for a scalar field uncoupled to the aether [16] (setting V(6, 0) = V()) and for ae-theory [8] (setting V(6, 0) = aO2, with c2 -> c2 + a). For convenience, from here on we will absorb \ Vqq into c2 and indicate the change with a tilde, e.g., á = ^ ci + 3c2 + c3 + m (8.56) 166 8 Lorentz Violation During Inflation and similarly for quantities like Gc. While this is convenient notation we should remember that Vgg and hence all tilded quantities are not necessarily constant, although they are nearly so during a slow-roll phase.7 We should first note that due to its direct coupling to the aether, the scalar field does source spin-1 perturbations, which is impossible in the uncoupled case as the scalar field itself contains no spin-1 piece. In pure ae-theory the spin-1 perturbations decay away as a~l [8]. We may wonder if the scalar-vector coupling can counteract this and generate a nondecaying spin-1 spectrum. Using the gauge freedom in the spin-1 Einstein equations, we choose to work in a gauge where H^1^ = 0; that is, we foliate spacetime with shear-free hypersurfaces. The i-j Einstein equation in the spin-1 case is unmodified from the ae-theory case [8] and gives a constraint relating the shift 5(±1^ and the spin-1 aether perturbation B<±i) = J,y<±i), (8.57) where Y = l67TGm2c13. (8.58) It is tempting to notice the similarities between the v = i aether field equation (8.52) and the CM Einstein equation (8.54), but this is just hinting at the underlying redundancy between the two equations. Indeed, using Eq. (8.52) to eliminate the scalar field term in Eq. (8.54) just leaves us with an identity. This is because, due to the constraint equation (8.57), the two perturbations B and V are related, and hence (by the Bianchi identities) these two equations have to contain the same content. We choose to use the CM Einstein equation to derive our equation of motion for the spin-1 perturbations. In this equation, the scalar field couples to the vector perturbations of the aether and the metric via ^Vg^cj)'. In the quadratically-coupled potential of Donnelly and Jacobson, which we discuss in detail in Sect. 8.5, the coupling Vg^ is exactly constant. In general, we will take Vg^ to be constant to first order in slow roll. Inserting the constraint into the CM Einstein equation we find 2—je2 —-— + c1—\ y(±1) mz mz a a ) + I |~(C1 _ C3) + _£!!_] /fc2y(±i) + Cl(2^y(±i)' + y(±D") 2 [ 1 - Y \ --2J^2 ) + -V^' a ) m Following Ref. [8], we define £ = aF(±1' to eliminate the first-derivative terms, so Eq. (8.59) becomes 7 A nonconstant Vqq requires cubic or higher order terms in the potential. For the quadratic Donnelly-Jacobson potential discussed in Sect. 8.5, Vgg is constant and can be freely set to zero by absorbing it into C2- 8.3 Spin-1 Cosmological Perturbations 167 \ mLc\ 2 mc\ J (8.60) where the no-coupling sound speed c^1-1 is the de Sitter propagation speed of the spin-1 aether and metric perturbations when the coupling to the scalar field is absent [8], ,(±1)2 _ 1 (1 - c3/c1) + 1 + c3/c1 l-y The background quantity A is defined by (8.61) A = 2Jf2 a = cTC — ift (8.62) and vanishes in exact de Sitter space. 8.3.1 Slow-Roll Limit The equation of motion (8.60) for the spin-1 aether and metric perturbations is difficult to solve in full generality. It was solved in pure de Sitter space (A = 0) in se-theory (i.e., in the absence of the scalar field) in Ref. [8]. In that limit, Eq. (8.60) is a wave equation with real frequency, so £ was found to be oscillatory. Therefore, in se-theory the spin-1 shift perturbation, 5(±1) = yi- /a, decays exponentially,8 leaving the post-inflationary universe devoid of spin-1 perturbations. To investigate whether the inflaton coupling term will change this conclusion, let us solve Eq. (8.60) in the slow-roll limit. We define the slow-roll parameters, s and rj, in the usual way, H J0" ' 7T 1 — <8'63) "= ~k = ^ (8'64) where for completeness we have included both the cosmic-time and conformal-time definitions. Slow-roll inflation occurs whenever s, rj <$C 1. In this limit, both parameters are constant at first order and we can find 8Here and in the rest of this chapter, "exponential" growth or decay should be taken to mean exponential in cosmic time, or as a power law in conformal time. 168 8 Lorentz Violation During Inflation ^(l+e), (8.65) Hx M'xi--(l + e). (8.66) x Taking conformal-time derivatives we can calculate the background quantity denned above, A = ^2-^' «-?r. (8.67) x Note that during slow roll, H is approximately constant but Jti? is not; therefore, even though we are working in conformal time, we will often choose to write the equations of motion and their solutions in terms of H, treating it as a free parameter which measures the energy scale of inflation. Using these relations, as well as the Klein-Gordon equation in the slow-roll limit and the fact that J$? = aH,we can write the £ equation of motion to first order in slow roll as /"-um 1 1 / a 1 1 + 2s - - \ r, + c ±1)2^ + - —e +---z^-VeM H = 0(e2)- x \m1c\ bmc\ Hi ) (8.68) V(6, 2. We proceed to demonstrate that just an instability arises. 8.3.2 Full Solution for the Vector Modes Noticing the similarity between Eq. (8.69) and the usual Mukhanov-Sasaki equation [16], which has solutions in terms of Bessel functions, we change variables to g = x~ll2^ with x = —c^^kx to recast Eq. (8.69) as Bessel's equation for g(x), x2^+x^ + (x2-v2)g = 0, (8.73) dxL ax with the order v given by v = - + A. (8.74) Depending on the sign and magnitude of A, the order v can be real or imaginary. We will find it convenient to write the general solution in terms of the Hankel functions as [akH^i-c^kx) + t3kHl2\-c^kx)]. (8.75) To determine the values of the Bogoliubov coefficients, and ft, we need to match this solution in the subhorizon limit, —c^^kx —>• oo, to the quantum vacuum state of the aether perturbations in flat space. This is desirable because we can assume that, at such short wavelengths, these modes do not "see" the cosmic expansion. In Sect. IV.B of Ref. [8] the quantum mode functions for the aether perturbation vl were demonstrated to satisfy Nk = —^=e-[kt. (8.76) ^V\ci\k This function is related to £ by Nk = = ^r£. The mode Nk is defined in Minkowski space, where a = 1 with t = x, so we only need to modify it by a factor of m to obtain £. Using the asymptotic formula 170 8 Lorentz Violation During Inflation lim H^2\-c^kr) = M , 1 gTJ^kr+S) (877) (iDfe^oo V 7t / r±l). with 5 = ^ (v + 1 /2), we find that in the subhorizon limit, % -> . a*e~,(c' *T+d) + ßke+,{c' kT+d} . (8.78) Matching to Eq. (8.76), and ignoring the unimportant phase factors e±lS, we see that we need a* = J-J^—r 4m y \c\ | fa = 0, (8.80) where we have (consistently) put in some factors of c^1-1 which do not appear in the flat-space calculation because it ignores gravity, but would have appeared if we had included gravity.9 Substituting in this value of ak, we find the full solution for the spin-1 perturbation V™ = iA/lr4rf^Z7^(1)(-^(±1)*r). (8.81) As a consistency check, if we turn off the scalar-aether coupling, we have v = 1/2, and (up to an irrelevant phase of —tt/2) we recover Eq. (91) in Ref. [8]. 8.3.3 Tachyonic Instability On superhorizon scales, the Hankel functions behave as lim H?\-cWkT) = ir(v) (^V^VV - (8.82) Plugging this into Eq. (8.81), we see that the large-scale vector perturbations to the aether and metric depend on time as yTo see this, consider Eq. (8.60) in the case a = 1, which is the spin-1 perturbation equation in flat space with gravitational perturbations turned on. Since this requires

2. In this case the coupling is large, but with the opposite sign to the previous case, and this large coupling drives runaway production of aether modes. This is precisely the tachyonic instability we anticipated in Sect. 8.3.1, as it results from the aether perturbations acquiring an imaginary effective mass. Since this growth is exponential (in cosmic time, or in number of e-folds), it seems quite probable that this growing vector mode will overwhelm the slow-roll background solution and therefore lead to an instability. In this subsection we will calculate the growth of a single vector mode and compare it to the background evolution. In order to maintain a homogeneous and isotropic background spacetime, the time-space term in the stress-energy tensor must be zero at the level of the background (f°i = 0). The spin-1 perturbations do contribute to these terms in the stress-energy tensor (8.54) through terms proportional to V^1^ In particular, we will focus on the scalar-aether coupling term T°i,k = ■■■ +mVe^Vk(±1)Y^1) + ■■■ , (8.84) which we will write as T^DmVe^vt1^. (8.85) While we focus on this term for simplicity, we note that there are many other terms in T°i which receive contributions from the vector modes, and some may even be larger than the term in Eq. (8.85). Our strategy will be to focus on a single mode, picking one of the larger modes available to us. Because Vk grows with decreasing k, we choose a mode which crosses the sound horizon at some early conformal time t, . Such a mode has wave number k= l1} ■ (8.86) Recall that, during inflation, r runs from — oo to 0. 172 8 Lorentz Violation During Inflation The perturbation Vfe(±1'1 is given by Eq. (8.81), which for a superhorizon perturbation becomes vf 1}(A0 = HV(v)2v (-t,-)5 e^N, (8.87) lit \m+J\c\ I where iV is the number of e-folds after the mode crossed the sound horizon. The mode function is given by [8] F.J1}(x) = -i- [(k x nj ^ i (k x nj ] eil*, (8.88) where n is a unit vector orthogonal to k. We can always choose three orthogonal coordinates such that kl = k8l i and nl = 8l2, so the mode function is V2 ik '(3c) = -^relK-x8*i. (8.89) This oscillates throughout space; we will choose x such that Re [(i ± l)e^] has its maximum value of 1. (The other terms in VJf^^Y^^ are all manifestly real.) Therefore, this particular mode has a contribution to the 0-i component of the stress-energy tensor which includes a term T\k D -mV0 3/2 (the condition we found above for Vfe(±1'1 to grow exponentially in cosmic time). While the derivatives of the potential in the numerator of Eq. (8.93) should be a few orders 8.3 Spin-1 Cosmological Perturbations 173 of magnitude smaller than the potential in the denominator due to slow roll, this is likely to be dwarfed by the exponential dependence on the number of e-folds, which even for the bare minimum length of inflation, ./V ~ 50-60, will be very large. Moreover, as we will see in Sect. 8.5, v can in principle be larger than 3/2 even by several orders of magnitude, hence the other terms with exponential dependence on v, as well as the gamma function, can be quite large as well. Therefore, when v > 3/2 the vector modes will generically drive the off-diagonal term in the stress-energy tensor far above the background density. This does not necessarily mean that isotropy is violated. As discussed in Sect. 8.4, the same physical process that drives VJf1^ will similarly pump energy into the spin-0 piece, V®\ which affects the perturbations to To as well as T°i. Consequently, background homogeneity and isotropy could still hold, but the slow-roll solution to the background Friedmann equations which we perturbed would be invalid. Either way our inflationary background becomes dominated by the perturbations. Note that this calculation was done for a single mode, albeit one of the largest ones available because Vk grows for smaller k. Integrating over all modes produced during inflation would of course exacerbate the instability. We will explore this instability in greater quantitative detail in Sect. 8.5, where we examine a specific potential for which we can elucidate the constraints on and V*0. 8.3.4 What Values Do We Expect for A? V(±1) has an effective mass-squared (8.72) which depends on both the theory's free parameters and derivatives of the scalar potential, and can be of either sign. When it is negative, the aether modes are tachyonic and y(±1) contains an exponentially growing mode. This occurs when the parameter A, defined in Eq. (8.70), satisfies A > 2. To leading order in the slow-roll parameters, A is written in terms of several free parameters: c\,m, H, and the potential derivatives Vg^ and V^, A = -7^% + ^)- (8-94) bmc\H J Hence A can span a fairly large range of orders of magnitude. However, there are several existing constraints on these parameters, most of which constrain several of them in terms of each other. There are two things we can do to clarify our expression for A. We generally expect that for a slow-roll phase, 4> <$C 3H

2. Let us first examine Eq. (8.100) to see the conditions under which it is positive. Most of the terms are manifestly positive. Positivity of the Hamiltonian for spin-1 perturbations in flat space requires c\ > 0 [8].11 Tachyonic stability of the scalar requires Mq to be real and positive. The timelike constraint on the aether requires that m be positive as well. Putting this all together, we find Vet 1/7 1/7 Mq ( m \~l „ A = sgn(^T/ mm K ° ~r= IT (XT) + (8-101) >0 >0 >0 >0 implying that in order for a to be positive,

) = ^M2(f>2 + /u,94> which we discuss in Sect. 8.5, both of these are constant, although one could construct inflationary potentials for which this is not true. The effective mass of the scalar during inflation, _ 1/2 M = , should be less than the Hubble rate in order to produce perturbations. Putting all this together, we are left with A = sgn(0) cr1'2 -^L - ( J1L. ) +^(e). (8.102) We can see that in order for A to be larger than 2, the aether VEV, m, needs to be at least a few orders of magnitude smaller than the Planck scale, m is effectively the Lorentz symmetry-breaking mass scale. It can therefore be quite a bit smaller than the Planck mass, although if it were below the scale of collider experiments, any couplings to matter could displace the aether from its VEV and Lorentz-violating effects could be visible. There are several experimental and observational results suggesting that m/MP\ should be quite small. Here we briefly discuss three strong constraints, arising from big bang nucleosynthesis, solar-system tests, and the absence of gravitational Cerenkov radiation, as well as a possible caveat. As mentioned in Sect. 2.2, the gravitational constant appearing in the Friedmann equations, Gc, and the gravitational constant appearing in the Newtonian limit, G#, are both displaced from the "bare" gravitational constant, G, by a factor that is, schematically, 1 + q(m/MPi)2. The primordial abundances of light elements such as helium and deuterium probe the cosmic expansion rate during big bang nucleosynthesis, which depends on Gc through the Friedmann equations. Therefore, by comparing this to G^ measured on Earth and in the solar system, c,m2 can be constrained. Assuming the q are &{X)}2 the BBN constraint implies m/MPi < 10_1 [7]. Slightly better constraints on Gc/G^ come from the cosmic microwave background (CMB) [19, 20]. The tightest bound, \GN/GC - 1| < 0.018 at 95% con- 12As the Ci are dimensionless parameters, this is perfectly reasonable. Note that even if m were order Mpi or larger and the constraints discussed in here are actually constraints on the smallness — 1/2 of the ci, A still depends on these parameters as Cj . 176 8 Lorentz Violation During Inflation fidence level, was computed using CMB data (WMAP7 and SPT) and the galaxy power spectrum (WiggleZ) in a theory closely related to the one described in this chapter, and should hold generally for ae-theory at the order-of-magnitude level [11]. These constrain m/MPi to be no greater than a few percent. There are yet stronger bounds on m/MPi through constraints on the preferred-frame parameters, a^2, in the parametrised post-Newtonian (PPN) formalism. These coefficients scale, to leading order, as q(m/MPi)2 [2, 21]. The observational bounds a\ < 10~4 and a2 < 4 x 10~7 therefore imply m/MPi <6x 10~4. Recent pulsar constraints on are even stronger than this [22], although they are derived in the strong-field regime and thus might not be directly applicable to the weak-field ae-theory results. Similarly, recent binary pulsar constraints on Lorentz violation [23] constrain m/MPi < 10_1, assuming q ~ The strongest constraints come from the absence of "gravitational Cerenkov radiation." Because the aether changes the permeability of the vacuum, coupled aether-graviton modes may travel subluminally, despite being nominally massless. Consequently, high-energy particles moving at greater speeds can emit these massless particles, in analogy to the usual Cerenkov radiation. This emission causes high-energy particles to lose energy, and at an increasing rate for higher-energy particles. Among the highest-energy particles known are cosmic rays, which travel astronomical distances and hence could degrade drastically due to such gravitational Cerenkov effects. Such a degradation has, however, not been observed; this generically constrains m/Mpi < 3 x 10"8 [24]. We should note that these constraints can be side-stepped if certain convenient exact relationships hold among the q, although crucially they cannot all be avoided in this way simultaneously without allowing for superluminal propagation of the aether modes [2]. The PPN parameters are identically zero when c3 = 0 and 2c\ = —3c2. The BBN constraint is automatically satisfied by requiring 2ci+3c2+c3 to vanish, as this sets Gc = [7]. Note that the PPN cancellations imply the BBN cancellation, though the reverse is not necessarily true.13 The Cerenkov constraints vanish if all five dynamical gravitational (metric and aether) degrees of freedom propagate exactly luminally. This happens when c3 = —c\ and c2 = c\/{\ — 2c\) [24]. Note that while a2 = 0 in this parameter subspace, a\ = —8ci(m/MPi)2, which would place a constraint on m/MPi of order 10~2. It is worth mentioning that the Cerenkov constraints on m will also be avoided if the mode speeds for some of the aether-me trie modes are superluminal. This includes a two-dimensional parameter subspace in which the PPN and BBN constraints are automatically satisfied [2]. Whether superluminal propagation is acceptable in ae-theory is somewhat controversial. It is a metric theory of gravity, so superluminality should imply violations of causality, including propagation of energy around closed timelike curves [8, 24]. The conditions for PPN and BBN to cancel can be relaxed by including a ca, term which describes a quartic aether self-interaction. We have ignored such a term in order to simplify the theory, although like the other three terms, it is permitted when that the aether equations of motion are demanded to be second order in derivatives. When c4 7^ 0, the vanishing of continues to imply that the BBN constraints are satisfied. 8.3 Spin-1 Cosmological Perturbations 177 However, this may be seen as an a posteriori demand, and some authors (see, e.g., Ref. [2]) do not require it. It is unclear what fundamental physical principle, if any, would cause the q to cancel in any of the aforementioned ways. Hence it seems to be a fairly general result that m must be several orders of magnitude below the Planck scale. If m/MPi is small enough compared to M/H and the other small parameters appearing in Eq. (8.102), A can easily be above 2 and the aether-inflaton coupling runs a serious danger of causing an instability. For a given m/MPi, this places a constraint on the size of the coupling V#0. We will discuss this constraint more quantitatively in Sect. 8.5 for a specific choice of the potential. 8.4 Spin-0 Cosmological Perturbations: Instability and Observability Let us now consider the spin-0 perturbations. Before getting bogged down in calcula-tional details, we first summarise this section. The spin-0 equations are complicated by the addition of 8

'&' - a2%84>) = (3Jf2 - A) O - 3JfV---k2^ - %nGccim2k2<$> G + %TtGccxm2k(y' + JtfV) - %TiGca3tf>kV + 4TTGcmaVe' - 3JT<5) (8.103) (kJf - kV') = -)' 8ttG 2 t" i - cim2— + -maVg^'V (8.104) a 2 AjtGc (' - a2%S^)) = (3J?2 - A) O + Ji?®' - 2JfV - *" - 87rGgOT g123fc2(0 + *) + 4ttGc — AVeee (3*' - 3JfO + kV). a - AnGcma \VB{3,^H + &') + Vg^'&4>] + 4nGcm2VeeiP [3AS - 0'(3*' - 3JfO + kV)]. (8.105) The off-diagonal i-j Einstein equation, unmodified by the coupling between the aether and scalar, gives a constraint, /t2(0 + *) = ya-2(a2kVY, (8.106) where y = \67tGm2c\3 was defined in Sect. 8.3. We may eliminate * and its derivatives by the constraint (8.106) and its conformal-time derivatives, = y{ak)~x (£" - At-) - <&', (8.107) = yiaky1 - 34? - A% + A$ ^JT - -^j - 0), as well as 8T[1V and the ^-equation of motion in ae-theory [8] in the limit V(0,) -> V(sj>). 8.4.2 The Instability Returns To lowest order in m/MPi, the constraint equation (8.106) tells us simply that the anisotropic stress vanishes: * = — O. Taking this into account, the 0-i Einstein equation at lowest order in m/MPi is (aO)' =AitGa4)'84). (8.110) The v = i aether equation of motion (8.109) is, dropping terms of Gim1 /Mp^), a2Vd(l)k where 2mc\ \ c\tnL J 2mc\ (0)2 = -125- = j2± lí + ďl \\ (8112) Clm2 + ay a \ \M2J J is the same spin-0 sound speed as in flat space (cf. Sect. 8.1) to first order in m/MPi. In de Sitter space this becomes, using Eq. (8.110) to replace (aO)' with 8 where C± are arbitrary constants, and \ c\inL) m c\ MP1 V0 2, for the spin-1 modes to be unstable. The real condition for instability may be slightly different, as A > 2 could violate our assumption that m/MPi is small; however, the additional ^>(m2/Mp1) terms would only change some multiplicative factors, and not by orders of magnitude. As in the spin-1 case, we can most easily see the effect of unstable aether modes on the metric perturbations through the off-diagonal i-j Einstein equation (8.106). If V blows up exponentially then so will O + and the metric perturbations will overwhelm the FLRW background. 8.4.3 The Small-Coupling Limit From here on we will assume that the aether perturbations are stable, so that A = Vh<^ , < 2. (8.117) 2mc\Hl This can be further split into two dominant cases, |A| <$C 1 and A < —1/4. There are regions in parameter space which are not covered by these cases, such as A ~ 1, but these are likely to be highly fine-tuned as many of the parameters which enter A 14The aether coupling will still enter the perturbed Klein-Gordon equation at this order through the potential terms. 8.4 Spin-0 Cosmological Perturbations: Instability and Observability 181 have no relationship to each other a priori. Consequently we should consider various values of A on an order-of-magnitude basis. | A | <$C 1 corresponds to the limit where the coupling | Vg^ | is small compared to the mass scale c\mH2/ 0 does smoothly go to ae-theory. The aether perturbation £ only appears, to &(m/Mp{), in the 0-i Einstein equation, O' + ^fO =AitG4)' ^S + ^j^^j ■ (8.119) The Vg

0 and substituting in the solution (8.114), this becomes m2 ( a \ -^-Cl 1 + -T Afp! V cirn1) (8.120) Mi The corrections enter at &(m2/Mpl) and are negligible for the purposes of this analysis. Therefore the limit | A| <$C 1 should only differ from ae-theory at &{m2/Mpj) < lO"15. It is worth mentioning that for small but finite A there will be new effects on extremely large scales, k < Vg^. These may or may not be observable, depending on the scales covered during inflation. 8.4.4 The Large-Coupling Limit: The 4> Evolution Equation One interesting case is left: a large coupling with opposite sign to 0, or A < 1 /4. We will consider this for the rest of this section. However, we should mention that the sign of '+ maVe4>)84>. (8.123) It is an interesting result that we can write the CM Einstein equation in geometrical terms as <&' + O = A8cj)/4>' (8.124) to both zeroth and first order in 0(m/Mp{). This does not hold, however, to higher orders, and might not hold away from quasi-de Sitter space or on subhorizon scales. Next we solve the metric perturbation f?' + 2 (3JT2 - A) O + AitGma V6 - 6JT8$ - 8(/)') - Ait Gma Vewft 8 + ... (8.125) where we have dropped terms of Gim1 /MpX) and higher. Equation (8.125) becomes an evolution equation for '). (8.126) Taking the conformal-time derivative, we find (dropping &{m2/Mpj) terms, as we do throughout) that (8.121) (8.122) 8.4 Spin-0 Cosmological Perturbations: Instability and Observability 183 a ~a - = (2--=— ^- H--=---hma%. (8.127) \ ' / ' ' Using the background Klein-Gordon equation, we obtain an expression for which includes contributions from the aether-scalar coupling, - 2a2y0 = + & + maVe^ Q^" - ^ - maV9'. (8.128) In deriving the previous two expressions we have made use of the assumption that ma^- ~ s '--- <$C 1. (8.129) (/)' Mpi H We can immediately use Eqs. (8.128) and (8.123) to remove and the 8

' + (2^f2 -2A- <*>] . (8.131) Simplifying, we find the evolution equation for aH, and imposing a Bunch-Davies vacuum state. The variable 8

Mq (which follows from k ^> aH since Mq <$C H), then k ^> \x as well, and the scalar at these scales behaves as it does in the case with no aether. We therefore see that the scalar and metric perturbations, 8

2> = <<*>gr> + 2 ( T7" ) The deviation from scale-invariance, ns, is denned by d In A" ns-l= (8.137) dmk where the dimensionless power spectrum is A| = ^/V (8.138) lit In GR, the deviation from scale-invariance is — 2s — rj. Using the results d In $L gr =_3_2e-n, (8.139) d In k din O2 ——2 =-3 + (ns-l)2, (8.140) dmk where (ns — 1)2 is the spectral index of ). The Planck sensitivity to r is about 10-1, and about 10"2 to ns - 1 [17, 18]. We see that the first corrections to O enter at Gim2 /Mp^). This is constrained by other experiments to be a tiny number, placing any coupling between

0 as the theory is invariant under the combined symmetry \x -> —\x and

—

2a-2-3^ 0'), (8.146) (8.147) (8.148) Normally, we can obtain a slow-roll inflationary solution to leading order by neglecting '2 in the Friedmann equation (8.146) and 4>15 in the scalar evolution Eq. (8.148). The same applies in this theory; we now briefly justify this. A slow-roll inflationary phase requires H to be changing slowly, and for inflation to be successful it needs to last at least 50-60 e-folds. This is guaranteed by making sure the slow-roll parameters H e = H2 e = 1 J0" He J^e' (8.149) (8.150) are both very small compared to unity. For convenience we will work in cosmic time (t = J adz) here. The slow-roll parameters are Ait Gr e = H2 X] = 2 e + (4>2 + mß ( 2

<$C 3H(p as usual. Note, however, that the usual identifications of s and r\ in terms of the potential will be changed if the scalar-aether coupling is large enough for y to be comparable to 51/2. In the slow-roll limit, the Friedmann and Klein-Gordon equations are, respectively, AttGc --Ma\• jxc, which corresponds to exact de Sitter expansion; for \x > \xc the slow-roll solution is unstable and grows without bound [4]. We will therefore always consider inflationary solutions with \x < \xc. There is an additional constraint on \xl\xc from the spin-0 stability constraint (8.34). Substituting the definition of \xc into this gives the constraint a2 9 24jtGm2ci23 ^- < 24jtGcm2cl23 = , , Q _ . (8.162) 8.5 Case Study: Quadratic Potential 189 The same constraint was derived along similar lines in Ref. [4]. 6 Since c123 < 1 and a > 0 (see Sect. 8.1, as well as Refs. [7, 8]), this is more restrictive than simply \x < ixc, unless m is comparable to, or greater than, the Planck scale—a possibility that seems to be ruled out by experiments, as discussed in Sect. 8.3.4. Since experiments suggest m/Mpi < 10~7, \xl\xc must be so small that inflationary dynamics would be effectively unchanged by the coupling, unless cancellations among the q conspire to weaken the bounds on m. 8.5.2 The Instability Explored Specialising to the Donnelly-Jacob son potential and using the slow-roll Eqs. (8.159) and (8.160), we can write the spin-1 equation of motion (8.69) to first order in the slow-roll parameters as £" + c5(±1)2£2| - 4? = 0' (8-163) t2 with A given by a s _ 1 jZfXc / + —) + 0(e), cf)Za + sgn(/i0)-L7^J —^ + eis + 3c2 + 0(e). (8.164) H2 \ s & V3ci V m: Here, as in Sect. 8.3.4, we have defined the dimensionless coupling a by ix2 = 2cmM2o = 2AitGcm2cmoix2c, (8.165) so that flat-space stability of the spin-0 modes implies a < 1. As with the general case, the solution (8.81) to Eq. (8.163) is written in terms of the first Hankel function of order v, where v2 = -+A (8.166) Repeating the analysis of Sect. 8.3.3, we pick a single mode which leaves the sound horizon at some conformal time t, , which we could take to be the start of inflation. We pick a mode which crosses the horizon early because Vkix) is largest at small k (with x held fixed), so this is one of the larger superhorizon modes available. We want to calculate the contribution of this mode to the 0-i component of the stress- 16As discussed in footnote 5, our action and potential differ from those in Ref. [4] because we give the aether units of mass while their aether is dimensionless. Taking the different definitions of c,, m, and /x into account, our constraint agrees with theirs. 190 8 Lorentz Violation During Inflation energy tensor. If it exceeds the background energy density, then this would indicate a violation of isotropy and signal an instability in the background solution which we found in Sect. 8.5.1. Using the slow-roll scalar equation and our expression (8.160) for V9 = MßH. 3 AnGf (sgn(0) + = M2H ^6mci23o- + Sgn(/i0)yi2ci23cr [Mp\ + (cB + 3c2)m2fj . (8.167) We can substitute this directly into Eq. (8.92) to find one of the terms in the contribution that this mode makes to T0,-, ci0) M M T°i,k/f°o D V — , sgn(/#)V^ + ypte^o m 12V3tt H yM2 + a y JM^ + a x r(v)2v"5 (-Tl-)2 e(v~^N83i. (8.168) We can now get a more quantitative handle on the argument made in Sect. 8.3.3. Assuming v > 3/2, the exponential in (v — 3/2)N is likely to overwhelm the other terms within the 50-60 or more e-folds that will occur after t,, which we take to be near the start of inflation. While several terms in Eq. (8.168) are likely to be several orders of magnitude smaller than unity, including M/H, m/MPi,17 and possibly M/Mpi, it is unlikely that these could be so small as to overwhelm the exponential terms and the gamma function. Hence, for v > 3/2, we expect that the slow-roll background solution we found in Sect. 8.5.1 is unstable, rapidly dominated by perturbations in the aether field generated by its coupling to the inflaton. In Sect. 8.3.4 we found that v can surpass 3/2, even by several orders of magnitude, if the aether VEV, m, is suitably small compared to the Planck scale. Armed with a specific form for the potential, we now briefly clarify that argument and use it to place constraints on the aether-scalar coupling parameter, \x. Ifv > 3/2 then A > 2, where A is defined in Eq. (8.164). It is not difficult to check that this is the same as the A we discussed for a general potential, Eq. (8.70), which we wrote in various forms in Sect. 8.3.4. There, we found that for A to be positive we needed jx4> to be positive. With the Donnelly-Jacobson potential, we have an expression for is only positive (assuming [i < iic) when ix4> is negative. We will take [i to be positive and then ask if 4> can be negative (the opposite case is trivial, as the theory has combined \x —>• —/x,

— 4> symmetry). This is not at all uncommon, and depends only on initial conditions. The dynamics for this inflationary model are encapsulated in ( 0, because when [i = 0, the equations for

—

—

0 half of the phase plane. Since \x <$c jxc, nearly half of all initial conditions leading to viable inflation have jx4> < 0. Considering each piece in A on an order-of-magnitude basis, and taking sgn(/i0) = —1, we have A = / M2 cf 2a - S ■s/3c\ «1 \ <1 &0) A Vö M2 (8.169) / Evidently, A will be greater than 2 if the smallness of m compared to the Planck scale exceeds the (square of the) smallness of the scalar mass, M, compared to the Hubble scale, Mpi ^ 2J3c~i (8.170) m where we have assumed that m/MPi <$c 1. While M/H should be small, there are no limits on how small m/MPi should be before the collider scale, and moreover, as discussed in Sect. 8.3.4, there are already likely to be stringent experimental constraints on m/Mpi (although these tend to depend on the q not cancelling out in particular ways). The tachyonic instability discussed here and in Sect. 8.3 is absent when \x and

0 is taken in the equation of motion, the ghost disappears as the equation becomes second-order. However, if we take the same limit in the solutions to Eq. (9.2), we do not obtain the correct solutions to the second-order equation of motion: the extra two modes do not decouple from the theory.3 For more details on this example, we refer the reader to Sect. 9.3 of [37]. 2We thank Angnis Schmidt-May for helpful discussions on the following points. 3 More precisely, for nonzero k the frequency of the additional modes goes to infinity in the limit e —> 0, and the actual value of the limit is not well-defined. In a theory where all modes have positive energy, infinite-frequency modes are impossible to excite with finite energy. An infinite-frequency negative-energy mode, which is what we have here, would however become even easier to produce [37]. 204 9 Discussion and Conclusions It is therefore not clear whether the presence of a ghost in doubly-coupled massive gravity and bigravity, even if we assume it is not excited at the energy scales for which we are solving, would lead to the same solutions as the ghost-free low-energy theory would. This should not be a problem with the FLRW solutions, for which the absence of the ghost at all scales has been proven [10], but is a sign that we need to continue to search for a doubly-coupled theory which is truly free of the Boulware-Deser ghost. At present it is unclear whether such a theory exists and, if so, what form it will take. If the coupling is not defined by minimally coupling matter to an effective metric, then the problems and potential solutions discussed in Chap. 5 may turn out to be quite relevant. However, a bird in the hand is worth two in the bush, and if we momentarily set aside our pining for a fully ghost-free doubly-coupled theory, we will notice that we have a double coupling, discussed in Chap. 6, which at the very least is good for cosmological solutions. Circumstantial evidence suggests that the new matter coupling should allow for cosmologically-stable models in a much larger region of the parameter space than the singly-coupled theory does. Recall from Chap. 3 that the instability which plagues most singly-coupled models appears specifically for small y. This is a problem in any finite-branch model (and most models only have viable solutions on the finite branch) because the quartic equation (2.58) requires y = 0 at early times. We have seen that in the doubly-coupled theory, y starts at /3/a and hence is always nonzero if yc ^ 0. While the modified Einstein equations will change the perturbation behaviour, if the rule of thumb that instabilities occur for small y holds, then double coupling should open up many more stable models. Moreover, recall, cf. equation (3.32), that the ft model is always stable in singly-coupled bigravity. The problem with this model is that it is ruled out by observations and theoretical conditions at the background level. In the doubly-coupled theory, the ft model has an acceptable background as long as P/a is above a threshold value, cf. Fig. 6.2. Finally, in one simple example, when y = yc at all times and as a result the metrics are proportional, we know that the perturbations must be well-behaved. We found that the effective Friedmann equation in this case reduces to that of ACDM. It can, moreover, be shown that for any solutions to the doubly-coupled theory in which gljL v and fljLv are related by a conformal factor, the theory reduces to general relativity,4 and that this equivalence to general relativity extends to linear perturbations [38]. This implies that the perturbations around the conformal cosmological solutions we have found must be the same as in general relativity, and hence are stable. A full investigation of the cosmological perturbations in this theory is therefore very well motivated in the search for cosmologically-viable models of bigravity. We are all in the gutter, but some of us are looking at the stars. Oscar Wilde, Lady Windermere's Fan 4In particular, matter couples only to the massless spin-2 field. References 205 References 1. C. de Rham, L. Heisenberg, R.H. Ribeiro, Quantum corrections in massive gravity. Phys. Rev. D88, 084058 (2013). arXiv: 1307.7169 2. M. von Strauss, A. Schmidt-May, J. Enander, E. Mortsell, S. Hassan, Cosmological solutions in bimetric gravity and their observational tests. JCAP 1203, 042 (2012). arXiv:1111.1655 3. Y. Akrami, T.S. Koivisto, M. Sandstad, Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality. JHEP 1303, 099 (2013). arXiv: 1209.0457 4. F. Konnig, A. Patil, L. Amendola, Viable cosmological solutions in massive bimetric gravity. JCAP 1403, 029 (2014). arXiv: 1312.3208 5. D. Comelli, M. Crisostomi, L. Pilo, Perturbations in massive gravity cosmology. JHEP 1206, 085 (2012). arXiv: 1202.1986 6. A. De Felice, A.E. Gumrukcuoglu, S. Mukohyama, N. Tanahashi, T. Tanaka, Viable cosmology in bimetric theory. JCAP 1406, 037 (2014). arXiv: 1404.0008 7. D. Comelli, M. Crisostomi, L. Pilo, FRW cosmological perturbations in massive bigravity. Phys. Rev. D90(8), 084003 (2014). arXiv: 1403.5679 8. D. Mattingly, Modern tests of Lorentz invariance. Living Rev. Rel. 8, 5 (2005). arXiv:gr-qc/0502097 9. Y. Yamashita, A. De Felice, T. Tanaka, Appearance of Boulware-Deser ghost in bigravity with doubly coupled matter. Int. J. Mod. Phys. D23, 3003 (2014). arXiv: 1408.0487 10. C. de Rham, L. Heisenberg, R.H. Ribeiro, On couplings to matter in massive (bi-)gravity. Class. Quant. Grav. 32, 035022 (2015). arXiv: 1408.1678 11. G. D'Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava et al., Massive cosmologies. Phys. Rev. D84, 124046 (2011). arXiv: 1108.5231 12. A.E. Gumrukcuoglu, C. Lin, S. Mukohyama, Open FRW universes and self-acceleration from nonlinear massive gravity. JCAP 1111, 030 (2011). arXiv: 1109.3845 13. A.E. Gumrukcuoglu, C. Lin, S. Mukohyama, Cosmological perturbations of self-accelerating universe in nonlinear massive gravity. JCAP 1203, 006 (2012). arXiv:1111.4107 14. B. Vakili, N. Khosravi, Classical and quantum massive cosmology for the open FRW universe. Phys. Rev. D85, 083529 (2012). arXiv: 1204.1456 15. A. De Felice, A.E. Gumrukcuoglu, S. Mukohyama, Massive gravity: nonlinear instability of the homogeneous and isotropic universe. Phys. Rev. Lett. 109, 171101 (2012). arXiv: 1206.2080 16. M. Fasiello, A.J. Tolley, Cosmological perturbations in massive gravity and the Higuchi bound. JCAP 1211, 035 (2012). arXiv: 1206.3852 17. A. De Felice, A.E. Gumrukcuoglu, C. Lin, S. Mukohyama, Nonlinear stability of cosmological solutions in massive gravity. JCAP 1305, 035 (2013). arXiv:1303.4154 18. L. Amendola, S. Fogli, A. Guarnizo, M. Kunz, A. Vollmer, Model-independent constraints on the cosmological anisotropic stress. Phys. Rev. D89, 063538 (2014). arXiv: 1311.4765 19. G.W Horndeski, Second-order scalar-tensor field equations in a four-dimensional space. Int. J. Theor. Phys. 10, 363-384 (1974) 20. M. Zumalacarregui, J. Garcia-Bellido, Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian. Phys. Rev. D89, 064046 (2014). arXiv: 1308.4685 21. J. Gleyzes, D. Langlois, F. Piazza, F. Vernizzi, Healthy theories beyond Horndeski. arXiv: 1404.6495 22. J. Gleyzes, D. Langlois, F. Piazza, F. Vernizzi, Exploring gravitational theories beyond Horndeski. arXiv: 1408.1952 23. S. Hassan, A. Schmidt-May, M. von Strauss, On partially massless bimetric gravity. Phys. Lett. B726, 834-838 (2013). arXiv: 1208.1797 24. C. de Rham, S. Renaux-Petel, Massive gravity on de sitter and unique candidate for partially massless gravity. JCAP 1301, 035 (2013). arXiv: 1206.3482 25. C. de Rham, K. Hinterbichler, R.A. Rosen, A.J. Tolley, Evidence for and obstructions to nonlinear partially massless gravity. Phys. Rev. D88(2), 024003 (2013). arXiv: 1302.0025 206 9 Discussion and Conclusions 26. E. Joung, W. Li, M. Taronna, No-go theorems for unitary and interacting partially massless spin-two fields. Phys. Rev. Lett. 113, 091101 (2014). arXiv: 1406.2335 27. S. Hassan, A. Schmidt-May, M. von Strauss, Particular solutions in bimetric theory and their implications. Int. J. Mod. Phys. D23, 1443002 (2014). arXiv: 1407.2772 28. S. Garcia-Saenz, R.A. Rosen, A non-linear extension of the spin-2 partially massless symmetry. arXiv: 1410.8734 29. A. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B39, 393-394 (1972) 30. C. Llinares, D. Mota, Releasing scalar fields: cosmological simulations of scalar-tensor theories for gravity beyond the static approximation. Phys. Rev. Lett. 110(16), 161101 (2013). arXiv: 1302.1774 31. C. Llinares, D.E Mota, Cosmological simulations of screened modified gravity out of the static approximation: effects on matter distribution. Phys. Rev. D89,084023 (2014). arXiv: 1312.6016 32. Y. Akrami, T.S. Koivisto, D.E Mota, M. Sandstad, Bimetric gravity doubly coupled to matter: theory and cosmological implications. JCAP 1310, 046 (2013). arXiv: 1306.0004 33. S. Hassan, A. Schmidt-May, M. von Strauss, On consistent theories of massive spin-2 fields coupled to gravity. JHEP 1305, 086 (2013). arXiv:1208.1515 34. J. Noller, S. Melville, The coupling to matter in massive, bi- and multi-gravity. JCAP 1501, 003 (2014). arXiv: 1408.5131 35. S. Hassan, M. Kocic, A. Schmidt-May, Absence of ghost in a new bimetric-matter coupling. arXiv: 1409.1909 36. C. de Rham, L. Heisenberg, R.H. Ribeiro, Ghosts and matter couplings in massive gravity, bigravity and multigravity. Phys. Rev. D90(12), 124042 (2014). arXiv: 1409.3834 37. R.P Woodard, Avoiding dark energy with 1/r modifications of gravity. Lect. Notes Phys. 720, 403-433 (2007). arXiv:astro-ph/0601672 38. A. Schmidt-May, Mass eigenstates in bimetric theory with ghost-free matter coupling. JCAP 1501, 039 (2014). arXiv: 1409.3146 Appendix A Deriving the Bimetric Perturbation Equations God does not care about our mathematical difficulties. He integrates empirically. Albert Einstein In Sect. 3.1.1 we presented the full linearised Einstein and fluid conservation equations for massive bigravity in a general gauge and without making a choice of the time coordinate (i.e., with a general lapse). Since these equations are arrived at by a fairly lengthy calculation, in this appendix we detail their derivation. The perturbations of the Einstein tensor are standard and can be found in, e.g., Ref. [1]. In order to calculate the fluid conservation equations we only need to know the linearised Christoffel symbols. For the g metric, these are N 1 --h -. N 2 roo = — + ^Eg T" = di(lE' + JjF') ij = ^2 [(#C1 + As) + \As - HEg) *y + l^jBg + Hdidj, N , (IN ■ \ * = 7a \2i;E' + F' + HF') r r --didjF« N (A.l) Note that in background, only r|J0, , and T0j are nonzero. Similarly we can find the /-metric Christoffel symbols, f(fp, © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7 207 208 Appendix A: Deriving the Bimetric Perturbation Equations 00 1 Oi 1 00 X 1 . — + -Ef X 2 / Y2 r' _ 1 0j ~ jk (\Ef + IFf) — [(k(1 + Af) + l-Af - KEf^j Stj + \didjBf + KdtdjBf X ,. /1 X . \ (K + \Af) siJ + \didJBf \ (s'jdkAf + VrfjAf - SjkdlAf + d'djdkBf) - X -didjFf :SjkdlFf. (A.2) The bulk of the work lies in calculating the perturbations of the mass term. We will focus on deriving the linearised field equations, i.e., calculating the matrices ^(n)v ramer than the second-order action. The metric determinants to linear order are det g = -N2a6 (l + Eg + 3Ag + V2Bg) , det / = -X2Y6 (1 + Ef + 3Af + V2Bf) . The matrix X = yj1 g~x f is defined in terms of the two metrics as pv Its background value is simply X°0 = x, »0 = Using this we can solve Eq. (A.5) to first order in perturbations to find JL°o = x(l + ^AE>j, X''0 1 Y x + y N 1 X x + y a (ydiFg - xdiFf) , (yd'Ff-xd'Fg), X-'■ = y (1 + \AA) S'j + ^djAB (A3) (A.4) (A.5) (A.6) (A.7) Appendix A: Deriving the Bimetric Perturbation Equations 209 The trace of this is [X] = x (l + l-AE^ + y [3 (l + l-AA^ + VAB (A.8) Similarly we can solve for the matrix Y = y/'f~lg, although we do not write its components here as they can be found by simply substituting (iV, a, g) with (X, Y, f) and vice versa.1 We now need the matrices X2 and X3 and their traces in order to compute the matrices appearing in the mass terms of the Einstein equations. For X2 we find (X2)°0 = xz(l + A£), C^2)°i = ^(yBiFg-xdiFf) QL2y0 = -(ydiFf-xdiFg), (X2)''; = y2 [(1 + AA)8' j + VdjAB], with trace [X2] = x2(l + Ef- Eg) + y2 [3(1 + AA) + V2AB] X3 is given by (X3)°0 = x3 [l + ^-AE (i+H' 77 (x + y ~ ^7) (ydiF* ~ xdiFf)' -(x + y- -^-) (yd'Ff - xd'Fg) , a \ x + y) v ' (X3yj=y3[(l + ^AASj (x3y 8l i + -dldjAB with trace [X3] = x3 (l + 3-AE^ + y3 ^3 (l + 3-AA^ + ^V2AB (A.9) (A. 10) (A. 11) (A.12) Y2 and Y3 can be determined trivially from these. It may also be calculated explicitly or by using the fact that Y is simply the matrix inverse of X, which can be easily inverted to first order. 210 Appendix A: Deriving the Bimetric Perturbation Equations Having calculated these we can determine the matrices Y^v(y/'g~xf) and ^(n)v(V' f~lS)- Two helpful intermediate results are 1 - ([X]2 - [X2]) = y2 [3(1 + AA) + V2AB] + xy [3(l + i(AA + A£))+i V2A5 (A.13) l- ([X]3 - 3[X][X2] + 2[X3]) = y3 ^1 + 3-AA + ^ V2A5^ ^3 ^1 + AA + ^A£^ + V2A5^ . + xy (A. 14) To obtain those intermediate results and the 0-0 and i- j components of the Y matrices, it saves a lot of algebra to write the traces, 0-0 components, and i-j components of the various X matrices in terms of c\ = x, 1 2 c2 = 3y, 1 2' 81 = -AE, 82 = - AA H—V AS, 53'_,- = fy% ~ ^J^2) Afl. (A.15) Finally, the matrices Y^v(y/g-1 f) defined in Eq. (2.41) are given by • n = 0: (A.16) • n = 1: (ydiFg-xdiFf) (v/8ZT7) = -3- p + jAAJ + ±V2AB (v^7) (1)0 (1)0 1 Y x + y N 1 X (ya'iv-xa'F,), (l + ±A*)*', x + y a —x 2y [(l + ^Aa) 5',. + 1 (^-V2 - 3%) A5 (A. 17) Appendix A: Deriving the Bimetric Perturbation Equations 211 • n = 2: Y'(2)o (v^1/) y2 [3(1 +AA) + V2Aß] 2y Y x + y N 2y X x + y a (ydtFg - xdiFf) , (yd'Ff-xd'Fg), 1 y + 2xy (1 + AA)S'j + - (8ljV2 - d'dj) AB ^1 + ^(AA + AE)^j 8'j + i - d'dj) AB (A. 18) • n = 3: (3)0 §» (-/Fl) 1 + -AA + -VZAB 2 y (3)0 y2 X x + y a 2 2 (ydiFg - xdiFf) .. (yVFf-xVFg) = -xy ^1 + AA + ^A£^ 8lj + i (5^-V2 - d'dj) AB (A.19) Plugging these into the field Eqs. (2.39) and (2.40), we obtain the full perturbation equations presented in Sect. 3.1.1. Reference [1] V.E Mukhanov, H. Feldman, R.H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions. Phys. Rep. 215, 203-333 (1992) Appendix B Explicit Solutions for the Modified Gravity Parameters As discussed in Sect. 4.2, the modified gravity parameters Q and r\ in singly-coupled massive bigravity have the Horndeski form, rj Q , (l+kzh4\ , n + k2h4\ + k2h3, while the growth of structures can be written in these terms as . 1 Q a2p 8 + H8- - — —hr8 = 0. 2 x] Ml (B.l) (B.2) (B.3) Hence the five ht coefficients allow us to determine all three modified gravity parameters we consider in Chap. 4. They are given explicitly by l + y2 (1 + y2) (A + 3/J3/ + (6ß2 - 2ß4) y3 + 3 (A - ß3) y2 -ßi + (3ß2 - At) y5 + (6ßi - 9ß3) y4 + (3ß0 - I5ß2 + 2ß4) y3 + (3ß3 - Tß^y2 ' (BA) (B.5) y h6 1 + y2 ßiy + - 2ß3ß4) yb + 3 {2ßx - 3ß3) ß3y> + (4ß0ß3 - 19ß2ß3 - ß4ß3 + 2ßxß4) y* + {-3ß2 - 18ß2 + 4ß0ß2 + 2ß2ß4) y3 + 3 ((A) - 3A) A + Al 0?4 - 5A)) y2 + {-Ißl + 2ß3ßx + 2 (A) - 3A) A) y-ßi (Ao + ß2) (B.6) ZI he 2ß3ß4y6 + 2 (3ß2 - ß4ß2 - 3 (Ai - 2A) A) / + (ßl (ßßi - 4ß4) + 3ß3 (-2ß0 + 9ß2 + ß4)) y4 + 2 (3ß2 - 2ß3ßx + 18A2 + 9A2 " 3A)A - 3A/S4) y3 + (37A/S2 + 27AA - 9ß0ß3 - 9A AO y2 + 2 (10A2 - 3A/Sl - 3 (/So - 3A) A) y + 3/?i (ßo + ß2) (B.7) © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7 213 214 Appendix B: Explicit Solutions for the Modified Gravity Parameters h6 ft 4ßißbU + ft (24ftft2 + {9ßl - Zßl) ß2 + 3ft (19A - 8ft) ft) yW + 2 (18ft4 - 12ftft3 + (117ft2 - 36ftft + 2ßl) ft2 + 6ft (4ft + 5ft) ftft +ft (99ft3 - 81ftft2 + 18ft2ft - 3ft2ft - 12ftftft - 8ftft2)) y9 - (72ft (ft - ft) ß\ + {-12ßl + 72ftft2 + (I17ft2 - 16ftf) ft + ft2 (85ft - 72ft)) ft +9ft (-60ft3 + 8ftft2 + 12ftft2 - 96ft2ft + 19ftft2 + 5ft2ft)) / - 2 (36ftft3 - (54ft2 - 36ftft + 69ft2 + 8ftf) ß\ -2ft (123ft2 - 76ftft + 3 (13ft2 + ft (4ft + ft))) ft -3 (72ft4 - 36ftft3 + (255ft2 + 4ft2) ft2 - 21ft2ftft - 9ft4 + 6ft2/?2 +ft (-12ft3 + 4ftft2 - 93ft2ft + 3ft2ft))) y1 + (24 (3ft - 2ft) ft3 + ft (-72ft + 507ft - 77ft) ft2 + (876ft3 - 508ftft2 + 600ft2ft + 48/S2ft - 3ft2ft + ft (-72ft2 + 48ftft - 69ft2)) ft +3ft {24ß2ß20 + (-228ft2 + 16ftft + 9ft2) ft + 9ft (48ft2 - 4ftft - 7ft2))) y6 + 2 (18ft4 + 45ftft3 + (477ft2 - 36ftft - 170ftft + 14ft2 + 9ft2) ß\ +3ft (126ft2 - 42ftft + 6ftft - 3ft2 + 2ftft) ft +6 (ft - 3ft) ft (-15ft2 + 3ftft + 2ftft + 6ft2)) y5 + ((441ft - 79ft) ß\ + ft (-33ft - 8ft + 33ft) ft2 + (648ft3 - 156ftft2 - 60ftft2 + 9ft2ft + 9ftft2) ft + 36 (ft - 3ft) ft2ft) / + 2ft (39ft3 - 26ftft2 + (167ft2 - 15ftft + 15ft2 - 3ft (ft + ft)) ft - 12ftftft) y3 + ft (36ft3 + 9ftftft + 3ft (3ft2 - 5ftft - 4ft2) + ß\ (112ft - 9ft)) y2 + 2ß\ (lift2 - 3ftft + 12ft2) y + 3ft3 (ft + ft) (B.8) where we have introduced two additional coefficients, and h-j, defined as h6 = 3m2a2 (1 + y2) {fa + fay2 + 2 fay) (p2 + fa fay5 + (3ft2 - fa fa - 3 (fa - 2 fa) fa) y4 + (3ftft + 12ftft - 3ftft - 2ft.ft) y3 + (3ft2 + ftft - 3 (ft - 3ft) ft) y2 + 5ft.ftj) , (B.9) (ft + y (2fa + fay)) (3fay3 + y2 (3fay {y2 - 5) + ft (3 - 9y2) - fay {y2 - 2j) + fa (6y4 - ly2 - 1)) hl =--• (B.10) While this notation is inspired by Refs. [1, 2], we have defined h\^j differently. References [1] L. Amendola, M. Kunz, M. Motta, I.D. Saltas, I. Sawicki, Observables and unobservables in dark energy cosmologies. Phys. Rev. D87, 023501 (2013). arXiv: 1210.0439 [2] L. Amendola, S. Fogli, A. Guarnizo, M. Kunz, A. Vollmer, Model-independent constraints on the cosmological anisotropic stress. Phys. Rev. D89, 063538 (2014). arXiv:1311.4765 Appendix C Transformation Properties of the Doubly-Coupled Bimetric Action Here we describe the transformation properties of the action (6.1) and how they determine the number of physically-relevant parameters for the doubly-coupled bigravity theory discussed in Chap. 6. C.l Rescaling the Action Let us write the action as Ml r A ,_ M2, s = J d4xJ-detgR (8)--^ J d4xJ-detfR (f) + m2M2g J d4xj-detgV (Vs"1/; ft) + J d4xj- det£effi?m (geS, O), (C.l) where 4 v (^ftrH; ft) = X ß"e" {^f) (C2) n=0 is the usual dRGT interaction potential and satisfies J-detgV (7^7; ft) = V-det/y (77^; A-*) • (C.3) Due to this property, the action is invariant under giiv fpv, Mg**Mf, a ft ft -> ft_„, (C.4) since the effective metric © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7 215 216 Appendix C: Transformation Properties of the Doubly-Coupled Bimetric Action gfv = « V + 2aßgßaXav + ß2fßV (C.5) is also invariant under this transformation, as shown below. Because the overall scaling of the action is unimportant, there is a related transformation which keeps the action invariant, but only involves the ratio of Mg and Mf, MSp ^Mff (Mf\4~nfi _> (Mf\ R MV^M*tf2 (C.6) These transformations reflect a duality of the action since they map one set of solutions, with a given set of parameters, to another set of solutions. Not all of the parameters Mg, Mf, a, ft and ft are physically independent. In effect, we can rescale these parameters, together with gliv and fliv, to get rid of either Mg and Mf or a, and ft In the end, only the ratio between Mg and Mf, or a and ft together with ft, are physically meaningful. The two parameter choices are physically equivalent and can be mapped to one another. We now describe the two scalings that give rise to the two parameter choices. Under the scalings -2 „ r . o— 2 r -\/r2 „,2 71*2 g/xv -> ol %v, -> b z//xv, m\ -> azMzg, M) -> f32M2, m2 -> a2m2, ft -> (^j ft, (C.7) the effective metric becomes gfv = gfiv + 2gfiaXav + fllv (C.8) while the action becomes Ml r . ,_ m2 j d4xJ-detgR (g)--fj d4xJ-detfR (f) S = 2 J ' ~ ~ 4 + m2M2 j d4Xy/-detgY^f3nen (TF1/) J n=0 + J d4xy/-detgeff^fm (geff, O). (C.9) The effective metric is thus uniquely denned in this parameter framework, while the ratio between Mg and Mf is the free parameter (in addition to the ft). For this choice of scaling, the action is invariant under gfiv ffiv, ft -> ft-rc, Mg Mf, (CIO) Appendix C: Transformation Properties of the Doubly-Coupled Bimetric Action 217 or, more generally, 4-n Mf (Mf\ (Mf\n If, instead, we apply the scalings ML _ M1 -Jeff J"eff (Mf\ § f M4 M1 Ml Meff Meff Meff then the effective metric is still of the form (C.5), while the action becomes m2 r m2 r S =--f- J d4Xyf^teTgR (g)--^j d4x,/^fetjR (f) + m2M2ff / d*xj-detg^pnen (V^F1/) ^ rc=0 + J d4x,J-detgeffj?m (ges, O). (C.13) For this choice of scaling, only the ratio between a and ft together with the ft, is physically important (the effective coupling M2ff can be absorbed in the normalisation of the matter content). Under this form, the action is invariant under gliv fpv, ft "> 04-n, a P- (C.14) To move from the framework with Mg and Mf to the one with a and ft one simply performs the rescaling , /a\" 4 m2M2 //xv -> P2f„v, ft -> ( - J ft, m4 -> —-f^, (C.15) Each of the parameter frameworks has its advantages. In the Mg and Mf framework, there is a unique effective metric, and it is the relative coupling strengths that determine the physics. In the a and ft framework, we have one single gravitational coupling, M2ff, and the singly-coupled limits are more apparent in the effective metric. Note that the ratio between a and ji only appears in the matter sector, whereas in the Mg and Mf formulation their ratio appears in both the matter sector and interaction potential. 218 Appendix C: Transformation Properties of the Doubly-Coupled Bimetric Action C.2 Symmetry of the Effective Metric In this section, we show that the effective metric is symmetric under the interchanges giiv^fiiv, a p. (C.16) In order to do this, we take advantage of the fact that g/xaX" is symmetric, i.e., gX = Xrg, as shown in Ref. [1]. We will find it useful to discuss the metrics in terms of their vielbeins, since we are dealing with square-root matrices and vielbeins are, in a sense, "square roots" of their respective metrics. We use Greek letters for spacetime indices and Latin letters for Lorentz indices. The g- and /-metric vielbeins are defined by (C.17) rfabLlLbv, (C.18) while the inverse metrics are given by g^v = r\abe^evb and similarly for f^v. The vielbeins of g^v are inverses of the vielbeins for gliv, e^e1^ = 8% and e° eva = 8V, and again similarly for the fliv vielbeins. We will assume the symmetry condition (also called the Deser-Van Nieuwen-huizen gauge condition) e*Lbll = e»Lail, (C.19) where Lorentz indices are raised and lowered with the Minkowski metric. It is likely, though it has not yet been proven, that this condition holds for all physically-relevant cases. In four dimensions, it holds when g~l f has a real square root (proven in Ref. [2], where it was conjectured that this result is valid also in higher dimensions). Assuming this condition, then it has been shown [3] that the square-root matrix is given by X£ = e£Lav. (C.20) The inverse of this is clearly (K-1)* = L*eav, (C.21) since then K^-'X = e%KLabebv = e*eav = 8». (C.22) The form of the inverse then implies (JF^y1 = ^fgZ^f, (C23) which will be a useful property when showing the symmetry of the effective metric. We also have fflV = Appendix C: Transformation Properties of the Doubly-Coupled Bimetric Action 219 g^Xav=eaileaaeabLbv = eailLav. (C.24) In order to show that gX = XT g, we must thus have £a(j,Lv = eavL . (C.25) Notice that this is not exactly the same as Eq. (C.19), since in the first case we contract over spacetime indices, whereas here we contract over Lorentz indices. The two symmetry conditions are, however, equivalent, as discussed in detail in [4]. An alternative way of seeing that gX = Xrg is as follows. Since /MaX" = L°Laae^Lj = L°Ltoe"Lj = /vaX^, (C.26) we have /X = Xr/. (C.27) But / = gX2, so Eq. (C.27) can also be written gX3 = XT gX2, which implies gX = XTg. (C.28) Using this property it is straightforward to show that the effective metric is symmetric under the interchange of the two metrics. The effective metric we study was introduced in Ref. [5] in the form gfv = a V + 2ofc(/xX^ + tS2/^. (C.29) Due to the symmetry property (C.28), we can write this without the explicit sym-metrisation, gfv = « V + 2af3gailXav + f32fliv, (C.30) Suppose now that we do the transformation giiv^fiiv, a f3. (C.31) The effective metric becomes gfv = a V + 2ujlflia{JT^~gTv + f32fllv. (C.32) This can be brought into the original form for g^ using the matrix property / (^T1/)"1 = gg~lf (^T1/)"1 = gVg^f- (C33) Combining this with Eq. (C.23) we get 220 Appendix C: Transformation Properties of the Doubly-Coupled Bimetric Action fJFH = gy/FU- (C34) Applying this to Eq. (C.32) we see that the effective metric is invariant under the duality transformation (C.31). This ensures that the entire Hassan-Rosen action treats the two metrics on entirely equal footing when matter couples to g^. Note that this duality does not hold for the single-metric (dRGT) massive gravity as it is broken by the kinetic sector. References [1] S. Hassan, A. Schmidt-May, M. von Strauss, On consistent theories of massive spin-2 fields coupled to gravity. JHEP 1305, 086 (2013). arXiv:1208.1515 [2] C. Deffayet, J. Mourad, G. Zahariade, A note on 'symmetric' vielbeins in bimetric, massive, perturbative and non perturbative gravities. JHEP 1303, 086 (2013). arXiv: 1208.4493 [3] P. Gratia, W. Hu, M. Wyman, Self-accelerating massive gravity: how zweibeins walk through determinant singularities. Class. Quantum Gravity 30, 184007 (2013). arXiv:1305.2916 [4] J. Hoek, On the Deser-Van Nieuwenhuizen algebraic vierbein gauge. Lett. Math. Phys. 6,49-55 (1982) [5] C. de Rham, L. Heisenberg, R.H. Ribeiro, On couplings to matter in massive (bi-)gravity. Class. Quantum Gravity 32, 035022 (2015). arXiv: 1408.1678 Appendix D Einstein-Aether Cosmological Perturbation Equations in Real Space In this appendix, we present the real-space equations of motion for the linear cosmological perturbations in Einstein-aether theory coupled to a scalar field as described in Chap. 8. We have for the v = 0 component of the aether field Eq. (2.77) - 6(ci3 + 2c2)^20 + 6c2 ^—^ O + J? [(2Cl + c2)V\i + c3(V\i + B\i) + 3c20' + 3(2c13 + c2W] - c3(0'S- - Bu- + V';,.) - c2iyi[i + 3*") + a2Sks + ^Vee ^6 - 2je2^j O + 3Jf('$> - Sep') -1 m 1 - 2 0'(3*' - 3jeO + V\t) + 3 ^— - 2je2^j 50 J = 0, (D.l) and the v = i component is © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7 221 222 Appendix D: Einstein-Aether Cosmological Perturbation Equations in Real Space |_ mL mL \ a J \ a J J + ^ [(ci + ^)<5>,i + 2ci(V/ - 5;)] 1 + -(-c3 + ci)B{iJ]J - ClVijJ - c^V'.ij + 1 a + ( - 1 -, 3m2 - /a" ~\ m - - -^4>'&4>,i + —Vee(--2^f2 (Vi- - 5,-) + -V^'CV,- - 5,-) aL aL \ a J a (D.3) m 8T'j = 2— a2 + a + (c13 + 3c2) 2 - 2 (D.4) OS' j - (c13 + 3c2)Jf'8i a^dV^kVj + (en + 3c2) W,- + -cn(V\j + V/ + iti)) -— (4>'2<5>-4)'84)' + a2V) = \fi02 + V(4>). This corresponds to pure ae-theory, with c2 rescaled to c2 + fa and a scalar field coupled only to gravity. The cosmological perturbations in that model are presented in [1]. Our equations agree with the literature in this limit, as we would expect. Reference [1] E.A. Lim, Can we see Lorentz-violating vector fields in the CMB?. Phys. Rev. D71, 063504 (2005). astro-ph/0407437 Curriculum Vitae Adam R. Solomon Department of Physics & Astronomy, 209 South 33rd Street, Philadelphia, PA 19104 Email: adamsol@physics.upenn.edu Web: https://web.sas.upenn.edu/adamsol/ Research Interests Modified gravity and its cosmological tests. Massive (bi)gravity, scalar-tensor theories, Lorentz-violating gravity. Employment History Sep. 2015-present University of Pennsylvania Postdoctoral Fellow Center for Particle Cosmology Apr. 2015-Jul. 2015 University of Heidelberg DAAD Visiting Fellow Institute for Theoretical Physics Dec. 2014-Feb. 2015 University of Cambridge Research Assistant Department of Applied Mathematics and Theoretical Physics Education 2011-2015 University of Cambridge - Ph.D. Department of Applied Mathematics and Theoretical Physics Thesis: Cosmology Beyond Einstein Supervisor: Prof. John D. Barrow © Springer International Publishing AG 2017 A.R. Solomon, Cosmology Beyond Einstein, Springer Theses, DOI 10.1007/978-3-319-46621-7 225 226 Curriculum Vitae 2010-2011 University of Cambridge - Master of Advanced Study in Mathematics Part III of the Mathematical Tripos (Distinction) Essay: Probing the Very Early Universe with the Stochastic Gravitational Wave Background Supervisors: Prof. Paul Shellard, Dr. Eugene Lim 2006-2010 Yale University - B.S. in Astronomy and Physics Thesis: The Sunyaev-Zel'dovich Effect in the Wilkinson Microwave Anisotropy Probe Data Supervisors: Prof. Daisuke Nagai, Dr. Suchetana Chatterjee Publications and Conference Proceedings Nersisyan, H., Akrami, Y., Amendola, L., Koivisto, T. S., Rubio, J., & Solomon, A. R., "On instabilities in tensorial nonlocal gravity." 2016, arXiv: 1610.01799 Carrillo-Gonzalez, M., Masoumi, A., Solomon, A. R., & Trodden, M. "Solitons in generalized galileon theories." 2016, arXiv: 1607.05260 Liiben, M., Akrami, Y, Amendola, L., & Solomon, A. R., "Aller guten Dinge sind drei: Cosmology with three interacting spin-2 fields." 2016, Phys. Rev. D 94,043530, arXiv: 1604.04285 Bull, P., Akrami, Y, et al., "Beyond ACDM: Problems, solutions, and the road ahead." 2016, Phys. Dark Univ. 12, 56-99, arXiv: 1512.05356 Akrami, Y, Hassan, S. E, Konnig, F., Schmidt-May, A., & Solomon, A. R., "Bimetric gravity is cosmologically viable." 2015, Phys. Lett. B 748, 37, arXiv: 1503.07521 Enander, J., Akrami, Y, Mortsell, E., Renneby, M., & Solomon, A. R., "Integrated Sachs-Wolfe effect in massive bigravity." 2015, Phys. Rev. D 91, 084046, arXiv: 1501.02140 Solomon, A. R., Enander, J., Akrami, Y, Koivisto, T. S., Konnig, F., & Mortsell, E. "Cosmological viability of massive gravity with generalized matter coupling" 2015, JCAP04 (2015) 027, arXiv: 1409.8300 Enander, J., Solomon, A. R., Akrami, Y, & Mortsell, E. "Cosmic expansion histories in massive bigravity with symmetric matter coupling." 2014, JCAP01 (2015) 006, arXiv: 1409.2860 Jazayeri, S., Akrami, Y, Firouzjahi, H., Solomon, A. R., & Wang, Y. "Inflationary power asymmetry from domain walls." 2014, JCAP11 (2014) 044, arXiv: 1408.3057 Konnig, F, Akrami, Y, Amendola, L., Motta, M., & Solomon, A. R. "Stable and unstable cosmological models in bimetric massive gravity." 2014, Phys. Rev. D 90, 124014, arXiv: 1407.4331 Solomon, A. R., Akrami, Y, & Koivisto, T. S. "Linear growth of structure in massive bigravity." 2014, JCAP10 (2014) 066, arXiv: 1404.4061 Akrami, Y, Koivisto, T. S., & Solomon, A. R. "The nature of spacetime in bigravity: two metrics or none?" 2014, Gen. Relativ. Gravit. 47, 1838, arXiv: 1404.0006 Curriculum Vitae 227 Solomon, A. R., & Barrow, J. D. "Inflationary Instabilities of Einstein-Aether Cosmology." 2014, Phys. Rev. D 89, 024001 Brooks, A. M., Solomon, A. R., Governato, F., McCleary, J., Mac Arthur, L., Brook, C, Jonsson, P., Quinn, T., & Wadsley, J. "Interpreting the Evolution of the Size-Luminosity Relation for Disk Galaxies from Redshift 1 to the Present." 2010, ApJ, 728,51 Cruz, K.L, Kirkpatrick, J. D., Burgasser, A. J., Looper, D., Mohanty, S., Prato, L., Faherty, J., & Solomon, A. "A New Population of Young Brown Dwarfs." 2008,14th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, ASP Conference Series, Vol. 384 Cruz, K. L., Reid, I. N., Kirkpatrick, J. D., Burgasser, A. J., Liebert, J., Solomon, A. R., Schmidt, S. J., Allen, P. R., Hawley, S. L., & Covey, K. R. "Meeting the Cool Neighbors. IX. The Luminosity Function of M7-L8 Ultracool Dwarfs in the Field." 2007, AJ, 133, 439 Talks June 2016: Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK Institute for Cosmology and Gravitation University of Portsmouth, UK May 2016: Institute for Theoretical Physics University of Heidelberg, Germany Nordic Institute for Theoretical Physics KTH Royal Institute of Technology and Stockholm University, Sweden Hot Topics in Modern Cosmology: Spontaneous Workshop X Institut d'Etudes Scientifiques de Cargese Cargese, France December 2015: Department of Physics Case Western Reserve University, USA September 2015: Department of Physics University of Delaware, USA April 2015: Institute for Theoretical Physics University of Heidelberg, Germany March 2015: Xth Iberian Cosmology Conference Aranjuez, Spain 228 Curriculum Vitae Extended Theories of Gravity Nordic Institute for Theoretical Physics KTH Royal Institute of Technology and Stockholm University, Sweden February 2015: London Relativity and Cosmology Seminar Queen Mary, University of London, UK November 2014: Department of Physics University of Nottingham, UK October 2014: Dark Energy Interactions Nordic Institute for Theoretical Physics KTH Royal Institute of Technology and Stockholm University, Sweden September 2014: UK Cosmology Department of Physics University of Oxford, UK University of California, Berkeley, USA May 2014: Institute for Theoretical Physics University of Heidelberg, Germany Department of Physics & Astronomy University of Pennsylvania, USA Center for Cosmology and Particle Physics New York University, USA Perimeter Institute for Theoretical Physics Waterloo, ON, CA February 2014: Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK January 2014: Nordic Institute for Theoretical Physics KTH Royal Institute of Technology and Stockholm University, Sweden Institute of Theoretical Astrophysics University of Oslo, Norway December 2013: 27th Texas Symposium on Relativistic Astrophysics Department of Physics University of Texas at Dallas, US Curriculum Vitae 229 September 2013: COSMO 2013 Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK Conferences, Workshops, and Schools • Hot Topics in Modern Cosmology: Spontaneous Workshop X (workshop -invited speaker) May 9th-14th, 2016 Institut d'Etudes Scientifiques de Cargese, France • New Frontiers in Entanglement (workshop) April 7^-8* 2016 University of Pennsylvania, US • New Frontiers in Particle Cosmology (workshop) December 3td-4th, 2015 University of Pennsylvania, US • Xth Iberian Cosmology Conference March SO^-April 1st, 2015 Aranjuez, Spain • Extended Theories of Gravity (Nordita program - invited participant) March 2nd-March 20th, 2015 Nordic Institute for Theoretical Physics, Stockholm, Sweden • UK Cosmo February 27th, 2015 Queen Mary, University of London, UK • Beyond ACDM January I3th-llth, 2015 Oslo, Norway • Dark Energy Interactions October lst-3rd, 2014 Nordic Institute for Theoretical Physics, Stockholm, Sweden • UK Cosmo September 23rd, 2014 University of Oxford, UK • COSMO 2014 August 25th-29th, 2014 Chicago, US • Cosmology and the Constants of Nature March 17th-19th, 2014 Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK • The Structure of Gravity and Spacetime (workshop - invited participant) February 6^-1^, 2014 University of Oxford, UK 230 Curriculum Vitae • 27th Texas Symposium on Relativistic Astrophysics December ^-I3th, 2013 Department of Physics University of Texas at Dallas, US • COSMO 2013 September 2nd-6th, 2013 Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK • Tales of Lambda July 1^-6^, 2013 Centre for Astronomy and Particle Theory University of Nottingham, UK • Infinities and Cosmology March lS^l81, 2013 Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK • Sixth TRR33 Winter School on Cosmology December 9th-14th, 2012 Pas so del Tonale, Italy • Gravity Beyond Einstein September 27th, 2012 Institute of Physics, London, UK • Cosmological Frontiers in Fundamental Physics (workshop) May l^-June 1st, 2012 Solvay Institute, Brussels, Belgium • The State of the Universe: Stephen Hawking 70th Birthday Conference January 5^-8*, 2012 Department of Applied Mathematics and Theoretical Physics University of Cambridge, UK Teaching Experience • Undergraduate - Cosmology (Fall 2011) - General Relativity (Spring 2012, 2013, 2014) • Graduate - Cosmology (Fall 2012, 2013, 2014) Major Awards and Fellowships 2015 DAAD Research Grant for PhD Students, Postdocs, and Junior Academics 2014 Cambridge Philosophical Society Research Studentship 2014 NORDITA Visiting PhD Student Fellowship Curriculum Vitae 231 2011 Gledhill Research Studentship, Sidney Sussex College, University of Cambridge 2011 Isaac Newton Studentship, University of Cambridge 2011 Tyson Medal, University of Cambridge For the best performance in subjects relating to astronomy. Awarded only at the discretion of the examiners. 2011 Jennings Prize, Wolf son College, University of Cambridge For performance on the Part III examinations. 2010 Ellsworth Prize, Jonathan Edwards College, Yale University For the best senior thesis in the sciences. 2008 Caltech Summer Undergraduate Research Fellowship (SURF) 2007 Caltech and Siemens Foundation Summer Undergraduate Research Fellowship (SURF) 2006 Intel Science Talent Search - Finalist (8th place) 2006 Davidson Fellow - $25,000 2005 Siemens Competition - Finalist (3rd place) Professional Activities Since 2015 Referee, Physics Letters B; General Relativity and Gravitation; Physics of the Dark Universe; European Physical Journal C. Reviewer, NASA Earth and Space Science Fellowship Program 2015 Group leader, Beyond ACDM conference, Oslo 2014 Organized bigravity meeting at Cambridge for collaborators from Oslo, Stockholm, and Heidelberg 2013-2015 Organizer, DAMTP cosmology journal club 2011-2015 Group leader, Part III (masters students) seminar series