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Cosmological tests of gravity Narimani, Ali 2016

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Cosmological Tests of GravitybyAli NarimaniB.Sc., Mechanical Engineering, Isfahan University of Technology, 2007B.Sc., Physics, Isfahan University of Technology, 2008M.Sc., Mechanical Engineering, Isfahan University of Technology, 2009M.Sc., Astronomy, The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Ali Narimani 2016AbstractGeneral Relativity (GR) has long been acclaimed for its elegance and simplicity, and hassuccessfully passed many stringent observational tests since it was introduced a centuryago. However, there are two regimes in which the theory has yet to be fully challenged.One of them is in the neighbourhood of very strong gravitational fields, and the other isthe behaviour of gravity on cosmological scales. While strong field gravity has challengedtheorists because of the desire to find consistency between GR and quantum mechanics,cosmology has motivated extensions to GR via the empirical discoveries of dark matter anddark energy.In this thesis, we study a diverse range of modifications to GR and confront them withobservational data. We discuss how a generic theory of modified gravity can be param-eterized for studies within cosmology, and we introduce a general parameterization thatis simpler than those that have been previously considered. This parameterization is thenapplied to investigate a specific theory, known as “gravitational aether”. The gravitationalaether theory was created to solve one of the theoretical inconsistencies that exists betweenGR and quantum mechanics, namely the fact that vacuum fluctuations appear not to grav-itate. Cosmology is unique in testing this theory, and we find that the gravitational aethersolution is excluded when all of the available cosmological data are combined. Neverthe-less, a generalization of this theory provides a consistent way to describe the strength ofcoupling between pressure and gravity, and we present the most accurate measurements ofthis coupling parameter.In addition, we discuss the constraints that can be placed on modified gravity models us-ing the latest data from cosmic microwave background (CMB) anisotropies, combined withiiAbstractseveral other probes of large-scale structure. Currently the most accurate CMB anisotropymeasurements come from the Planck 2015 CMB power spectra, which we use, along withother cosmological data sets, to perform an extensive study of modified theories of gravity.We find that GR remains the simplest model that can explain all of the data. We end witha discussion of the prospects for future experiments that can improve our understanding ofgravity.iiiPrefaceThree chapters of this dissertation are based on previously published work. My contribu-tions to each work are explained below.• Chapter 3 is based on a published paper in Physical Review D titled “Minimal param-eterizations for modified gravity”. I performed the numerical calculations, made theplots, and wrote a first draft of the paper. Douglas Scott provided crucial guidanceand significant improvement to the text. We received helpful feedbacks from JamesZibin.• Chapter 4 is based on a published paper in Journal of Cosmology and AsroparticlePhysics. The analytical and numerical calculations, statistical analyses, and plotswere made by me. I received guidance and feedback from Niayesh Afshordi, andDouglas Scott. I wrote the first draft of the paper which was edited by Douglas Scottand Niayesh Afshordi. Niayesh also added important parts to the conclusion chapter.• Chapter 5 is a reformatted version of published work in Astronomy and Astrophysics.The original paper was among the 2015 primary publication and data release of thePlanck collaboration. The numerical calculations, plots, and first draft of the includedsections were all done by me. I received feedback from Martin Kunz and Valeria Pet-torini in our weekly teleconferences, and the text was edited by the Planck editorialboard.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Ancient history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Modern cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Evidence for hot big bang cosmology . . . . . . . . . . . . . . . . . . . . 91.3.1 Expansion of the Universe . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Big bang nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Relic radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Perturbations in the Universe . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Metric perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.1 Classification of metric perturbations . . . . . . . . . . . . . . . . 121.5.2 Choosing a gauge . . . . . . . . . . . . . . . . . . . . . . . . . . 15vTable of Contents1.5.3 Matter fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Qualitative understanding of the perturbation equations . . . . . . . . . . . 171.7 Observational evidence for big bang cosmology at the perturbation level . . 181.7.1 CMB anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . 191.7.2 Baryon acoustic oscillations and other data sets . . . . . . . . . . . 211.8 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 An Introduction to Modified Gravity . . . . . . . . . . . . . . . . . . . . . . 232.1 Gravity beyond GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.1 Scalar-Tensor theories . . . . . . . . . . . . . . . . . . . . . . . . 252.1.2 f (R) theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Notes on modified gravity studies, their limitations and uses . . . . . . . . 283 Minimal Parameterization for Modified Gravity in Cosmology . . . . . . . . 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Modified gravity formulation . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Connection with other methods of parameterization . . . . . . . . . . . . . 353.4 Numerical calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.1 Effects of A and B on the power spectra . . . . . . . . . . . . . . . 393.4.2 Markov chain constraints on A and B . . . . . . . . . . . . . . . . 433.4.3 Alternative powers of k in B . . . . . . . . . . . . . . . . . . . . . 463.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 How Does Pressure Gravitate?Cosmological Constant Problem Confronts Observational Cosmology . . . . 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Equations of motion at the background and perturbative level . . . . . . . 544.3 Cosmological constraints on GGA . . . . . . . . . . . . . . . . . . . . . . 584.3.1 Consistency checks on CAMB . . . . . . . . . . . . . . . . . . . 594.3.2 Cosmological constraints . . . . . . . . . . . . . . . . . . . . . . 63viTable of Contents4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5 Conclusions, and open questions . . . . . . . . . . . . . . . . . . . . . . . 715 Planck 2015 Results. XIV. Dark Energy and Modified Gravity . . . . . . . . 745.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Why is the CMB relevant for modified gravity? . . . . . . . . . . . . . . . 775.3 Models and parameterizations . . . . . . . . . . . . . . . . . . . . . . . . 785.4 Modified gravity and effective field theory . . . . . . . . . . . . . . . . . . 795.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.5.1 Planck data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.5.2 Background data combination . . . . . . . . . . . . . . . . . . . . 835.5.3 Perturbation data sets . . . . . . . . . . . . . . . . . . . . . . . . 845.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.6.1 Perturbation parameterizations . . . . . . . . . . . . . . . . . . . 865.6.2 Modified gravity: EFT and Horndeski models . . . . . . . . . . . 885.6.3 Further examples of particular models . . . . . . . . . . . . . . . 925.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103viiList of Tables1.1 A list of the cosmological observables and their information content. WLand RSD stand for weak lensing and redshift space distorsion data sets,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 The mean likelihood values together with the 68% confidence interval forthe usual six cosmological parameters, together with constant A and B, us-ing CMB constraints only. . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1 Mean likelihood values together with the 68% confidence intervals for theusual six cosmological parameters (see Ref. [1]), together with the GGAparameter GN/GR. “WP” refers to WMAP-9 polarization, which has beenused to constrain the optical depth, τ. “HighL” refers to the higher multipoledata sets, ACT and SPT. The PPN parameter, ζ4 can be obtained throughζ4 = GR/GN − 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Mean likelihood values together with the 68% confidence intervals for theusual six cosmological parameters, plus r (the tensor-to-scalar ratio), to-gether with the GGA parameter GN/GR. The data are as in Table 4.1, butnow including BICEP2 measurements of the B-mode CMB polarization.GR is still favoured over GA if we include HighL CMB or BAO measure-ments. However, even the conventional seven parameter GR model (thatincludes r), is disfavoured at around the 3σ level when one considers BI-CEP2, as well as Planck, and HighL data. The PPN parameter, ζ4 can beobtained through ζ4 = GR/GN − 1. . . . . . . . . . . . . . . . . . . . . . . 66viiiList of Tables5.1 Marginalized mean values and 68 % CL intervals for the EFT parameters,both in the linear model, αM0, and in the exponential one, {αM0, β} (seeSect. 5.6.2) Adding CMB lensing does not improve the constraints, whilesmall-scale polarization can more strongly constraint αM0. . . . . . . . . . 935.2 95 % CL intervals for the f (R) parameter, B0 (see Sect. 5.6.3). While theplots are produced for log10 B0, the numbers in this table are produced viaan analysis on B0 since the GR best fit value (B0 = 0) lies out of the boundsin a log10 B0 analysis and its estimate would be prior dependent. . . . . . . 97ixList of Figures1.1 Measurements of the CMB temperature anisotropies from the Planck satel-lite [2], and the predictions of the best-fit ΛCDM model (red curve). Errorbars show 68% confidence intervals. The low-` range is plotted in a loga-rithmic axis, with each multipole plotted, while the high-` range os plottedon a linear axis, with points binned together. . . . . . . . . . . . . . . . . . 202.1 A very schematic chart of the different theories in physics, and their realmof relevance. While Newtonian mechanics works well on everyday scales,things change rather drastically on very small/large scales, or high energies. 293.1 Effects of a constant, non-zero A and B on the ISW effect. This plot showsthe ISW effect for a specific scale of k = 0.21 Mpc−1. . . . . . . . . . . . . 413.2 Effects of a constant B on the CMB power spectra. The plot shows thedifference in power for the case of zero B minus the best fit non-zero B,using WMAP9 and SPT12 data, while keeping all the rest of the parametersthe same. The error band plotted is based on the reported error on the binnedCMB power spectra from the WMAP9 [3] and SPT12 [4] groups. . . . . . 423.3 Effects of a constant, non-zero A or B on the CMB power spectra. One cansee that the two parameters have quite different effects. . . . . . . . . . . . 433.4 68 and 95 percent contours of the constants A and B using WMAP9-yeardata alone (left) and SPT12 (right) without including lensing effects, andneglecting late time growth effects. . . . . . . . . . . . . . . . . . . . . . . 44xList of Figures3.5 68 and 95 percent contours of the constants A and B using WMAP9-yeardata alone (left) and WAMP9 + SPT12 (right), with lensing effects included. 453.6 The strong anti-correlation between parameter A and the initial amplitude,As, makes the constraints on either one of these two parameters weaker. . . 463.7 Effect of a non-zero B0 on the CMB power spectra, with the choice B =H B0/k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1 Effects of non-radiation modifying terms compared to the effects of radia-tion, at k = 0.04 Mpc−1 (see Eq. (4.29)). This shows that the non-radiationterms are smaller by more than a factor of 20. . . . . . . . . . . . . . . . . 604.2 Checking the code at the perturbation level by comparing the CMB anisotropiespower spectra for the models P1 := {GRN = 1/3GN,Neff = 3.04,T 40 =(2.7255)4} and P2 := {GR withNeff = 4/3 × 3.04, T 40 = 4/3 × (2.7255)4}(left panel). There is a small difference between the two at around the firstpeak. This can be explained by considering the effects of a non-zero ω (di-vergence of the aether four-velocity). The two models completely coincidewith each other by setting ∆ω to zero in the P1 model (right panel). . . . . 614.3 Checking the code at the perturbation level by comparing the the modelsP1 := {GRN = 1/3GN,Neff = 3.04,T 40 = (2.7255)4} andP2 := {GR withNeff =4/3×3.04, T 40 = 4/3×(2.7255)4} (left panel). The small difference betweenthe two can be explained by considering the effects of a non-zero ω. Thetwo models completely coincide with each other by setting ∆ω to zero inthe P1 model (right panel). . . . . . . . . . . . . . . . . . . . . . . . . . 624.4 Comparing general relativity versus gravitational aether predictions for theCMB power spectrum. The values of the input parameters for the left panelare taken from the Planck analysis [1]. The right panel compares the best-fit predictions of the two theories with all cosmological parameters alsoallowed to vary. GA predicts less power at higher `s, as one can see fromthe right panel (this difference is more evident in the residual plot in Fig. 4.5). 64xiList of Figures4.5 Comparing general relativity (bottom panel) and gravitational aether (toppanel) predictions for the CMB power spectrum with Planck and SPT datasets. Here we plot D` ≡ `(` + 1)C`1/2pi residuals, along with ±1σ errorbars, from Refs. [1] and [5]. While the two theories can both fit the lower-`observations, GR fits the data points significantly better than GA for ` &1000, at the > 4σ level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.6 Confidence intervals (68% and 95%) for the GGA parameter and the cos-mological parameters it is most degenerate with. The ratio GN/GR is plot-ted on the left axes and ζ4 on the right axes. The horizontal dashed linesindicate the GR (top line) and GA (bottom line) predictions. . . . . . . . . 674.7 A pictorial comparison of marginalized GN/GR = (1 + ζ4)−1 measurements.We have plotted the central values and ±1σ error bars using different datasets. The GR and GA predictions are shown as vertical dashed lines. . . . . 685.1 Marginalized posterior distributions at 68 % and 95 % C.L. for the two pa-rameters αM0 and β of the exponential evolution, Ω(a) = exp(Ω0 aβ) − 1.0,see Sect. 5.6.2. Here αM0 is defined as Ω0β and the background is fixedto ΛCDM. αM0 = 0 corresponds to the ΛCDM model also at perturbationlevel. Note that Planck means Planck TT+lowP. Adding WL to the data setsresults in broader contours, as a consequence of the slight tension betweenthe Planck and WL data sets. . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Marginalized posterior distribution of the linear EFT model backgroundparameter, Ω, with Ω parameterized as a linear function of the scale factor,i.e., Ω(a) = αM0 a, see Sect. 5.6.2. The equation of state parameter wde isfixed to −1, and therefore, Ω0 = 0 will correspond to the ΛCDM model.Here Planck means Planck TT+lowP. Adding CMB lensing to the data setsdoes not change the results significantly; high-` polarization tightens theconstraints by a few percent, as shown in Tab. 5.1. . . . . . . . . . . . . . . 92xiiList of Figures5.3 68 % and 95 % contour plots for the two parameters, {Log10(B0), τ} (seeSect. 5.6.3). There is a degeneracy between the two parameters for PlanckTT+lowP+BSH. Adding lensing will break the degeneracy between thetwo. Here Planck indicates Planck TT+lowP. . . . . . . . . . . . . . . . . 955.4 Likelihood plots of the f (R) theory parameter, B0 (see Sect. 5.6.3). CMBlensing breaks the degeneracy between B0 and the optical depth, τ, resultingin lower upper bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96xiiiGlossaryCMB Cosmic Microwave BackgroundBAO Baryon Acoustic OscillationsWMAP Wilkinson Microwave Anisotropy ProbeACT Atacama Cosmology TelescopeSPT South Pole TelescopePPN Parameterized Post-NewtonianPPF Parameterized Post-FriedmannCAMB Code for Anisotropies in the Microwave BackgroundEFT Effective Field TheoryxivAcknowledgementsThere are a few people I should thank, without whom aiming for a PhD, and writing thisthesis would have been impossible. First, I want to thank my eighth-grade science teacher,Reza Khalifeh-Mahmoodi, for all the things he did for me (from inviting my parents toschool for a private conversation, to his one-on-one lessons that enabled me pass the en-trance exams of the best high schools in my home-town). His encouragement and beliefin me was so influential that I sometimes think I could have been somewhere else doingsomething totally different without his presence in my life.Next, I should thank my Bachelor’s and Masters degree advisers and supervisors inIran, Mansour Haghighat, Mojtaba Mahzoon, Behrouz Mirza, and Saeed Ziaei-Rad.The biggest thank you goes to my Masters and PhD supervisor, Douglas Scott. I cannotthank him enough for his persistent effort in guiding me through my research projects,connecting me with other researchers in the field, and creating the opportunity for me tobecome a Planck member. I benefited multiples of times from his generous travel supports,and his endless patience in reading and editing my reports, paper drafts, and, of course, thisthesis.I want to thank Niayesh Afshordi for offering me one of his research projects and payingfor my visit and stay in Perimeter Institute. We had a fruitful collaboration that resulted ina paper in JCAP, and is the basis for a whole chapter of this thesis.I would also like to thank Lloyd Knox for offering me a travel support to UC-Davis. Ilearnt a lot from his expertise during my visit, and our weekly telecons with him, MariusMillea, Silvia Galli, Martin White, and Douglas Scott.My office mate, Anna Hughes, helped me in translating the two poems at the beginningxvAcknowledgementsof the Introduction, and the end of the Conclusion chapters. Thanks Anna for all your help,and all the political-social thought provoking discussions.I would want to sincerely thank my beloved parents and sisters. Thank you so muchfor giving me the freedom to choose my path independently, and being supportive of all thedecisions I have made for my life.This thank-you-list misses a significant member without the love of my life, Mina.Thanks darling for all your care, love, and support during these my best five years of life.xviChapter 1Introduction1Chapter 1. Introduction... If love falls in the heart of a stoneIt reaches out its hands, like a lover, to embrace a gemFor if magnetism was not in loveWhy should it grab an iron so enthusiasticallyAnd, with no love going onAmber would not always seek for a strawThere are so many stones and gems on the EarthAnd they neither absorb iron, nor strawThe essences, that are countless in numberAll have a tendency towards a centreCreatures do not know anything but to absorb each otherAnd this absorption is called “love” by philosophersWith no love in the heavensThere would be no life on the Earthfrom “Khosro and Shirin” by Nezami Ganjavi, Iranian poet, 12th century AD21.1. Ancient history1.1 Ancient historyThe struggle for understanding the sky above us, and the effort to tie all of the seeminglyunrelated phenomena around us on Earth, along with what is going on in the skies, into ameaningful and coherent story, has a long history, probably as long as humanity. Among theearliest theories (or stories) accessible to us, are the works of Thales, from the 7th centuryB.C., communicated to us through other philosophers [6]. Thales’s main concern, like otherthinkers of his time, was to find the element that is the origin of everything we see on Earth.According to him, Earth is flat and floating on a great ocean, the water of which is theorigin of everything. The ocean beneath is not supported by anything, and the world aboveis limited by a vault, to which the stars are attached. He makes no distinction between thesolar system planets and the distant stars.The real breakthrough, at least among the ancient Greeks, comes with the Pythagoreanschool, 6th-4th century B.C. They, famously, held the rather eccentric idea that numbers arethe origin of everything. Although strange by modern scientific standards, their obsessionwith numbers led to a better picture of the ‘Kosmos’. They believed that Earth, togetherwith all of the other celestial bodies, has a spherical shape, because a sphere possesses themost perfect shape of all the objects. They were the first to figure out that the harmonyin musical sounds is governed by a regular pattern of intervals, and that the planets andthe Sun are also going around the Earth in regularity. As for their view of the large-scaleUniverse, they dissected the Universe into three spherical pieces [6]: the Heavens, whichincluded the Earth, was bounded with the sphere of the Moon, and was the home for lifeand change; the Kosmos, which encompassed the Moon, the Sun, and the planets and wasthe place of regular motion; and the Olympus, the sphere of distant fixed stars, which wasthe place for elements in their pure form. They also had a few odd beliefs, even judged bythe standards of their own time, which stemmed from their unusual mathematical principle.They believed that there exists a planet that is not observable for us on Earth, because it isalways behind the Sun, and named that planet antichthon, or anti-Earth [7]. They added thisnew planet because the sum of the other bodies in the sky, i.e. the Sun, the Moon, Earth,31.1. Ancient historythe five planets, and the distant stars is nine, which is not a perfect number. Their solutionwas to add one more planet to complete the system of ten objects.Pure thinking and mythology occasionally led ancient Greeks to notions that were lateradopted by modern physics. The idea of an “aether” or pure air (sometimes referred toas quintessence) that gods breathe, first appeared in Greek mythology, and was later intro-duced to philosophy through the works of Plato and Aristotole [8]. These great philosophersintroduced aether as the fifth classical element. These five elements were proposed to in-terpret the nature of all that exists, and aether was proposed to explain the circular motionof the celestial bodies. Aether came back to modern physics at times when the known con-stituents of the Universe were not sufficient for explaining the world, e.g. in defining thespeed of light with respect to a reference frame, or for what drives the expansion of theUniverse (as we will discuss at length later). It is also remarkable that Leukippus and hisdisciple Demokritus, 5th century B.C., believed that the building blocks of all the materialshould be indivisible bodies which they called “atoms” [6], an idea that is also more or lessequivalent to our modern atomic view.Following this story further would require a thesis of its own, but it is probably suffi-cient to mention that things had changed a lot by the beginning of the twentieth century.A series of observations and insights by Brahe, Galileo, and Kepler, had led Newton to hisremarkable discovery of Newtonian gravity. Two new planets, other than the ones knownsince ancient times, had been found, the discoveries of which were a source of admirationand debate in science and philosophy and provided further tests of Newtonian gravity [9].It was common knowledge among scientists by then that Earth is not at the centre of every-thing, but is actually going around the Sun. The existence of other galaxies was still justa speculative idea, and the common belief among many was that the Milky Way galaxy isthe only structure in the Universe [10]. Earth was not at the centre of everything, but stilloccupied a special place, since the solar system was thought to be very close to the centre ofthe Milky Way. Although these discoveries were all effectively tests of the Newtonian ideasof gravity, there was still a fundamental problem that there was no consistent gravitational41.1. Ancient historytheory for describing the whole Universe.One major difference between twentieth century scientists and ancient Greek philoso-phers, which is probably more remarkable than their factual discoveries, lies in the world-view of the scientists. The main concern of early 20th century scientists was to explainnewly discovered phenomena, based on other known facts and well-tested theories, ratherthan wrapping them in metaphysical stories or unchangeable rules. This is the key to prac-tising science even today.We have now come far from those earlier ideas of antiquity, or even the somewhat mis-guided beliefs about the Universe a century ago. We now know that the Milky Way is onlyone galaxy among an incredibly large (perhaps infinite) number of them. We can, however,only access a subset of these galaxies, the ones that are inside the “observable Universe”.We also know that the Universe is full of a “dark energy” that forms approximately 70% ofthe energy content, and causes an ever increasing expansion rate for the Universe. Of theremaining 30%, in terms of energy budget, most is in a substance dubbed “dark matter”,for it can be neither touched nor seen, even though it is responsible for the formation ofgalaxies and large-scale structures in the Universe, among other things. All that can beobserved, touched, or felt with our senses consists of about 1/6 of the matter, a 5% fractionof the whole Universe, that is in the form of atoms or regular matter. Among all of thesestructures and substances of the Universe, the solar system holds no special place, since itis in an unremarkable part of the disk of a fairy normal galaxy, which is one of billions ofgalaxies in the observable Universe.Even though it may appear that we have invented a new set of “mystical” ideas forexplaining the newly discovered cosmological phenomena, there is a big difference betweenthese set of modern ideas, and those of antiquity. The real merit of these ideas is in theircapability to explain a very diverse range of empirical phenomena with high precision whilebeing amenable to tests using further observations, and, maybe most importantly, to guideus to a whole new set of questions and directions for understanding the physical world.The main goal of this thesis is to explore one of these directions, namely the theoretical51.2. Modern cosmologyspace of “modified theories of gravity” and to test their predictions using cosmological data.Modified theories of gravity alter Einstein’s gravity in order to either find an explanation forone of the two dark entities in the Universe, or to solve some of the theoretical difficultiesthat face the theory when it is combined with quantum mechanics. Such ideas will beconstrained by performing tests on the behaviour of gravity on the largest scales that wecan probe. We will explain Einstein’s gravity and the standard model of cosmology in therest of this chapter, and introduce modifications to this theory, as well as discussing therelevance of cosmology for testing those modified theories in the following chapter. Notethat we use units in which the speed of light is unity, c = 1, throughout this thesis.1.2 Modern cosmologyThe birth of modern cosmology was made possible only after the introduction of generalrelativity (GR) by Einstein in 1915 [11]. GR links the structure of space-time with its energycontents through the field equations,Gµν + Λgµν = 8 piG Tµν . (1.1)The left-hand side of this equation is a specific nonlinear second-order differential equationfor the space-time metric, gµν, while the right-hand side holds the energy momentum tensorof the constituents of that space-time, Tµν (times a constant). Here, G is Newton’s grav-itational constant and Λ, or the cosmological constant, is a free parameter of the theory,whose non-zero value is also the best currently available candidate for dark energy. The{µ, ν} indices run from zero to three, representing time and the three spatial coordinates.Even though the Universe looks very lumpy and anisotropic on very small scales, (e.g.Earth has a very high density compared to the space between the planets) it becomes moreand more homogeneous and isotropic as we go to larger and larger scales, e.g. hundreds ofmillions of light years, compared to the solar system scales that are hundreds of light hoursat most.61.2. Modern cosmologyIsotropy and homogeneity of the Universe on its largest scales was once a mere as-sumption, called the “cosmological principle”, but it now has roots in cosmological ob-servables (e.g. see [12]). The assumption of homogeneity and isotropy dictates the form ofthe energy-momentum tensor and the metric [13]. The energy-momentum tensor will onlycontain perfect fluids, and the metric is the Friedman-Robertson-Walker (FRW) metric,ds2 = gµν dxµ dxν = −dt2 + a2(t)(dr21 − k r2 + r2dΩ2). (1.2)Here, k is a parameter that determines the curvature of the Universe, with positive k valuescorresponding to spherical geometry, negative to hyperbolic, and k = 0 to a flat Universe.We set k = 0 in the rest of this thesis, since this is what the most recent data points to [2].The quantity a(t), with a = 1/(1+ z) (z being the observed redshift), is a free function calledthe “scale factor” that simply scales all lengths at different times.The FRW metric [14, 15] is the basis of big bang cosmology, leaving us to determinek and a(t) through observations. The only dynamical element of this metric is the scalefactor, whose evolution is governed by Eq. (1.1). The evolution equations were first derivedby Friedmann in 1922 [16] and areH2 =8 piG ρ + Λ3,a¨a= −4 piG3( ρ + 3 p) +Λ3. (1.3)Here, H ≡ a˙/a is called the Hubble constant, with a dot meaning a derivative with respectto the coordinate time, t. The quantities ρ and p are the total density and pressure of all thefluids in the energy momentum tensor and Λ is again the cosmological constant.ΛCDM modelIn order to fully specify the theory, one has to define the fluids in the energy-momentumtensor. Based on the currently accepted model, the energy momentum tensor contains fivedifferent types of substance: baryons, photons, neutrinos, cold dark matter, and dark energy71.2. Modern cosmology[17, 18]. The most fundamental difference between these different types of fluids lies intheir equation of state, p = w ρ, where p stands for pressure, ρ is the density of the fluid andw is the equation of state parameter.All of the particles that are formed from electrons, protons, and neutrons are groupedunder “baryons” with an equation of state where w = 0 (actually not quite zero, but the ther-mal pressure is negligible in density units). While photons and massless neutrinos share thesame equation of state, w = 1/3, massive neutrinos start from this state at early times, butgradually move towards w = 0 as the Universe evolves. Dark matter has the unique prop-erty that it either has a very weak, or entirely vanishing interaction through any force otherthan gravity with any particle (including itself), and therefore w = 0. The cosmologicalconstant, on the other hand, has a constant density and pressure, with p = −ρ, and thereforew = −1. As was mentioned before, the cosmological constant, Λ, is the best (i.e. sim-plest) mathematical model of dark energy as of today, although possibilities with w , 0 andnon-constant w(a) are also consistent [19].This “standard” model, ΛCDM, is named after its two strange components, Λ for darkenergy, and CDM for cold dark matter. The model is fully specified through its six param-eters: {Ωbh2,ΩDMh2,H0, τ, As, ns}. The first two parameters of the list are proportional tothe energy density of baryons and cold dark matter,Ωxh2 ≡ 8 piG ρx3 ∗ (100)2 . (1.4)Here, x stands for either baryons (b), or CDM (DM). τ is the optical depth of the Universe,the probability that a given photon scatters before reaching the observer. {As, ns} are the ini-tial condition parameters, and determine the current lumpiness of the Universe on differentscales. As we shall see, this six-parameter model fits a wide range of data extremely well.81.3. Evidence for hot big bang cosmology1.3 Evidence for hot big bang cosmologyThere are three classical pieces of evidence for the hot big bang model, which we nowdescribe in turn.1.3.1 Expansion of the UniverseEq. (1.3) states that, in contrast to the generally held belief of the early 20th century, theUniverse cannot be static but is instead expanding. This expansion was discovered by EdwinHubble in 1929 [20] (the story of this discovery is complicated, Kragh and Smith [21]present a rather long discussion of who should be credited for this discovery). The Hubbleconstant, defined in the previous section, measures the expansion rate of the Universe atthe present time. The current estimate of this parameter from the local Universe, measuredfrom Cepheid data [22] (including reassessment of uncertainties [23]) isH0 = (70.6 ± 3.3) km s−1 Mpc−1. (1.5)Adding a ‘0’ subscript is a standard notation for referring to the present time.Further observations determined that the expansion is accelerating [18], giving con-straints on the behaviour of a(t). But the basic idea of an expanding Universe has beenexamined and confirmed by a huge set of observations over the last decades [24]. Thisalso tells us that the Universe has a finite age in the big bang model, with the best currentestimate being t0 = 13.813 ± 0.038 Gyr[2].1.3.2 Big bang nucleosynthesisUsing Eq. (1.3), along with an equation of state for each of the fluids, it is clear that theUniverse should have been in a hot and dense state during the earliest stages of its evolution.This temperature was so high that the protons and neutrons could not bind together to formany of the elements in the periodic table. The high density and hot temperature of theUniverse at early times meant very frequent collisions between particles, keeping them in91.3. Evidence for hot big bang cosmologyequilibrium with each other, until the collision rate became comparable to the expansionrate of the Universe. This was when the out-of-equilibrium process of the formation ofheavier elements began. Alpher et. al. [25] calculated the ratio of hydrogen to heliumatoms based on the big bang theory in 1948. Their predictions were later confirmed by acontinuous set of observations [26], extending to the present day. This explains why, onaverage, the Universe consists of about 25% helium (by mass) and 75% hydrogen, withabout one part in 105 of the hydrogen in the form of deuterium [27].1.3.3 Relic radiationAs the Universe expanded, its temperature cooled adiabatically and the first neutral atomsstarted to form a few hundred thousand years after the big bang. As these atoms formed, thecollision rate between photons and baryons dropped significantly and the photons startedtheir “free streaming” phase. This epoch is called “recombination”, even though it was thefirst time that electrons and protons combined together and formed neutral atoms. The pre-diction that the remnant of the early radiation should now be detectable on Earth accordingto the first estimates, by Alpher et. al in 1948 [25] (although the story is again complicated,see the book by Peebles et al. [28]), and particularly its discovery in 1964 by Penzias andWilson [29] had a major impact on the general acceptance of the hot big bang theory. Thisrelic radiation has a blackbody spectrum, its temperature is the best currently measurednumber in cosmology [30], T0,CMB = (2.7255 ± 0.0006)K, and it is usually referred to asthe “Cosmic Microwave Background” or CMB. It is a “cosmic” background, since it is re-ceived as a very nearly isotropic radiation field that cannot be attributed to any individualstar, galaxy, or other localized structure. Although almost isotropic, as we shall show, smallamplitude variations in the CMB are a critical observable that is used to precisely pin downour cosmological model, as well as to constrain theories of modified gravity.Although we have described the three main pieces of evidence used to support the hotbig bang model, there have been a huge number of additional observations that back upthis picture, including studies of objects at high redshift, clustering of galaxies, and clouds101.4. Perturbations in the Universealong the line of sight to distant objects [31].1.4 Perturbations in the UniverseThough generally successful, the classical big bang theory had a few problems or missingingredients. One of the most important shortcomings was the lack of a clear recipe formaking galaxies and all the other structures we see in the Universe. It also potentiallysuffered from a singularity at the bang moment, t = 0 and no explanation for the expansion.Additionally there were issues with explaining flatness, near-isotropy of the CMB sky andlack of magnetic monopoles. These problems were solved by the addition to the theory of anearly era of exponential expansion. Based on this new picture, the Universe is presumed tobe dominated by the potential energy of a scalar field during an “inflationary” era [32, 33].The quantum fluctuations of this scalar field provide the initial perturbations in the Universe,which then grow through gravitational instabilities and form the structures, such as galaxiesand galaxy clusters, which we see today. We will complete this chapter by describing thebasics of the theory at the perturbation level, and some of the most relevant observationsthat are used to test the theory.1.5 Metric perturbationsEven though the FRW metric (which describes a homogeneous and isotropic Universe) cansuccessfully explain a good number of observations, the existence of galaxy clusters andvoids show that the Universe is far from being homogeneous and isotropic on all scales. Thegeneral idea for explaining the structures in the Universe is to assume that they are formedfrom minuscule perturbations in the density field of the fluids. These early perturbations actlike seeds of structure formation. The regions that are a little overdense will expand moreslowly, and hence the effect of gravity is to make them increasingly more dense comparedto the expanding background. This positive feedback loop continues until the formation ofvery first stars and structures in the Universe. Since the deviations from homogeneity are111.5. Metric perturbationsbelieved to be relatively small at early times (on all scales), it is reasonable to adopt linearperturbation theory for studying their evolution. We lay out the foundations of cosmologicalperturbation theory in this section, and will explain the observational tests that follow fromits use in the following section.In order to perturb the gravitational fields in GR, we need to consider metric perturba-tions. The first order background metric was written based on the observed symmetries ofour Universe, i.e. homogeneity and isotropy. We now break these symmetries by addingsome general perturbation to the background metric:gµν = gµν + hµν, hµν  1. (1.6)For the rest of this chapter, a bold quantity will mean a tensor that contains both the back-ground and perturbation terms. In words, Eq. (1.6) means that gµν, which was a good toolfor measuring the distance in a fully isotropic space-time, is slightly modified to gµν. Thequantity gµν will take us one step closer to reality, because it is capable of describing phe-nomena beyond perfect homogeneity, and will allow us to address observables that relate toinhomogeneity and anisotropy.1.5.1 Classification of metric perturbationsLet us use the symmetries of the background metric to classify the metric perturbationsdefined in Eq. (1.6). Consider the following transformation of the space-time coordinates: t → t′ = t;xi → x′i = ∆i(x j).(1.7)Here ‘i’ runs from one to three and ∆i is a function consisting of either a (constant) rotationor displacement.1 Following the general rule of coordinate transformation for tensors we1 The rotation or the displacement transformations are not infinitesimal in general.121.5. Metric perturbationsfind that the space-time metric transforms asg′µν = g′µν + h′µν =∂xα∂x′µ∂xβ∂x′νgαβ=∂xα∂x′µ∂xβ∂x′ν(gαβ + hαβ)= gµν +∂xα∂x′µ∂xβ∂x′νhαβ. (1.8)The last line follows because the background metric is invariant under the coordinate trans-formations defined in Eq. (1.7), i.e. under rotations and displacements. The zeroth-orderequation then simply reads g′µν = gµν, and the first-order terms giveh′00 = h00,h′0i =∂xa∂x′ih0a,h′i j =∂xa∂x′i∂xb∂x′ jhab.(1.9)Therefore, h00 transforms like a scalar, h0i like a spatial vector, and hi j like a tensor. Aspatial vector can be further decomposed into a scalar and a vector, while a tensor, like hi j,can be decomposed into two scalars, a divergenceless vector and a traceless divergencelesstensor:h00 ≡ −A,h0i ≡ a(t) (∂iB + Ei), ∂iEi = 0,hi j ≡ a2(t) (Cδi j + ∂i∂ jD + ∂iF j + ∂ jFi + Ji j), {∂iFi = 0, Jii = 0, ∂iJi j = 0}.(1.10)Hence, under the transformations defined in Eq. (1.7), the set {A, B,C,D} behaves likescalars, {Ei, Fi} transform like vectors, and Ji j like a tensor. The metric perturbation thushas four degrees of freedom in scalars, four in vectors,2 and two in tensors,3 a total of 10degrees of freedom (like one would expect from a four by four symmetric tensor).2 There are two three-dimensional vectors, each of which have one constraint (they are divergenceless).3 Ji j is a 3×3 symmetric tensor, which means six degrees of freedom. However, there are also four constraintson the components of this tensor, defined in Eq. (1.10).131.5. Metric perturbationsUnder a completely general (infinitesimal) coordinate transformation, t → t′ = t + α(xµ), α  1,xi → x′i = xi + δi j∂ jβ(xµ) + ξi(xµ), ∂iξi = 0, {β, ξi}  1,(1.11)the perturbed metric transforms asg′µν = g′µν + h′µν =∂xα∂x′µ∂xβ∂x′νgαβ +∂xα∂x′µ∂xβ∂x′νhαβ=∂xα∂x′µ∂xβ∂x′νgαβ + hµν. (1.12)Here the last line follows because we only keep terms up to first-order. Although one canstill classify the terms into perturbation and background, it is not obvious how to separatea real perturbation, i.e. deviations from a homogeneous isotropic background, from a ficti-tious space-time dependency that is merely due to a minuscule coordinate transformation.To make some progress, one can define the components of the perturbed metric in anycoordinate system asg00 ≡ −1 − A,g0i ≡ a(t)(∂iB + Ei),gi j ≡ a2(t)(1 + Cδi j + ∂i∂ jD + ∂iF j + ∂ jFi + Ji j), (1.13)and take any space-time dependency (besides the a(t)) as a perturbation. Our previous setsof scalars, vectors, and tensors do not behave as they should under these general coordinatetransformations. However, one can prove that for first-order perturbations, the scalars, vec-tors and tensors evolve independently under Einstein’s equations, Eq. (1.1), thanks to therotational symmetry of the background space-time (see [34] for a nice short proof of this).To summarize, the perturbed metric defined in Eq. (1.13), is the sum of the backgroundmetric defined in Eq. (1.2) and the perturbation defined in Eq. (1.10). Under a general141.5. Metric perturbationscoordinate transformation, the perturbed metric transforms asg′µν = g′µν + h′µν=∂xα∂x′µ∂xβ∂x′νgαβ= gµν + ∆hµν + hµν. (1.14)Here: g′µν and gµν are equal to each other and are both equal to the metric defined inEq. (1.2); h′µν and hµν have the same general structure defined in Eq. (1.10), and are relatedto each other, since h′µν = hµν + ∆hµν; and ∆hµν holds the change caused by the coordinatetransformation and at first-order is completely defined via the general coordinate transfor-mation parameters {α, β, ξi}. After a coordinate transformation, the metric perturbationstransform asA′ = A − 2α˙,B′ = B +αa− a β˙,C′ = C + 2Hα,D′ = D − 2 β,E′i = Ei + 2 a ξ˙i,F′i = Fi − ξi,J′i j = Ji j. (1.15)1.5.2 Choosing a gaugeSince all of the coordinate systems follow the same field equations in GR (in contrast toNewtonian mechanics, where only inertial frames follow Newton’s second law), any arbi-trary choice of the {α, β, ξi} functions will generate a new legitimate set of metric variablesat the perturbation level. Fixing the parameters of the coordinate system are usually referredto in the literature as “fixing the gauge” [35, 36]. In particular, starting from a general non-151.5. Metric perturbationszero set of metric variables, one can use Eq. (1.15) and choose a set of {α, β, ξi} functionsthat can set two of the scalars and one of the vectors to zero in the transformed coordinatesystem. This is in fact not a choice but a necessity, since the Einstein equations providetwo independent equations for each of the scalar, vector, and tensor components, whilethe metric perturbation has ten independent components (as was explained earlier). Oneshould therefore use this freedom of choosing a coordinate system to eliminate the fourextra degrees of freedom. Among the two widely used gauges for scalar-perturbations incosmology, one of them sets {B,D} to zero (known as the Newtonian gauge) while the otherone chooses {C,D} as its non-zero metric fields (synchronous gauge). These two metricsare explained further in Chapter 3.1.5.3 Matter fluctuationsThe fluids at the background level were completely specified with two numbers (at anygiven time), their density and pressure. In this homogeneous picture, the particles of a fluidmoved with the same speed (called the “Hubble flow”), and there was therefore no shearstress between the particles. This picture is significantly enhanced at the perturbation levelby adding spatial and temporal deviations to the pressure and density field of the fluids,allowing deviations from the Hubble flow, and non-zero shear stress.The perturbations of the energy momentum tensor from a homogeneous backgroundcan be decomposed into scalar, vector, and tensor perturbations in a similar way as for themetric perturbations. The scalar part of the energy momentum tensor can then be definedas [37]T00 = −ρ¯(1 + δρ),T0i = (ρ¯ + p¯)∂iV,Ti j = δi j( p¯ + δp) + (∂i∂ j −13δi j∇2)σ. (1.16)Here, a bar means a background quantity, ρ and p stand for the density and pressure of the161.6. Qualitative understanding of the perturbation equationsfluids, respectively, V is the amplitude of velocity perturbations, and σ is the source foranisotropic shear stress.After this point, one can use Einstein’s equations, along with a set of initial conditions,to solve the differential equations of the matter and metric perturbations. This is donein many standard texts e.g. in Ref. [37] for the two widely used coordinate systems incosmology.1.6 Qualitative understanding of the perturbation equationsThe perturbation equations form a system of coupled first-order ordinary differential equa-tions. The number of these coupled equations varies depending on the approximations thatare used and typically grows to as many as a hundred in numerical integrators. Approxi-mate analytical solutions of these equations have been extensively studied (e.g. in Hu andSugiyama [38] or Dodelson [39]), and exact numerical solutions are available in publiccodes such as CAMB [40]. In this section, we provide a qualitative description of theseequations and their solution.The general picture to have in mind is an expanding Universe that embodies differentfluids, and is affected by each of these fluids in turn. The first one that dominates theUniverse by density (and is therefore the most important for gravity) is radiation. As theUniverse evolves, radiation gives its place to matter domination, which is then substitutedby dark energy at recent times.The differential equations describe the effect of two rival forces on the perturbationsduring the times that radiation plays a significant role. One is gravity, which is effective onall causal scales (with the limit typically known as the “Hubble scale”), and the other oneis pressure, which is only important on scales smaller than the sound horizon. Hubble scaleis roughly equal to the speed of light multiplied by the age of the Universe, and the soundhorizon is determined from the speed of sound in the fluid. Since the sound speed is alwayssmaller than the speed of light (e.g. is equal to (1/√3) for radiation), the sound horizon isalways smaller than the Hubble scale. These two scales determine the effective forces for171.7. Observational evidence for big bang cosmology at the perturbation leveldifferent scales at every moment in time.At early times when radiation is the dominant fluid, the temperatures are so high thatelectrons do not efficiently bind with the hydrogen or helium nuclei and all the atoms areeffectively ionized. The photons are thus tightly coupled with electrons (which are in turncoupled with the rest of the baryons) through electromagnetic forces, and the fluctuationsin one fluid are also reflected in the other. On scales smaller than the sound horizon, thetwo opposing gravity and pressure forces induce a harmonic oscillatory motion in the per-turbations of the density fields, while the perturbations on larger scales (but smaller than theHubble scale) grow slightly due to gravity. The largest scale perturbations are not affected atall by either pressure or gravity. As the Universe evolves and adiabatically cools, it becomesfavourable for electrons to abandon photons and “recombine” with the positively chargedhydrogen and helium atoms. As this happens, the interaction between electrons and photonsdrops significantly, and the photons start to “free stream” in the Universe, while carryingthe oscillatory structure of early times within themselves. A map of the perturbations of therelic (CMB) photons is therefore like a picture of the Universe when it was still young.Similar to CMB anisotropies, baryon acoustic oscillations (BAO) are the imprints ofperturbations on the matter density field [39]. Unlike CMB photons that free-stream fromthe epoch of recombination to now, baryons have gravitational (and other) interactions withdark matter and themselves, leading to the formation of stars, galaxies, and the other struc-tures in the Universe. The imprints of those early oscillations is visible in the matter powerspectrum on the largest scales, where the evolution of perturbations is still linear.1.7 Observational evidence for big bang cosmology at theperturbation levelAs was explained before, small field perturbations in the very early Universe led to anisotropiesand inhomogeneities in the fluids. Since the origin of these perturbations is believed to bequantum mechanical in nature, it is impossible to predict their exact form, due the so called181.7. Observational evidence for big bang cosmology at the perturbation level“uncertainty principle”. The 1-dimensional analogue of this situation is a particle in an in-finite potential well; while it is impossible to predict the exact location of such a particle ata given moment, the average position of an ensemble of similarly prepared particles, andthe variance of these particle positions, are determined by the width of the potential well,and the energy state of the particles. These ideas are related to two of the main pieces ofobservational evidence for the ΛCDM model, which confirm the theory at the perturbationlevel.1.7.1 CMB anisotropiesThe prediction of small anisotropies in CMB radiation, and their detection by the COBE [41],WMAP [3], and Planck [42] satellites, is one of the greatest successes of modern cosmol-ogy. As well as defining the parameters of the ΛCDM model, CMB anisotropies also play asignificant role in testing theories of gravity beyond Einstein’s theory, as will be explainedin later chapters of this thesis.The anisotropies in CMB temperature can be simply defined asT (xi, t) = T0 + δT (xi, t, nˆ). (1.17)Here the coordinates (xi, t) specify the time and location of the observations, which are typ-ically taken to be today (in a cosmological sense) and on Earth, while nˆ specifies the obser-vation direction. Similar to the particle in a 1-dimensional well, it is intrinsically impossibleto determine the exact shape of the fluctuations, δT (xi, t, nˆ). However, one can instead de-termine the average power-spectrum of these fluctuations, P ≡〈δT (xi, t, nˆ)δT (x′i, t′, nˆ′)〉.It is common to decompose the temperature anisotropies into spherical harmonics, Y`m,δT (xi, t, nˆ) =∑`,ma`m(xi, t) Y`m(nˆ), with 〈a`m〉 = 0, (1.18)191.7. Observational evidence for big bang cosmology at the perturbation leveland define the power spectrum asC` ≡ δ``′δmm′ 〈a`ma`′m′〉 . (1.19)Fig. 1.1 shows the best-fit ΛCDM model prediction, together with data from the Plancksatellite [2]. The imprints of the early harmonic fluctuations of the density fields are dis-tinctively visible in this figure.CMB anisotropies have perturbation amplitudes much smaller than one, δT/T ∼ 10−5,and hence the mathematical equations for describing them are linear and straightforward.Because of this, and the fact that measurements have been made in many different multipolemodes, the CMB provides the most powerful constraints on the cosmological parameters ofthe ΛCDM model. Comparison with Planck data will form a large part of Chapter 5.10 300100020003000400050006000500 1000 1500 2000Multipole `DTT`≡`(`+1)C`/2piFigure 1.1. Measurements of the CMB temperature anisotropies from the Planck satel-lite [2], and the predictions of the best-fit ΛCDM model (red curve). Error bars show 68%confidence intervals. The low-` range is plotted in a logarithmic axis, with each multipoleplotted, while the high-` range os plotted on a linear axis, with points binned together.201.7. Observational evidence for big bang cosmology at the perturbation level1.7.2 Baryon acoustic oscillations and other data setsThe study of BAO signatures is a fairly recent field in modern cosmology. The earliestfoundations of the modern ideas about BAO, and their detection were proposed and dis-cussed around 2003 by a number of different researchers [43–46], and their first detectionwas confirmed in the Sloan Digital Sky Survey, and the 2-degree Field galaxy redshift sur-vey [47, 48] in 2005.Besides CMB and BAO data sets that have established the ΛCDM model, there areother data sets, such as galaxy weak lensing [49, 50], and redshift space distortions [51],which can provide stringent constraints on modified theories of gravity and will be used inlater chapters. Galaxy weak lensing data measures the deflection of light that is emittedfrom distant galaxies. It is called “weak” since the lensing effects are considerably smallerthan in “strong” lensing, and cannot be distinguished by eye. Redshift space distortions, onthe other hand, measure the standard deviation of fluctuations in the matter-density field.This is done via apparent distortions that happen in the redshift space. Distance of galax-ies are measured through their redshift, but the redshift of a galaxy is caused by both itsreceding velocity (due to the Hubble flow), and also its “peculiar velocity” (caused by theenvironment of the galaxy). The redshift space distortions are caused by peculiar velocities,and are good proxies of matter-density fluctuations in the Universe.These data sets prove to be useful since they test theories at the perturbation level andat late times. Since deviations from GR tend to grow over time in many models of modifiedgravity, the late time probes of structure formation, such as redshift space distortions andgalaxy weak lensing, provide very strong constraints on these types of models.Table 1.1 summarizes the different data sets that are used in cosmology, and their rele-vance in testing the theory of gravity.211.8. Structure of this thesisObservation . . . . . . . . . . . . . . . InformationBBN . . . . . . . . . . . . . . . . . . . . . Background evolution at a ∼ 10−10CMB, (` . 800) . . . . . . . . . . . . . Background evolution since z ∼ 1100 and perturbations at z ∼ 1100CMB, (` & 800) . . . . . . . . . . . . . Some of the information contained in ` . 800 plus perturbations at recent timesBAO . . . . . . . . . . . . . . . . . . . . . Perturbations at the decoupling epoch, and background evolution since decoupling up to a ∼ 1WL and RSD . . . . . . . . . . . . . . . Perturbations at recent times (a ∼ 1)Table 1.1. A list of the cosmological observables and their information content. WL andRSD stand for weak lensing and redshift space distorsion data sets, respectively.1.8 Structure of this thesisAfter this brief summary of modern cosmology, the big bang theory, its predictions, andobservational evidence, we will continue the thesis by reviewing some common theoreticalapproaches for modifying GR. In Chapter 2, we will explain the motivations behind mod-ified gravity, and the possible ways that the theory can be modified within the context ofthe language of Lagrangians. We will then proceed in chapter 3 by introducing a generalparameterization scheme that can capture any generic modification of gravity in cosmology.We will use this parameterization to study a specific theory of modified gravity that aimsat solving the so called “old cosmological constant problem” in chapter 4. A new effectivefield theory approach to cosmology is presented in chapter 5, along with tests from the latestobservations of CMB anisotropies by the Planck satellite, combined with other cosmologi-cal data. We conclude the thesis in chapter 6 with some discussion of directions for futurestudy.22Chapter 2An Introduction to Modified GravityGeneral relativity has been extremely successful up to now in describing the behaviourof gravity in experiments, as well as introducing new physical phenomena, and providingquantitative predictions for them. Effects such as gravitational time delay [52], gravitationalwaves [53], and gravitational lensing were all predicted by Einstein as a consequence of GRfield equations. When combined with astrophysics, GR has also established the foundationsof fields like cosmology, and the study of compact objects.Despite all of its successes, however, there are a few regimes where general relativitydoes not provide a satisfactory answer, even though gravity plays a significant role in them.Specifically, these regimes are the Universe at its earliest times, and a full, proper treatmentof black holes, both of which may ultimately require a theory of gravity that is also quantummechanical by construction. There are also observational phenomena, such as dark energy,where the answer provided by GR is still the best existing solution (in this case a cosmolog-ical constant with a very small size), but is not widely accepted, since it is not compatiblewith the general expectations of quantum mechanics (see Chapter 4 for further discussionon cosmological constant problems).The answer to all of these problems and shortcomings may one day come from a co-herent theory of quantum-gravity. Combining general relativity and quantum mechanics,however, appears to be a prohibitively hard problem. Theories such as string theory or loopquantum gravity claim success in building a quantum theory of gravity, but they both fallshort of providing precise quantitative predictions, at least as of today [54, 55].Part of the problem for building a new theory of gravity is the lack of any clear obser-vational deviations from the predictions of GR. Here is where theories that modify GR can232.1. Gravity beyond GRbe at least partially helpful. Instead of an attempt to build a fully coherent and completetheory that resolves the inconsistencies of quantum theory and GR, modified theories ofgravity set about to perform a systematic search for observational deviations from GR byproposing specific theoretical extensions for investigating the theory space that lies close toGR itself. Several different ideas have been proposed (see e.g. [56, 57] for reviews). Wewill introduce some of the main classes of modified gravity in this chapter, and will presentcosmological tests for some of these theories in the next three chapters.2.1 Gravity beyond GRGeneral relativity is a unique theory with a strikingly simple Lagrangian:L = 116 piG√−g (R − 2 Λ) . (2.1)Here, R is the Ricci scalar and g is the determinant of the metric. More explicitly, Lovelock’stheorem [58] states that field equations of GR (Eq. (1.1)) are the only second-order, localdifferential equations derivable from the Lagrangian L = L(gµν), in 4-dimensional freespace (i.e. with Tµν = 0). This theorem not only puts GR at a special position in the spaceof theories, but also provides a clear method for classifying modified theories of gravity.According to the theorem, the only possible ways to extend GR are one of (or a combinationof) the following:• extending the degrees of freedom in the Lagrangian and including scalar, vector, ortensor degrees of freedom;• including higher-order terms in the Lagrangian;• writing the theory in higher dimensions;• breaking locality;• dropping the requirement of obtaining field equations from a Lagrangian at all.242.1. Gravity beyond GRDifferent proposals that have been discussed in the literature break one or more of theseconditions. In this context, we will describe some of the most well-known theories ofmodified gravity in the rest of this section.2.1.1 Scalar-Tensor theoriesScalar-Tensor theories were among the very first proposed theories of modified gravity.They evade Lovelock’s theorem by adding a new degree of freedom in the form of a scalarfield. The simplest version of this idea, known as the “Brans-Dicke theory” [59], was intro-duced in 1961. The main goal of the Brans-Dicke theory was to incorporate Mach’s prin-ciple [60] into the theory of relativity.4 The authors argued that a theory of gravity wouldfollow Mach’s principle only if the Newton gravitational constant becomes an environment-dependant variable. They therefore substituted the constant G with a scalar field, and wrotethe following Lagrangian density for what is now known as the Brans-Dicke theory:L = 116 pi√−g(φR − ωφ∇µφ∇µφ)+Lm(gµν,Ψ). (2.2)Here, φ is a scalar field, R is the Ricci scalar, g is the determinant of the metric gµν, Lmis the Lagrangian of the matter fields Ψ, and ω is a free parameter of the theory. GR is aspecial case of the theory with ω→ ∞ (very large values for ω would suppress the gradientterms in the Lagrangian (∇µφ) and would hence correspond to a constant φ field, which willmake the Lagrangian equivalent to GR). In the paragraph following the first appearance ofthis Lagrangian, the authors state that “In any sensible theory ωmust be of the general orderof magnitude of unity”. However, the tightest constraints on this parameter now come fromthe Cassini satellite [61], implying ω > 40000 at (95% confidence limit5). This means thatif we have such a theory, then the modifications to GR have to be very small.A more general version of the theory has also been considered in the literature [62, 63]:4 The Brans-Dicke paper explains Mach’s principle in a few pages. In short, one can state the principle as:“the large-scale structure of the Universe determines the local inertial frame.”5 Abbreviated to “CL” in the remainder of this thesis.252.1. Gravity beyond GRL = 116 pi√−g(φR − ω(φ)φ∇µφ∇µφ + V(φ))+Lm(gµν,Ψ). (2.3)Here the parameter ω has become a function of the scalar field φ. One can apply the con-formal transformation gµν = g˜µν/φ and write this theory in the so-called “Einstein frame”,where the scalar field is not coupled to the Ricci scalar [56]L = 116 pi√−g˜ (R˜ − 12∇˜µψ ∇˜µψ + V(ψ))+Lm(gµν,Ψ). (2.4)Here, φ and ψ are related to each other according to∂ψ∂χ=√3 + 2ω4 pi, with e2 χ = φ−1. (2.5)The first term of this new Lagrangian looks very similar to the Einstein-Hilbert action,Eq. (2.1), plus a scalar field, and therefore the field equations for the metric are similar tothe field equations of GR. However, one should notice that the matter fields are coupledto the original metric, gµν. This means that the energy momentum tensor is not conservedwith respect to the transformed metric, g˜µν, and test particles do not follow the geodesicsof this metric. There has been some debate in the literature on the validity of using the“Einstein frame” and the physical meaning of the solutions in this space (see e.g. [64–66]).This debate is inherently tied to the definition of units, and becomes philosophical at somepoint (see e.g. [67] for a discussion of the importance of units, and particularly the use ofdimensionless quantities in modified gravity studies).Horndeski [68] has further generalized the scalar-tensor Lagrangian and has writtenthe most general Lagrangian of a local, second-order theory of scalar-tensor gravity in 4-dimensions. This Lagrangian is constructed from the metric-field, a scalar-field, and fourcompletely general functions of the scalar. It is a particularly long Lagrangian and we referthe interested reader to the original paper, or the notationally more modern versions of itin Ref. [69]. The Lagrangian has second-order terms that are coupled with each other, andit is therefore not immediately clear that the resulting field equations will be second order262.1. Gravity beyond GRas well. However, these second-order terms in the Lagrangian are carefully added so thatthe higher-order terms cancel each other in the field equations. We will come back to thistheory in Chapter 5, where we constrain a subclass of Horndeski theories with cosmologicaldata.2.1.2 f (R) theoriesAnother approach is to write some f (R) in the Lagrangian in place of R, i.e. L = √−g(R +f (R)).6 These f (R) theories break another assumption of the Lovelock theorem. They areamong the so-called “higher-order” theories, where the equations of motion have more thantwo derivatives. Lagrangians for these theories are in some sense the most obvious way forextending GR, but then there is still freedom to choose the form of f .The special case of f (R) = cR2 was proposed as one of the earliest models of infla-tion [32, 70] and still appears to be consistent with the data [71].Most higher-order theories of gravity exhibit “ghost-like” degrees of freedom. In thecontext of modified gravity, a “ghost” refers to degrees of freedom that show up with thewrong sign for the kinetic term in the Lagrangian. As a result, ghosts have the undesiredproperty of increasing their kinetic energy as they climb up the potential. They should alsobe avoided for quantum mechanical reasons, because of the lack of a well defined groundstate. However f (R) theories, despite being “higher-order”, are equivalent to a version ofthe scalar-tensor theory in Eq. (2.3) with no kinetic terms, and are therefore ghost-freetheories.Cosmology proves to be a strong tool for constraining f (R) models, as we will explainin more detail in Chapter 5. Cosmological data provide unique tests because of the time-dependent background metric, and due to the various fluids that dominate the Universe atdifferent epochs. Starting from radiation domination with R = 0, the Universe transits intomatter domination with an evolving Ricci scalar, and ends with a (close to) cosmologicalconstant dominated state with a constant (but non-zero) value of the Ricci scalar. Each6 It is convenient to define the f (R)-Lagrangian this way, and separate standard GR from the modificationterms in f (R).272.2. Notes on modified gravity studies, their limitations and usesof these different eras of cosmology constrain the form of the f (R) function in a uniquemanner, and dictate that the form of the function has to be very close to that of GR. Generaldeviations from GR are parameterized via the functionB =fRR1 + fRHR˙H˙ − H2 (2.6)in cosmology. Here, fR and fRR are the first and second derivatives of f (R), and a dotmeans a derivative with respect to conformal time, dτ = dt/a. GR corresponds to B ≡ 0,and the tightest constraints on the present value of this parameter come from CMB datacombined with the matter power spectrum, B0 < 0.83 × 10−4 (95 % CL). We will explainthe cosmological constraints on these theories in more detail in Chapter 5.Almost all possible ways of escaping from the Lovelock theorem have been examined inthe literature. We have only discussed two of the most well-known ideas of modified gravityhere, because they have shown some merit in providing an explanation for the inflationaryera, and because they will be used later in this thesis. A complete list of even just the namesof the different proposed models would not fit this page, so we refer the interested reader tothe extensive and comprehensive review of Ref. [56].2.2 Notes on modified gravity studies, their limitations and usesIn a remarkable contrast to the theories of general and special relativity, which redefinefundamental notions such as mass, energy, space, and time, the modified theories of gravityaccept all of these relativistic notions and simply reformulate their interactions. Modi-fied theories of GR are thus like modified versions of Newtonian gravity that use differentfunctions of distance for the gravitational force, without establishing a new paradigm toreformulate physics. They are, therefore, incapable of introducing genuinely new physicaleffects (like GR did by introducing gravitational waves, gravitational lensing, etc.), and canonly modify the quantitative predictions for the set of observables that were already intro-duced by GR. Perhaps one day we will have a fundamentally different way of extending282.2. Notes on modified gravity studies, their limitations and usesGR, with a full theory of quantum gravity, and entirely new phenomena will emerge. Butfor now the ambitions of those who study “modified gravity” are more modest.Despite the above-mentioned deficiency, modified gravity deserves some credit for theway it allows us to systematically search for deviations from GR. This is a search that coversa diverse range of scales in distance and energy, and scrutinises GR predictions under manydifferent conditions, as we will see in the following chapters.When discussing different physical regimes, it is common practice to chart the wholeset of physical observables in terms of energy and distance, and put the focus of the searchfor new physics at the extremes of this chart (e.g. see Fig. 2.1). Even though this sameexercise was historically fruitful for the cases of special relativity and quantum mechanics,it does not always hold. As an example, let us look at this type of argument in the light ofthe well-known story of the precession of Mercury. Newtonian  MechanicsQuantumMechanicsField TheoryGeneralRelativity ?SpecialRelativityEnergySizeFigure 2.1. A very schematic chart of the different theories in physics, and their realmof relevance. While Newtonian mechanics works well on everyday scales, things changerather drastically on very small/large scales, or high energies.The excess in precession of Mercury was discovered around 1859 by Leverrier [72].In terms of distance, Mercury is not special, since Newtonian gravity works well frommetre scales on Earth up to tens of AU in the solar system. In terms of gravitational energy,292.2. Notes on modified gravity studies, their limitations and usesMercury also does not hold a special place, since the amount of energy stored in the Jupiter-Sun system is the largest among the planets in the solar system. Mercury’s gravitationalacceleration is much smaller than values we experience everyday on Earth, and even thoughthe force of gravity between the Sun and Mercury is much stronger than what is achievablehere, it is not the strongest among the planets, because Venus, Earth, Jupiter, and Saturn allexperience stronger gravitational pulls from the Sun than Mercury. The only thing that issomewhat special for Mercury is the Sun’s gravitational potential at its position. It is thedeepest among the solar system planets, and certainly deeper than the gravitational potentialon the Earth’s surface. However, even though the gravitational potential at Mercury is thedeepest, releasing a test particle from Mercury’s orbit and tracking its trajectory will showvery good agreement with Newtonian mechanics. The only thing that made Mercury aunique example for testing gravity was a combination of a relatively strong gravitationalpotential, a periodic orbital motion in that potential, and observations that spanned a 150-year time interval. But despite all of this, observations of the precession of Mercury werekey to establishing the validity of GR, which overthrew Newtonian gravity entirely. Perhapsin a similar sense, modified theories of gravity might be able to discover some deviationfrom GR in conditions that are not necessarily extreme in terms of distance or energy scales.In summary, no modified theories of gravity seem to be observationally favoured at themoment, and they appear to lack clear philosophical justification. Nevertheless, a search forobservational deviations from the predictions of GR under different conditions is a valuableexercise, and modified theories of gravity provide us with systematic tools for developingsuch observational tests (as we will see in the next three chapters).30Chapter 3Minimal Parameterization forModified Gravity in Cosmology3.1 IntroductionGeneral Relativity (GR) has been confronted with many theoretical and experimental testssince its birth in 1915. From the gravitational lensing experiments in 1919 [73] up to theextensive studies and tests in the 1960s and 1970s [74–78], the theory has been confirmedobservationally and theoretically bolstered in many different respects.Cosmology has challenged GR with two, yet to be fully understood discoveries: darkmatter and dark energy [79, 80]. Along with these two phenomena, the lack of renormaliz-ability in GR [56] and the apparently exponential expansion in the very early Universe [81]are usually taken as signs for the incompleteness of the theory at high energies. Due to theseshortcomings in GR the study of modified gravity has become a broad field. Scalar-tensortheories [59, 82], f (R) modifications [83, 84], Horava-Lifshitz theory [85], multidimen-sional theories of gravity [86, 87], and many other suggestions have been made in the hopeof finding, or at least deriving, some hints for, a fully consistent theory that can successfullyexplain the observations and satisfy the theoretical expectations. (Ref. [56] has an extensivereview).The new data coming from various experiments such as the WMAP and Planck satel-lite measurements of the cosmic microwave background (CMB) anisotropies [88], and theWiggleZ [89] or Baryon Oscillation Spectroscopic Survey [90] measurements of the matterpower spectrum, provide us with an opportunity to test specific modified theories of grav-313.2. Modified gravity formulationity. However, since there are many different modified theories, all with their own sets ofparameters, there has recently been some effort to come up with a way to describe genericmodified theories using only a few parameters, and to try to constrain those parameters withgeneral theoretical arguments and by direct comparison with cosmological data.The parameterized post-Friedmann (PPF) approach, as described in Ref. [91], is aneffective way to parameterize many of the modified theories of gravity. However, it is notreally feasible to constrain its more or less dozen additional free functions, even with thepower of Markov Chain codes such as CosmoMC [92]; there are just too many degrees offreedom to provide useful constraints in the general case. In this chapter we will describea somewhat different way to parameterize modified theories of gravity in which we try toretain only a small number of parameters, which we then constrain using WMAP 9-year [3]and South Pole Telescope (SPT) data [4].In the next section we will describe the formulation of this new parameterization, andwill show its connection with PPF and other approaches in Sect. 3.3. In Sect. 3.4 we willdiscuss the results of a numerical analysis using CAMB [40] and CosmoMC, and we willconclude the chapter in Sect. 3.5 with a brief discussion.3.2 Modified gravity formulationThere are two common strategies for modifying gravity. One can start from the point ofview of the Lagrangian or from the equations of motion. The Lagrangian seems like themore obvious path for writing down specific new theories, where one imagines retainingsome desired symmetries while breaking some others. We will pursue this further in Chap-ter 5. The equations of motion, on the other hand, provide an easier way in practice toparameterize a general theory of modified gravity, especially in the case of first-order per-turbations in a cosmological context.The evolution of the cosmological background has been well tested at different redshiftslices, specifically at Big Bang nucleosynthesis and at recombination through the CMBanisotropies. It therefore seems reasonable to assume that the background evolution is not323.2. Modified gravity formulationaffected by the gravity modification, with the only background level effect being a possibleexplanation for a fluid behaving like dark energy.The linearized and modified equations of motion for gravity can be written in the fol-lowing form in a covariant theory:δGµν = 8piG δTµν + δUµν. (3.1)Here, δGµν is the perturbed Einstein tensor around a background metric, δTµν is the first or-der perturbation in the energy-momentum tensor and δUµν is the modification tensor sourcefrom any term that is not already embedded in GR.Since we will be using CAMB for numerical calculations, we will choose the syn-chronous gauge from now on, and focus only on the spin-0 (scalar) perturbations. This willmake it much more straightforward to adapt the relevant perturbed Boltzmann equations.The metric in the synchronous gauge is written asds2 = a2(τ)[−dτ2 + (δi j + hi j)dxidx j],hi j =∫d3kei~k.~x{kˆi kˆ j h(~k, τ) + (kˆi kˆ j − 13δi j) 6 η(~k, τ)} , ~k = kkˆ, (3.2)where ~k is the wave vector. Putting this metric into Eq. (3.1) results in the following fourequations [37]:k2η − 12a˙ah˙ = −4 piG a2δρ + k2 A(k, τ); (3.3)kη˙ = 4 piG a2(ρ¯ + p¯) V + k2B(k, τ); (3.4)h¨ + 2a˙ah˙ − 2 k2 η = −24 piG a2(δP) + k2 C(k, τ); (3.5)h¨ + 6η¨ + 2a˙a(h˙ + 6η˙) − 2 k2η = −24piG a2(ρ¯ + p¯)Σ + k2D(k, τ). (3.6)333.2. Modified gravity formulationHere we have used the following definitions:δT 00 = −δρδT 0i = (ρ¯ + p¯) ViδT ii = 3 δPDi jδT i j = (ρ¯ + p¯)Σ,,,,a2δU00 = k2 A(k, τ),a2δU0i = k2 B(k, τ),a2δU ii = k2 C(k, τ),a2Di jδU i j = k2 D(k, τ),(3.7)with ρ¯ and p¯ being the background energy density and pressure, respectively, and a dotrepresenting a derivative with respect to τ. The factors of k are chosen to make the mod-ifying functions, {A, B,C,D}, dimensionless. The quantity Di j is defined as kˆikˆ j − 13δi j.The parameterization described here has a very close connection in practice with the PPFmethod explained in Ref. [91]. The most important differences are that we have groupeda number of separate parameters into a single parameter, and have used the synchronousgauge in Eqs.( 3.3) to (3.6).As was mentioned earlier in Chapter 1, Einstein’s equations provide six independentequations. For the case of first order perturbations in cosmology, two of these six equationsare for the two spin-2 (tensor) degrees of freedom, two of the equations are for the spin-1(vector) variables and only two independent equations are left for the spin-0 (scalar) degreesof freedom. This means that Eqs. (3.3) to (3.6) are not independent and one has to imposetwo consistency relations on this set of four equations. These consistency relations of coursecome from the energy-momentum conservation equation, ∇µ(T µν + Uµν) = 0. Assumingthat energy conservation holds independently for the conventional fluids, ∇µT µν = 0, (seeRef. [91] for the strengths and weaknesses of such an assumption) one then obtains thefollowing two consistency equations:2 A˙H + 2 A −2 k BH + C = 0 ; (3.8)6 B˙ + k C + 12H B − k D = 0 . (3.9)Here we have definedH ≡ a˙/a and dropped the arguments of the functions A to D.343.3. Connection with other methods of parameterizationEqs. (3.3) to (3.6), together with Eqs. (3.8) and (3.9), show that two general functionsof space and time would be enough to parameterize a wide range of modified theories ofgravity. This approach of course does not provide a test for any specific modified theory.However, given the current prejudice that GR is the true theory of gravity at low energies(e.g. see Ref. [93] for a discussion), the main question is whether or not cosmological datacan distinguish between GR and any other generic theory of modified gravity. Clearly,if we found evidence for deviations from GR, then we would have a parametric way ofconstraining the space of allowed models, and hence hone in on the correct theory.3.3 Connection with other methods of parameterizationIn this section we show the connection between the {µ, γ} parameterization of modifiedgravity, the PPF parameters and the parameterization introduced in Sect. 3.2.The parameters defined in Sect. 3.2 are related to the PPF parameters according toA(k, τ) = A0Φˆ + F0Γˆ + α0χˆ +α1k˙ˆχ, (3.10)B(k, τ) = B0Φˆ + I0Γˆ + β0χˆ +β1k˙ˆχ, (3.11)C(k, τ) = C0Φˆ +C1k˙ˆΦ + J0Γˆ +J1k˙ˆΓ + γ0χˆ +γ1k˙ˆχ +γ2k2¨ˆχ, (3.12)D(k, τ) = D0Φˆ +D1k˙ˆΦ + K0Γˆ +K1k˙ˆΓ + 0χˆ +1k˙ˆχ +2k2¨ˆχ. (3.13)While the authors of Ref. [91] have insisted on the modifications being gauge invariant7,it is good to keep in mind that there is nothing special about the use of gauge invariantparameters, as is shown in Ref. [94]. The important issue is to track the degrees of freedomin the equations. There are originally four free functions for the spin-0 degrees of freedomin the metric, but the gauge freedom can be used to set two of them to zero. Using only twogauge invariant functions instead of four, means that the gauge freedom has been implicitlyused somewhere to omit the redundant variables.7 They have slightly modified the tone of their paper in its later versions.353.3. Connection with other methods of parameterizationAll 22 of the parameters on the right hand side of Eqs. (3.10) to (3.13) are in fact two-dimensional functions of the wave number, k, and time. A hat on a function means that itis a gauge invariant quantity. The symbol χˆ is the gauge invariant form of any extra degreeof freedom that can appear, for example, in a scalar-tensor theory, or in an f (R) theory asa result of a number of conformal transformations (see section D.2 of Ref. [91] for furtherexplanation). Φˆ and Γˆ are related to the synchronous gauge metric perturbations through:Φˆ = η − H2 k2(h˙ + 6 η˙); (3.14)Ψˆ =12 k2(h¨ + 6 η¨ +H (h˙ + 6 η˙)); (3.15)Γˆ =1k( ˙ˆΦ +H Ψˆ). (3.16)One needs to add more parameters to the right hand side of Eqs. (3.10) to (3.13) if thereis more than one extra degree of freedom, or if the equations of motion of the theory arehigher than second order and the theory cannot be conformally transformed into a secondorder theory. The reason this many parameters were introduced in Ref. [91] is that there isa direct connection between these parameters and the Lagrangians of a number of specifictheories, like the Horava-Lifshitz, scalar-tensor or Einstein Aether theories. Therefore, inprinciple, constraining these parameters is equivalent to constraining the theory space ofthose Lagrangians.However, there are a number of issues that may encourage one to consider alternativesto the PPF approach for parameterizing modifications to gravity. First of all, it is practicallyimpossible to run a Markov chain code for 22 two-dimensional functions. One can reducethe number of free functions to perhaps 15 using Eqs. (3.8) and (3.9), but there is still a hugeamount of freedom in the problem. The second reason is that the whole power of the PPFmethod lies in distinguishing among a number of classically modified theories of gravitythat are mostly proven to be either theoretically inconsistent, like the Horava-Lifshitz theory[56], or already ruled out observationally, like TeVeS (at least for explaining away darkmatter) [95]. While it is certainly important and useful to check the GR predictions with the363.3. Connection with other methods of parameterizationnew coming data sets, it does not appear reasonable at this stage to stick with the motivationof any specific theory. For the moment it therefore seems prudent to consider an evensimpler approach, as we describe here.There is another popular parameterization in the literature, described fully in Refs. [96–98]. This second parameterization is best described in the conformal Newtonian gauge, viathe following metric:ds2 = a2(τ)[−(1 + 2ψ)dτ2 + (1 − 2 φ)δi jdxidx j]. (3.17)The modifying parameters, {µ, γ}, are defined through the following:k2ψ = −µ(k, a)4piGa2{ρ¯∆ + 3(ρ¯ + p¯)Σ} ; (3.18)k2[φ − γ(k, a)ψ] = µ(k, a)12piGa2(ρ¯ + p¯)Σ . (3.19)Here ∆ = δρ + 3Hk (1 + p¯/ρ¯)V , and all of the matter perturbation quantities are in theNewtonian gauge.In order to see the connection between this method of parameterization and the onedescribed in the previous section through Eqs. (3.3) to (3.6), one needs to use the modifiedequations of motion in the Newtonian gauge:k2φ + 3H(φ˙ +Hψ)= −4piGa2δρ + k2AN ; (3.20)k2(φ˙ +Hψ)= 4piGa2(ρ¯ + p¯)k V + k3BN ; (3.21)k2(φ − ψ) = 12piGa2(ρ¯ + P¯)Σ + k2DN . (3.22)The parameters {AN, BN,DN} are the modifying functions in the Newtonian gauge. These373.3. Connection with other methods of parameterizationparameters are related to γ and µ viaα ≡ 1 − µ , (3.23)β ≡ γ − 1, (3.24)4piGa2α {ρ¯∆ + 3(ρ¯ + p¯)Σ} = k2(AN − 3Hk BN + DN), (3.25)βψ = 12 piGa2 αΣ + k2 DN, (3.26)where one can choose between using the functions {AN, BN,CN,DN}, along with two con-straint equations similar to the Eqs. (3.8) and (3.9), or using the two parameters γ and µ andtrying to remain consistent in the equations of motion.It is argued in Ref. [99] that the {γ, µ} choice is not capable of parameterizing secondorder theories in the case of an unmodified background and no extra fields. To show thisthe authors use the fact that, in the absence of extra fields, all of the Greek coefficients inEqs. (3.10) to (3.13), i.e. {α0, ..., 2}, have to be zero. Furthermore, they argue that in thecase of second order theories, F0 and I0 have to be zero, and therefore the constraints ofEqs. (3.8) and (3.9) show that J0 and K0 are zero as well. After all of this, one can see thatEq. (3.22) can be written as the following in this special case:k2(φ − ψ) = 12piGa2(ρ¯ + P¯)Σ + k2(D0φ +D1kφ˙). (3.27)Ref. [99] then shows that the absence of a term proportional to the metric derivative willlead to an inconsistency. However, this conclusion is valid only if one assumes that β inEq. (3.26) is a function of background quantities, which usually is not the case. Otherwise,one can use Eq. (3.26) to define β:β ≡ 12 piGa2 αΣ + k2(D0φ + D1k φ˙)ψ, (3.28)383.4. Numerical calculationleaving no ambiguity or inconsistency.8It is also claimed in Ref. [100] that the {γ, µ} parameterization becomes ambiguous onlarge scales, while none of these shortcomings apply to the PPF method. However, thesecriticisms do not seem legitimate, since, as was shown in this section, there is a directconnection between {γ, µ}, and the PPF parameters.9 For any given set of functions for thePPF method, one can find a corresponding set of functions {γ, µ}, using Eqs. (3.10) to (3.13)and (3.26), that will produce the exact same result for any observable quantity. One onlyneeds to ensure the use of consistent equations while modifying gravity through codes suchas CAMB.Although we believe that there is no ambiguity in the {γ, µ} parameterization, we alsobelieve that our {A, B,C,D} parameterization can be implemented much more easily inBoltzmann codes. Furthermore, there is a potential problem for the {γ, µ} parameterizationon the small scales that enter the horizon during the radiation domination era. The metricperturbation ψ will oscillate around zero a couple of times for these scales and that makesthe γ function blind to any modification at those instants of time. This behaviour also hasthe potential to lead to numerical instabilities.3.4 Numerical calculationIn this section we will constrain the parameterization described in Sect. 3.2 using the CMBanisotropy power spectra. We will describe the effects of the modifying parameters on theCMB and show the results of numerical calculations from CAMB and CosmoMC.3.4.1 Effects of A and B on the power spectraBefore showing numerical results, we first describe some of the physical effects of havingnon-zero values of A or B. So far we have not placed any constraints on these quantities,8 Note that this might be troublesome if ψ goes to zero at some moments of time. This can happen for thescales that enter the horizon during radiation domination.9 In particular there is nothing wrong with the {γ, µ} parameterization on large scales, since ψ is certainlyalways non-zero.393.4. Numerical calculationwhich are in general functions of both space and time. There are some effects that can beexplicitly seen from the equations of motion and energy conservation. For example, a posi-tive A enhances the pressure perturbation and anisotropic stress, while reducing the densityperturbation. On the other hand, a positive B will enhance the momentum perturbation andreduce the pressure perturbation and anisotropic stress.There are also some other effects that need a little more algebra to see, and we nowdiscuss four examples.Neutrino moments:The neutrinos’ zeroth and second moments, {Fν0, Fν2}, are coupled to the modifyinggravity terms according to the Boltzmann equations [37] and Eqs. (3.3) to (3.6):F˙ν0 = −k Fν1 − 23 h˙; (3.29)F˙ν2 =25k Fν1 − 35k Fν3 +415(h˙ + 6η˙). (3.30)Here h˙ is modified according to Eq. (3.3), and the term h˙+6η˙ is coupled to A and B throughEqs. (3.3) and (3.4):h˙ + 6η˙ =2 k2 η + 8 piG a2δρH + 24 piG a2(ρ¯ + p¯)Vk− 2 k2 AH + 6 k B. (3.31)Therefore, modified gravity can have a significant effect on the neutrino second moment.Photon moments:While the same thing is valid for the photons’ second moment after decoupling, thesituation is different during the tight coupling regime. The Thomson scattering rate is sohigh in the tight coupling era that it makes the second moment insensitive to gravity. Inother words, the electromagnetic force is so strong that it does not let the photons feelgravity.ISW effect:The integrated Sachs-Wolfe (ISW) [101] effect is proportional to φ˙+ψ˙ in the Newtonian403.4. Numerical calculationgauge. In the synchronous gauge this isφ˙ + ψ˙ =˙¨h + 6 ˙¨η2 k2+ η˙. (3.32)Here η˙ is modified according to Eq. (3.4) and, therefore, a subtle change in the functionB(k, τ), can have a considerable influence on the ISW effect. Fig. 3.1 shows the effects ofa constant non-zero A and B on the ISW effect. For the case of a constant non-zero B, theISW effect is always present, since the time derivative of the potential is constantly sourcedby this function. This will result in more power on all scales, including the tail of the CMBpower spectrum (see Fig. 3.2). Fig. 3.2 shows that if a non-zero B is favoured by CMB data,it will happen in the large `s, (` > 1500), and will be due to its anti-damping behaviour.-5e-06 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 550  900  1250ISWConformal time (Mpc)A=0 , B=0A=0.05 , B=0-0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 550  900  1250ISWConformal time (Mpc)A=0 , B=0A=0 , B=0.005Figure 3.1. Effects of a constant, non-zero A and B on the ISW effect. This plot shows theISW effect for a specific scale of k = 0.21 Mpc−1.Fig. 3.3 shows the CMB power spectra for the case of a constant but non-zero A or B.Note how a constant non-zero B raises the tail of the spectrum up. One might also point tothe degeneracy of A and the initial amplitude (usually called As) mostly by looking at the413.4. Numerical calculation-12-10-8-6-4-20240 500 1000 1500 2000 2500 3000`1/2∆C`/C`Multipole `(A=B=0) - (A=0 , B = 0.00074)error bandFigure 3.2. Effects of a constant B on the CMB power spectra. The plot shows the dif-ference in power for the case of zero B minus the best fit non-zero B, using WMAP9 andSPT12 data, while keeping all the rest of the parameters the same. The error band plottedis based on the reported error on the binned CMB power spectra from the WMAP9 [3] andSPT12 [4] groups.height of the peaks.Matter overdensity:The Boltzmann equation for cold dark matter overdensity in the synchronous gaugereads [37]δ˙CDM ≡(δρCDMρ¯CDM).= −12h˙. (3.33)Using Eqs. (3.3), (3.4), and the Friedmann equation, H2 = 8 piG a23ρ¯, and assuming amatter-dominated Universe with no baryons, one obtains the following equation for the423.4. Numerical calculation010002000300040005000600070000 500 1000 1500 2000 2500 3000`(`+1)C`/2piMultipole `A=0 , B=0A=0.05 , B=0A=0 , B=0.005Figure 3.3. Effects of a constant, non-zero A or B on the CMB power spectra. One can seethat the two parameters have quite different effects.cold dark matter overdensity:H δ˙CDM = −3H22δCDM + k2A − k2η,i.e. δ¨CDM +(H˙H +32H)δ˙CDM + 3 H˙δCDM = − k3H B +k2H A˙. (3.34)The above equation clearly shows the role of A and B as driving forces for the matteroverdensity. The k3 prefactor makes the first term on the right hand side dominant on smallscales and this will therefore have a significant effect on the matter fluctuation amplitude atlate times. The matter power spectrum will therefore be expected to put strong constraintson modified gravity models.3.4.2 Markov chain constraints on A and BSince A(k, τ) and B(k, τ) are free functions, we need to choose some simple cases to investi-gate. We choose here to focus on the simple cases of A and B being separate constants (i.e.433.4. Numerical calculationindependent of both scale and time). We do not claim that this is in any sense a preferredchoice — we simply have to pick something tractable. With better data one can imag-ine constraining a larger set of parameters, for example describing A and B as piecewiseconstants or polynomial functions.We used CosmoMC to constrain constant A and B, together with the WMAP-9 [3] andSPT12 [4] CMB data. The amplitudes of the CMB foregrounds were added as additionalparameters and were marginalized over for the case of SPT12. The resulting constraints anddistributions are shown in Fig. 3.4. Here we focus entirely on the effects of A and B on theCMB. Hence we turn off the post-processing effects of lensing [102], and ignore constraintsfrom any other astrophysical data-sets.ConstantBConstant AWMAP9Constant AWMAP9+SPT12-0.0015-0.001-0.000500.00050.0010.00150.0020.0025-0.02 -0.01 0 0.01 0.02-0.0015-0.001-0.000500.00050.0010.00150.0020.0025-0.02 -0.01 0 0.01 0.02Figure 3.4. 68 and 95 percent contours of the constants A and B using WMAP9-year dataalone (left) and SPT12 (right) without including lensing effects, and neglecting late timegrowth effects.443.4. Numerical calculationOne might conclude from Fig. 3.4 that general relativity is ruled out by nearly 3σ usingCMB alone, since a non-zero value of B is preferred. However, adding lensing to the picturewill considerably change the results. As was shown in Eq. (3.34), a non-zero B will changethe matter power spectrum so drastically that in a universe with non-zero B, lensing will beone of the main secondary effects on the CMB. The results of a Markov chain calculationthat includes the effects of lensing (i.e. assuming B was constant not only in the CMB era,but all the way until today) is shown in Fig. 3.5, and are entirely consistent with GR.ConstantBConstant AWMAP9+lensingConstant AWMAP9+SPT12+lensing-0.0002-0.00015-0.0001-5e-0505e-050.00010.000150.0002-0.02 0 0.02 0.04 0.06 0.08-0.0002-0.00015-0.0001-5e-0505e-050.00010.000150.0002-0.02 0 0.02 0.04 0.06 0.08Figure 3.5. 68 and 95 percent contours of the constants A and B using WMAP9-year dataalone (left) and WAMP9 + SPT12 (right), with lensing effects included.The broad constraint on A is mainly due to a strong (anti-)correlation between A andthe initial amplitude of the scalar perturbations. Two-dimensional contour plots of A versusAs, the initial amplitude of scalar curvature perturbations, are shown in Fig. 3.6.The mean of the likelihood and 68% confidence interval for the six cosmological pa-453.4. Numerical calculationlog(1010As)Constant A2.9533.053.13.15-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07Figure 3.6. The strong anti-correlation between parameter A and the initial amplitude, As,makes the constraints on either one of these two parameters weaker.rameters together with A and B are tabulated in Table 3.1. Note that the simplified case weare considering here treats CMB constraints only. If we really took B as constant for alltime, then there would be large effects on the late time growth, affecting the matter powerspectrum, and hence tight constraints coming from a relevant observable, such as σ8 fromcluster abundance today.3.4.3 Alternative powers of k in BExamining Eq. (3.34) reveals that the only term modifying the matter power spectrum inthe case of constant A and B, is k3B/H . This term is important for two reasons. Firstly,this is the only term introducing a k dependence in the cold dark matter amplitude at latetimes and at sufficiently large scales where one can completely ignore the effect of baryons463.5. DiscussionParameter . . . . . . . . . . . . . . . . . WMAP9 WMAP9+SPT12 WMAP9+SPT12+lensing100 Ωbh2 . . . . . . . . . . . . . . . . . . 2.22 ± 0.05 2.10 ± 0.03 2.22 ± 0.04ΩDMh2 . . . . . . . . . . . . . . . . . . . . 0.118 ± 0.005 0.122 ± 0.005 0.115 ± 0.004100 θ . . . . . . . . . . . . . . . . . . . . . 1.038 ± 0.002 1.040 ± 0.001 1.042 ± 0.0010τ . . . . . . . . . . . . . . . . . . . . . . . . 0.086 ± 0.014 0.076 ± 0.012 0.084 ± 0.012log(1010As) . . . . . . . . . . . . . . . . 3.1 ± 0.2 3.1 ± 0.2 3.0 ± 0.4ns . . . . . . . . . . . . . . . . . . . . . . . 0.96 ± 0.01 0.93 ± 0.01 0.96 ± 0.01100A . . . . . . . . . . . . . . . . . . . . . 1 ± 10 0.02 ± 10 2.7 ± 1.71000B . . . . . . . . . . . . . . . . . . . . 0.44 ± 0.41 0.74 ± 0.24 −0.0097 ± 0.016Table 3.1. The mean likelihood values together with the 68% confidence interval for theusual six cosmological parameters, together with constant A and B, using CMB constraintsonly.on the matter power spectra. Secondly, the k3 factor enhances this term significantly onsmall scales in the case of a constant B. Since the amplitude of matter power spectrum (vialensing effects) was the main source of constraints on B, it would seem reasonable to chooseB = H B0/k, where B0 is a dimensionless constant. This should avoid too much power inthe matter densities on small scales, and therefore reduce lensing as well. However, thischoice will lead to enormous power on the largest scales, as is shown in Fig. 3.7.In order to match with data, one could choose a form in which B switches from B =H B0/k to B = H B0/k0, where k0 is some small enough transition scale. We discuss thissimply as an alternative to the B = constant case. There is clearly scope for exploring awider class of forms for the functions A(k, τ) and B(k, τ).3.5 DiscussionSince a constant A is essentially degenerate with the initial amplitude of the primordialfluctuations, the CMB alone cannot constrain this parameter. On the other hand, constantB seems to be fairly well constrained by the CMB data. However, if B was an oscillatingfunction of time, changing sign from time to time, its total effect on the CMB power spectrawould become weaker and the constraints would be broader. According to Eq. (4.19), aconstant B will change η monotonically, while the effect of an oscillating B will partially473.5. Discussion05000001e+061.5e+062e+062.5e+0610 100 1000`2(`+1)C`/2piMultipole `A=B0=0A=0, B0=0.005Figure 3.7. Effect of a non-zero B0 on the CMB power spectra, with the choice B =H B0/k.cancel some of the time. Together, the results of Sect. 3.4 show that the ISW effect and thegrowth at relatively recent times (driving the amplitude of matter perturbations) can havehuge constraining power for many generic theories of modified gravity. (See Ref. [103] fora recent example). One can consider different positive or negative powers of (H/k) as partof the dependence of B in order to get around the matter constraints, as was discussed inSect. 3.4.3.We have seen that when considering CMB data alone, there seems to be a mild pref-erence for non-zero B. This is essentially because it provides an extra degree of freedomfor resolving a mild tension between WMAP and SPT. Neveretheless it remains true thata model with B constant for all time would be tightly constrained by observations of thematter power spectrum at redshift zero. We leave for a future study the question of whetherthere might be any preference for more general forms for A(k, τ) and B(k, τ) using a com-bination of Planck CMB data and other astrophysical data-sets.48Chapter 4How Does Pressure Gravitate?Cosmological Constant ProblemConfronts Observational Cosmology4.1 IntroductionOne of the most immediate puzzles of quantum gravity (i.e., applying the rules of quantummechanics to gravitational physics) is an expectation value for the vacuum energy that is 60–120 orders of magnitude larger than its measured value from cosmological (gravitational)observations. This is known as the (now old) cosmological constant problem [104], andhas been thwarting our understanding of modern physics for almost a century [105]. Thediscovery of late-time cosmic acceleration [106, 107], added an extra layer of complexityto the puzzle, showing that the (gravitational) vacuum energy, albeit tiny, is non-vanishing(now dubbed, the new cosmological constant problem).Gravitational Aether (GA) theory is an attempt to find a solution to the old cosmologicalconstant problem [108, 109], i.e., the question of why, in lieu of fantastic cancellations, thevacuum quantum fluctuations do not appear to source gravity. The approach is to stopthe quantum vacuum from gravitating by modifying our theory of gravity, as we describebelow. In this way the (mean density of ) quantum fluctuations will have no dynamicaleffect in astrophysics or cosmology (see [110] for one of the very first steps and [111] foran alternative but related attempt for solving the problem).Although GA is a very specific proposal for modifying gravity, it may serve as an ex-494.1. Introductionample of more general theories. As we will see below, a generalized version may representa broader class of theories in which the gravitational effects of pressure (and includinganisotropic stress) might be different from those of GR.It is important to be clear that this theory does not have any solution for the “new” cos-mological constant problem, i.e., the empirical existence of a small vacuum energy densitywhich now dominates the energy budget of the Universe, driving the accelerated expansionand making the geometry of space close to flat. Hence in GA theory it is assumed that thevacuum quantum fluctuations (the old problem) and the small but non-zero value of Λ (thenew problem) are two separate phenomena that should be explained independently (but see[112]).The Einstein field equations in the GA theory (in units with c = 1, and with metricsignature=(–+++)) are modified to(8 piGR)−1(Gµν + Λgµν) = Tµν − 14Tααgµν + T′µν, (4.1)with T ′µν = p′(u′µu′ν + gµν). (4.2)Most significantly, the second term on the right hand side of Eq. (4.1), − 14 Tααgµν, solves theold cosmological constant problem by cancelling the effect of vacuum fluctuations in theenergy momentum tensor. The third term, T ′µν, is then needed to make the field equationsconsistent, and is dubbed gravitational aether.The form used for T ′µν in Eq. (4.2) is a convenient choice, but is probably not unique,although it is limited by phenomenological and stability constraints [108]. However, p′and u′µ, the pressure and four velocity unit vector of the aether, are constrained through theterms in the energy-momentum tensor by applying the Bianchi identity and the assumptionof energy-momentum conservation, i.e.,∇µT ′µν =14∇νT. (4.3)The only free constant of this theory, as in General Relativity (GR), is GR, although, as504.1. Introductionwe will see, this is not the same as the usual Newtonian gravitational constant, GN. In addi-tion, of course, there are parameters describing the constituents in the various tensors, i.e.,the cosmological parameters. In cosmology, the energy-momentum tensor, Tµν, consists ofthe conventional fluids, i.e., radiation, baryons, and cold dark matter, plus a contributiondue to vacuum fluctuations,Tµν = T Rµν + TBµν + TCµν + ρvacgµν, (4.4)T ≡ Tαα = −(ρB + ρC) + 4 ρvac, (4.5)where neutrinos are included as part of radiation, and their mass is set to zero in this paper.Equations (4.1) and (4.2) that describe GA are drastic modifications of GR with noadditional tunable parameter. Therefore, one may wonder whether GA can survive all theprecision tests of gravity that have already been carried out. These tests are often expressedin terms of the parameterized post-Newtonian (PPN) modifications of GR, which are ex-pressed in terms of 10 dimensionless PPN parameters [113]. While these parameters do notcapture all possible modifications of GR, they are usually sufficient to capture leading cor-rections to GR predictions in the post-Newtonian regime (i.e., nearly flat space-time withnon-relativistic motions), in lieu of new scales in the gravitational theory. It turns out thatonly one PPN parameter, ζ4, which quantifies the anomalous coupling of gravity to pressurehas not been significantly constrained empirically, as the existing precision tests only probegravity in vacuum, or for objects with negligible pressure. Indeed, since only the sourcingof gravity is modified in GA, the vacuum gravity content is identical to GR, and the onlyPPN parameter that deviates from GR is ζ4 = 1/3 (as opposed to ζ4 = 0 in GR) [108, 109].The idea that ζ4 could be non-zero runs contrary to the conventional wisdom that relatesgravitational coupling to pressure on the one hand, to the couplings to internal and kineticenergies on the other [114], both of which are already significantly constrained by experi-ments. However, this expectation is based on the assumption that the average gravity of agas of interacting point particles, is the same as the gravity of a perfect fluid that is obtained514.1. Introductionby coarse-graining the particle gas.10 This connects with the whole issue of the assumptionof the continuum approximation for cosmological fluids, where the particle density is low,so that the average distance between particles is a macroscopic scale. Gravity is only well-tested on scales & 0.1 mm [115], which are larger than the distance between particles inmost terrestrial or astrophysical precision tests of gravity. Therefore, there is no guaranteethat the same laws of gravity apply to microscopic constituents of the continuous media inwhich gravity is currently tested. Indeed, GA could only be an effective theory of gravityabove some scale λc . 0.1 mm, implying that sources of energy-momentum on the righthand sides of Eqs. (4.1) or (4.3) should be coarse-grained on scale λc.At first sight, it might appear that the dependence of gravitational coupling on pressuresignals a violation of weak and/or strong equivalence principles (WEP and/or SEP). How-ever, WEP is explicitly imposed in GA, as all matter components couple to the same metric.Moreover, SEP is so far only tested for gravity in vacuum (e.g. point masses in the solarsystem), where GA is equivalent to GR, as aether is not sourced, and thus vanishes (in lieuof non-trivial boundary conditions; see e.g. [112]).What goes against one’s intuition in the case of the GA modification of Einstein gravity,compared to e.g. scalar-tensor theories, is that even in the Newtonian limit, comparableeffects come from the change in couplings and the gravity of the energy/momentum of theaether. In contrast, the additional fields in the usual modified gravity theories carry littleenergy/momentum in the Newtonian regime, while they could modify couplings by orderunity. If the change in the gravitational mass (due to the dependence of G on the equation ofstate) is by the same factor as the change in energy/momentum (due to the additional termson the RHS of Einstein equations), then the ratio of gravitational to inertial mass remainsunchanged.A more intuitive picture might be to consider aether (minus the trace term) as an exoticfluid bound to matter, similar to an electron gas for example, within ordinary GR. Like the10 This would not be the case in the GA theory, since the aether tracks the motion of individual particles, dueto the constraint of Eq. (4.3). Therefore, the nonlinear back-reaction of the motion of the aether would be lostin the coarse-grained perfect fluid.524.1. Introductionelectron gas, the effect will be to modify the gravitational field source, by the amount of en-ergy/momentum in the exotic fluid. However, unlike the electron gas, the non-gravitationalenergy/momentum exchange between matter and the exotic fluid is tuned to zero, which en-sures WEP, at least at the classical level. Moreover, the action-reaction principle (Newton’s3rd law) for gravitational forces should include the momentum in, and interaction with theexotic fluid.Another conceptual issue with Eqs (4.1–4.2) is that, at least to our knowledge, they donot follow from an action principle. However, an action principle may not be necessary(or even possible) for a low energy effective theory, such as in the case of Navier-Stokesfluid equations, even if the fundamental theory does follow from an action principle. Giventhe severity of the cosmological constant problem, it seems reasonable that we might beprepared to relax requirements that are not absolutely necessary for a sensible effectivedescription of nature.There are two obvious places in the Universe to look for the gravitational effect ofrelativistic pressure, and thus constrain ζ4:1. The first situation involves compact objects, particularly the internal structure of neu-tron stars [116, 117]. While, in principle, mass and radius measurements of neutronstars can be used to constrain ζ4, at the moment the constraints are almost completelydegenerate with the uncertainty in the nuclear equation of state (not to mention otherobservational systematics). However, future observations of gravitational wave emis-sion from neutron star mergers (e.g., with Advanced LIGO interferometers) mightbe able to break this degeneracy [117]. It may also be possible to develop tests thatprobe near the hot accretion disks of black holes or during the formation of compactobjects in supernova explosions.2. The second situation is the matter-radiation transition in the early Universe. Ref. [118]studied constraints arising from the big bang nucleosynthesis epoch. However, moreprecise measurements come from various cosmic microwave background (CMB)anisotropy experiments, such as the Wilkinson Microwave Anisotropy Probe (WMAP)534.2. Equations of motion at the background and perturbative level[119], Planck [120], the Atacama Cosmology Telescope (ACT) [121], and the SouthPole Telescope (SPT) [5], amongst other cosmological observations. The constraintson GA were studied in detail in Ref. [109], with the data sets available at that time.While GA might arguably ease tension among certain observations, such as the Ly-αforest, primordial Lithium abundance, or earlier ACT data, it was discrepant with oth-ers, such as Deuterium abundance, SPT data, or low-redshift measurements of cosmicgeometry. The aim of this paper is to carefully revisit these tensions in observationalcosmology, in light of the significant advances within the past three years.With this introduction, in Sect. 4.2, we move on to derive the equations for the cosmo-logical background, as well as linear perturbations, in the GA theory. Similar to Ref. [109],we use the Generalized Gravitational Aether (GGA) framework, which interpolates be-tween GR and GA, to quantify the observational constraints. This framework depends onthe ratio of the gravitational constant in the radiation and matter eras, GR/GN = 1+ζ4, whichis 1 + 13 =43 (1 + 0 = 1) for GA (GR). Sect. 4.3 discusses our numerical implementationof the GGA equations, and the resulting constraints from different combinations of cosmo-logical data sets, some of which appear to exclude GA at the 4–5σ level, while others areequally (in)consistent with GR or GA at about the 3σ level. Finally, Sect. 4.5 summarizesour results, discusses various open questions, and highlights avenues for future inquiry.4.2 Equations of motion at the background and perturbativelevelBaryons, radiation, and cold dark matter can be considered as perfect fluids with simpleequations of state, p = wρ, at the background level. The following p′ and u′ will solve544.2. Equations of motion at the background and perturbative levelEqs. (4.1–4.3) in this case:p′ =∑i(1 + wi)(3 wi − 1)4ρi; (4.6)u′µ =∑i(1 + wi)(1 − 3 wi)2uiµ. (4.7)Here, “i” stands for either baryons, radiation, cold dark matter, or vacuum fluctuations.Based on Eqs. (4.6) and (4.7), p′ = −(ρB + ρC)/4, and u′µ = uCµ at the background level.Substituting these relations back into Eq. (4.1), the field equations will take the followingform in terms of the conventional fluids in Tµν:(8 pi)−1(Gµν + Λgµν) = GR T Rµν +34GR (T Bµν + TCµν). (4.8)One of the clearest testable predictions of this theory is that space-time reacts differ-ently to matter and to radiation: a spherical ball full of relativistic matter curves the space-time more than a spherical ball of non-relativistic substance (of the same size and density).Defining GR ≡ 4 GN/3, where GN is the usual Newtonian gravitational constant, and usingthe FRW metric, ds2 = a2(−dτ2 + dx2), the Friedmann equation in the GA theory will be:H2 = 8piGNa23(ρ +13ρR) , ρ = ρR + ρB + ρC + ρΛ. (4.9)H is defined as a˙/a here, and a dot represents a derivative with respect to the conformaltime, τ.The Friedmann equation can be used to calculate the predictions of the theory for bigbang nucleosynthesis (BBN) (see e.g., Ref. [109]). Although the different effective valueof G in the early Universe means that the BBN predictions are different from the standardmodel, uncertainties in the consistency of the light element abundances suggest that thecomparison with data cannot be considered as fatal for the theory. Therefore, one needs togo one step further and calculate the first-order perturbations to determine the predictionsfor observables such as the CMB anisotropies, or the matter power spectrum.554.2. Equations of motion at the background and perturbative levelBefore dealing with the perturbations, it is worth noticing that the GA theory can betreated as a special case of a more general framework. We shall call this the GeneralizedGravitational Aether (GGA), which has the following field equations:(8 pi)−1(Gµν + Λgµν) = GRTµν −GRNTααgµν + 4 GRNT ′µν. (4.10)Here GRN ≡ GR − GN = ζ4GN is the difference in gravitational constants between radiationand matter. GR and GN are both free constants and one will recover the gravitational aetherby setting GR = 4GN/3. General relativity is also a special case of GGA, with GRN = 0. TheFriedmann equation in GGA will be:H2 = 8pia23(GN ρ + GRN ρR). (4.11)Using GGA as a framework, we then have a family of models, parameterized by ζ4, withζ4 = 0 corresponding to GR and ζ4 = 1/3 being GA.It is fairly straightforward to calculate the perturbation equations in the general (GGA)framework, which will then contain GA and GR as special cases. We will use the cold darkmatter gauge (see e.g., Ref. [37]) with the following metric for the first order perturbations:ds2 = a2(τ)[−dτ2 + (δi j + hi j)dxidx j];hi j =∫d3kei~k.~x[kˆikˆ jh(~k, τ) +(kˆikˆ j − 13δi j)6η(~k, τ)], ~k = kkˆ. (4.12)We will also use the following definitions for the perturbation parts of the energy momen-tum tensor:δT 00 = −δρ; (4.13)δT 0i = (ρ¯ + p¯) Vi; (4.14)δT ii = 3 δp; (4.15)Di jδT i j = (ρ¯ + p¯)Σ. (4.16)564.2. Equations of motion at the background and perturbative levelThe barred variables refer to background quantities andDi j is defined as kˆikˆ j − 13δi j. Onceagain, the fluids in Tµν are baryons, cold dark matter, radiation, and vacuum quantum fluc-tuations. We will follow the conventions of Ref. [109] and define the perturbations in theaether density and four velocity asδp′ = p′ −(−ρM4), δu′µ = u′µ − uCµ . (4.17)Here ρM is the total matter density, i.e., baryons plus cold dark matter, and the quantities ρMand uCµ consist of both their background and perturbation parts. Using the above definitionsand the metric defined in Eq. (4.12), we obtain four equations of motion from the GGAfield equations:k2η − 12H h˙ = −4 piGN a2δρ + k2 A(k, τ); (4.18)kη˙ = 4 piGN a2(ρ¯ + p¯) V + k2B(k, τ); (4.19)h¨ + 2H h˙ − 2 k2 η = −24 piGN a2(δP) + k2 C(k, τ); (4.20)h¨ + 6η¨ + 2H (h˙ + 6η˙) − 2 k2η = −24piGN a2(ρ¯ + p¯)Σ + k2D(k, τ). (4.21)The four functions, {A, B,C,D}, areA(k, τ) =−4 piGRN a2 δρRk2, (4.22)B(k, τ) =4 piGRN a2 (i kiδT 0i − ρ¯M ω)k3, (4.23)C(k, τ) =−8 piGRN a2(δρR + 12 δp′)k2, (4.24)D(k, τ) =−24 piGRN a2Di jδT Ri jk2. (4.25)Here ω is defined as the divergence of the aether four velocity perturbation: ω ≡ ikiδu′i/a.One can equally use Eqs. (4.18) to (4.21), or use Eq. (4.3) to derive the following two574.3. Cosmological constraints on GGAconstraints for the aether parameters:3Ha∂τ(aω) + k2 ω = k2ρ¯B0ρ¯M0θB; (4.26)δp′ =ρ¯M12H (ω −ρ¯B0ρ¯M0θB). (4.27)Here ρ¯B0 and ρ¯M0 are the current background density in baryons and matter, respectively, andθB is the divergence of the baryon velocity perturbation: θ ≡ iki VBi . At very early times,when k  H , one can ignore the right hand side of Eq. (4.26), together with the k2 ω factoron the left hand side. The initial condition for the divergence should therefore be deducedfromω˙ +H ω = 0 . (4.28)Any non-zero initial condition on ω will be damped as a−1, and it is therefore reasonableto assume the initial condition ω = 0 at all scales. It is also interesting to notice that, sincewe are using the cold dark matter gauge, ω will once again be washed out for very largescales, k  H , at late times when baryons fall into the potential well of the cold dark matterparticles and start co-moving with them.The physical meaning of the four modifying terms, {A, B,C,D}, is explained in Ref. [122]for an even more general theory. In short, the second term and the time derivative of thefirst term will act as driving forces for matter overdensities, while the second term and thetime derivative of the fourth term are important in the integrated Sachs-Wolf (ISW) [123]effect.We will confront the GGA theory with cosmological observations in the next section.4.3 Cosmological constraints on GGAWe have modified the cosmological codes CAMB [40] and CosmoMC [92] in order to test thepredictions of GGA against cosmological data. Before confronting the theory with data, itis necessary to make sure that the codes are internally consistent and error-free. We will584.3. Cosmological constraints on GGAlist a number of consistency checks we have made on CAMB in the next subsection, and thenreport the constraints on the GGA parameter.4.3.1 Consistency checks on CAMBOne of the relatively trivial tests on the modified CAMB code is that it should reproduce theC`s of the non-modified code after setting GRN = 0. The next obvious thing is a test at thebackground level. The GA theory is completely degenerate at the background level witha GR model that has one third more radiation (see Eq. (4.9)). In the standard picture eachlight neutrino species adds 0.23 times as much radiation as the photons. Therefore, thefollowing models should result in exactly the same a(τ) andH(τ) functions: B1 := {GRN =1/3GN,Neff = 3.04}11 andB2 := {GR with Neff = 5.54}, whereNeff is the effective numberof light neutrinos.The effect of GGA at the perturbation level is evident through the four modifying func-tions {A, B,C,D}. Using constraint equations such as Eq. (4.3), one can see that the func-tions {C,D} are linear combinations of the first two functions {A, B} and their time deriva-tives. Therefore, the two functions {A, B} are sufficient for tracing the perturbative effectsof GGA. Between the two, A is purely dependent on radiation at the perturbation level (seeEq. (4.22)). Looking closer at B in Eq. (4.23) we see thatk3B =43piGRN a2(3∆ω + 4 ρ¯νθν + 4 ρ¯γθγ). (4.29)Here ∆ω is defined as ( ρ¯B θB − ρ¯M ω), which is proportional to the time derivative of ω,according to Eq. (4.26), and is therefore smaller than the radiation terms (see Fig. 4.1). θνand θγ are the neutrino and photon first moments, respectively [37].Putting this together, we find that the GGA effects are almost degenerate with extraradiation, even when we consider perturbations. However, the GGA-Neff degeneracy doesnot hold exactly at the perturbation level, since δργ/δρν and θγ/θν are both time- and scale-dependent, contrary to the previous case at the background level, where ρ¯γ/ρ¯ν was a con-11 Here B is for background, and P will be for perturbations.594.3. Cosmological constraints on GGA10−2 10−1 100 101a/aeq−0.02−0.010.000.010.020.030.040.05(ρ¯BθB−ρ¯Mω)/(ρ¯νθν+ρ¯γθγ)k = 0.04(Mpc)−1Figure 4.1. Effects of non-radiation modifying terms compared to the effects of radiation,at k = 0.04 Mpc−1 (see Eq. (4.29)). This shows that the non-radiation terms are smaller bymore than a factor of 20.stant number through time and for every scale.The final test of the modified CAMB code is at the perturbation level when the GA pa-rameter, GRN, is not set to zero and GA is fully effective. Based on the discussion in thepreceding paragraphs the code should produce the same C`s for the following two mod-els: P1 := {GRN = 1/3GN,Neff = 3.04,T 40 = (2.7255)4} and P2 := {GR withNeff =4/3 × 3.04, T 40 = 4/3 × (2.7255)4}, when the non-radiation terms, i.e., terms in ∆ω areignored. We use the value of the present CMB temperature, T0, from Ref. [124]. However,one needs to be careful while performing this test, since T0 appears in many parts of thecode that are totally irrelevant to gravity (see Ref. [125] for a related discussion), e.g., thesound speed of the plasma before last scattering depends on the photon-to-baryon densityratio and hence on T0. Figs. 4.2 and 4.3 show these tests of our calculations for GGA. The604.3. Cosmological constraints on GGAright panel of these figures tests the code at the perturbation level, while the left panelsshow the effect of non-radiation fluids on the CMB anisotropies and matter power spectra.Ignoring the effect of non-radiation fluids, is crucial in reducing the GA Eqs. (4.1–4.3), toEq. (4.8) at the perturbation level, and hence the GRN–ζ4 correspondence.0 500 1000 1500 2000Multipole ℓ01000200030004000500060007000D ℓ≡ℓ(ℓ+1)Cℓ/2πMp1ModelMp2Model0 500 1000 1500 2000Multipole ℓ01000200030004000500060007000Mp1Model , ∆ω = 0Mp2ModelFigure 4.2. Checking the code at the perturbation level by comparing the CMBanisotropies power spectra for the models P1 := {GRN = 1/3GN,Neff = 3.04,T 40 =(2.7255)4} and P2 := {GR withNeff = 4/3 × 3.04, T 40 = 4/3 × (2.7255)4} (left panel).There is a small difference between the two at around the first peak. This can be explainedby considering the effects of a non-zero ω (divergence of the aether four-velocity). The twomodels completely coincide with each other by setting ∆ω to zero in the P1 model (rightpanel).Figure 4.4 shows the CMB anisotropy power spectrum predictions from GR and GA(with other GGA models interpolating between the two). The input parameters of the leftpanel are the same for both theories and are taken from Ref. [1]. We see on the left panelthat the positions of the peaks are consistently shifted towards smaller scales, i.e., higher614.3. Cosmological constraints on GGA10−3 10−2 10−1k(hMpc−1)102103104105P(h−3Mpc3)Mp1ModelMp2Model10−3 10−2 10−1k(hMpc−1)102103104105Mp1Model , ∆ω = 0Mp2ModelFigure 4.3. Checking the code at the perturbation level by comparing the the modelsP1 := {GRN = 1/3GN,Neff = 3.04,T 40 = (2.7255)4} and P2 := {GR with Neff = 4/3 ×3.04, T 40 = 4/3 × (2.7255)4} (left panel). The small difference between the two can beexplained by considering the effects of a non-zero ω. The two models completely coincidewith each other by setting ∆ω to zero in the P1 model (right panel).`s. This is because the Universe is younger at recombination in the GA theory, which inturn is due to having effectively more radiation at the background level of the GA theorycompared to GR. There is also an enhanced early ISW effect [126] in the GA theory due thepresence of the the two modifying functions, B and D, as was explained in chapter 3. Thiscan be understood more intuitively using the fact that GA is effectively degenerate at thebackground and perturbation level with a GR model with one third more radiation. Sincethe ISW effect is proportional to e−τ (where τ is the optical depth), and the time derivativeof the metric potentials (that are non-zero only during the matter radiation transition andat very late times), then having more radiation in the Universe will delay the radiation tomatter transition to later times with smaller τ and enhance the ISW effect.624.3. Cosmological constraints on GGAIn order for the GA model to match GR and hence fit the data, since there is a verygood match between data and GR predictions, one needs to change the matter to radiationdensity ratio to get the right position for the peaks. This can be done by either deductingfrom the radiation density, or adding more matter to the GA model. The first option ishighly restricted from the CMB temperature data [124]. The second option can be doneeither through adding baryons or cold dark matter, or both. Since the density of baryons isconstrained through helium abundance ratio (see e.g. [127]), the only remaining option isto add cold dark matter to the theory. This is also limited by the ratio of even to odd peaksin the CMB power spectra, but is the last resort! The best fit value for the cold dark matterdensity in the GA theory, using CMB data only, is: ΩDMh2 = 0.147 ± 0.004.After fixing the position of the peaks, one needs to get the right amplitude for the spec-tra. The relative amplitude of the high-` to low-` multi-poles is highly affected by the earlyISW effect that was explained before and is evident in the left panel of Fig. 4.4 by com-paring the ratio of the power of the two curves in ` ∼ 250, and ` ∼ 2000. This relativemismatch in the amplitude can be fixed by choosing higher values of the spectral index, ns.The best fit value of this parameter in the GA theory is: ns = 1.042 ± 0.008.The best-fit predictions of the two theories are compared in the right panel of Fig. 4.4.We see that the best fit GA theory predicts less power at high `s compared to GR. Thebest-fit predictions of the two theories are compared with Planck and SPT data in Fig. 4.5.4.3.2 Cosmological constraintsWe now turn to deriving precision constraints on GGA from cosmological observations.We assume that GN is equal to the Newtonian gravitational constant measured in Cavendish-type experiments (see e.g. Ref. [128]) using sources with negligible pressure. Then CosmoMCcan be used for sampling GR using different combinations of the following cosmologicaldata.1. The first data release of the all-sky CMB temperature anisotropy power spectrum,634.3. Cosmological constraints on GGA0 500 1000 1500 2000Multipole ℓ01000200030004000500060007000D ℓ≡ℓ(ℓ+1)Cℓ/2πGRbest fitGAwithGRbest fit0 500 1000 1500 2000Multipole ℓ0100020003000400050006000GRbest fitGAbest fitFigure 4.4. Comparing general relativity versus gravitational aether predictions for theCMB power spectrum. The values of the input parameters for the left panel are taken fromthe Planck analysis [1]. The right panel compares the best-fit predictions of the two theorieswith all cosmological parameters also allowed to vary. GA predicts less power at higher `s,as one can see from the right panel (this difference is more evident in the residual plot inFig. 4.5).measured by the Planck [129] satellite.122. The 9-year (and final) data release of the WMAP satellite CMB temperature and po-larization anisotropy power spectra, which we denote as WMAP-9 [119] (with “WP”indicating the large angle polarization data only).3. Three seasons of high resolution CMB temperature anisotropy measurements fromthe ACT experiment [121].4. 790 deg2 of high resolution CMB temperature anisotropy measurements from the12 http://pla.esac.esa.int/pla/aio/planckProducts.html644.3. Cosmological constraints on GGA500 1000 1500 2000 2500 3000Multipole ℓ−0.2−0.10.00.10.20.3∆D ℓ/Dℓ(Planck − GA)/P lanck(SPT − GA)/SPT500 1000 1500 2000 2500 3000Multipole ℓ−0.2−0.10.00.10.20.3∆D ℓ/Dℓ(Planck − GR)/P lanck(SPT − GR)/SPTFigure 4.5. Comparing general relativity (bottom panel) and gravitational aether (toppanel) predictions for the CMB power spectrum with Planck and SPT data sets. Herewe plot D` ≡ `(` + 1)C`1/2pi residuals, along with ±1σ error bars, from Refs. [1] and[5]. While the two theories can both fit the lower-` observations, GR fits the data pointssignificantly better than GA for ` & 1000, at the > 4σ level.SPT experiment [5].5. Sloan Digital Sky Survey (SDSS) [130] and other estimates of the BAO length scale[131–133].6. The first claimed detection of the amplitude of primordial gravitational waves, basedon B-mode polarization anisotropy band-powers detected by the BICEP2 experimentat degree scales [134].There are two special cases of particular interest, which are GR = GN (standard GeneralRelativity; ζ4 = 0) and GR = 43 GN (Gravitational Aether theory; ζ4 =13 ). If the data areconsistent with the GR/GN = 4/3 case, or favour this theory over GR, then that would be654.3. Cosmological constraints on GGAParameter . . . . . . . . . WMAP-9 WP + Planck WP+Planck+HighL WP+Planck+BAOΩbh2 . . . . . . . . . . . . . 0.0226 ± 0.0005 0.0227 ± 0.0005 0.0225 ± 0.0004 0.0223 ± 0.0003ΩDMh2 . . . . . . . . . . . . 0.14 ± 0.03 0.128 ± 0.006 0.124 ± 0.005 0.125 ± 0.005100 θ . . . . . . . . . . . . . 1.038 ± 0.003 1.0421 ± 0.0008 1.0418 ± 0.0007 1.0415 ± 0.0006τ . . . . . . . . . . . . . . . . 0.088 ± 0.014 0.097 ± 0.015 0.096 ± 0.015 0.090 ± 0.013log(1010As) . . . . . . . . 3.10 ± 0.04 3.11 ± 0.03 3.10 ± 0.03 3.10 ± 0.03ns . . . . . . . . . . . . . . . 0.979 ± 0.019 0.987 ± 0.017 0.975 ± 0.014 0.970 ± 0.009GN/GR . . . . . . . . . . . 0.86 ± 0.18 0.913 ± 0.048 0.951 ± 0.043 0.959 ± 0.035Table 4.1. Mean likelihood values together with the 68% confidence intervals for the usualsix cosmological parameters (see Ref. [1]), together with the GGA parameter GN/GR. “WP”refers to WMAP-9 polarization, which has been used to constrain the optical depth, τ.“HighL” refers to the higher multipole data sets, ACT and SPT. The PPN parameter, ζ4can be obtained through ζ4 = GR/GN − 1.Parameter . . . . . . . . . WP+Planck+BICEP2 WP+Planck+HighL+BICEP2 WP+Planck+BAO+BICEPΩbh2 . . . . . . . . . . . . . 0.0229 ± 0.0005 0.0228 ± 0.0004 0.0223 ± 0.0003ΩDMh2 . . . . . . . . . . . . 0.132 ± 0.006 0.128 ± 0.005 0.127 ± 0.005100 θ . . . . . . . . . . . . . 1.0425 ± 0.0008 1.0422 ± 0.0007 1.0416 ± 0.0006τ . . . . . . . . . . . . . . . . 0.101 ± 0.015 0.101 ± 0.015 0.090 ± 0.013log(1010As) . . . . . . . . 3.11 ± 0.03 3.11 ± 0.03 3.10 ± 0.03ns . . . . . . . . . . . . . . . 1.001 ± 0.016 0.991 ± 0.015 0.976 ± 0.009r . . . . . . . . . . . . . . . . 0.18 ± 0.04 0.18 ± 0.04 0.16 ± 0.03GN/GR . . . . . . . . . . . 0.871 ± 0.045 0.905 ± 0.040 0.938 ± 0.034Table 4.2. Mean likelihood values together with the 68% confidence intervals for the usualsix cosmological parameters, plus r (the tensor-to-scalar ratio), together with the GGAparameter GN/GR. The data are as in Table 4.1, but now including BICEP2 measurementsof the B-mode CMB polarization. GR is still favoured over GA if we include HighL CMBor BAO measurements. However, even the conventional seven parameter GR model (thatincludes r), is disfavoured at around the 3σ level when one considers BICEP2, as well asPlanck, and HighL data. The PPN parameter, ζ4 can be obtained through ζ4 = GR/GN − 1.evidence that GA theory provides a better description of the cosmological data.From a broader perspective, any unequal values for GN and GR would be interesting,because this is a way of parameterizing general deviations from the matter-radiation equiv-alence principle. The MCMC constraints on GGA, excluding the recent BICEP2 data re-lease, are summarized in Table 4.1. It is important to allow the usual cosmological param-eters to vary while constraining GN/GR. This is because there could be (and indeed are)664.3. Cosmological constraints on GGAFigure 4.6. Confidence intervals (68% and 95%) for the GGA parameter and the cosmo-logical parameters it is most degenerate with. The ratio GN/GR is plotted on the left axesand ζ4 on the right axes. The horizontal dashed lines indicate the GR (top line) and GA(bottom line) predictions.degeneracies in the new 7-parameter (or 8-parameter when the tensor-to-scalar ratio r isincluded) space. Some of these degeneracies between the GGA parameter, GN/GR, and theconventional parameters of cosmology are shown in Fig. 4.6.In fact, we find that if one omits the BICEP2 data, then GN/GR = 1 provides a good fitand the cosmological parameters hardly shift from their best-fit GR values. On the otherhand, adding BICEP2 data shifts the results towards GA by about 1σ. This may be pointingto some tension in data, or a mild inconsistency between GR and the existing data sets. Ofcourse the most exciting possibility that any such tension is due to missing physics ratherthan systematic effects. Table 4.2 shows these constraints, while Fig. 4.7 presents a pictorialcomparison of constraints on GN/GR using different data sets.674.4. Discussion0.75 0.80 0.85 0.90 0.95 1.00GN/GR = (1 + ζ4)−1WP+PlanckWP+Planck+HighLWP+Planck+BAOWP+Planck+BICEP2WP+Planck+HighL+BICEP2WP+Planck+BAO+BICEP2GravitationalAetherGeneralRelativity0.33 0.11 0.0ζ4Figure 4.7. A pictorial comparison of marginalized GN/GR = (1 + ζ4)−1 measurements.We have plotted the central values and ±1σ error bars using different data sets. The GRand GA predictions are shown as vertical dashed lines.4.4 DiscussionAs we can see in Fig. 4.7, although GR is generally preferred over GA, different combina-tions of data sets appear to give constraints for the GGA parameter (or anomalous pressurecoupling), which are discrepant by as much as 2σ. Perhaps most intriguingly, the com-bination of Planck temperature anisotropies and polarization from WMAP-9 and BICEP2(which represents the state of the art for CMB anisotropy measurements above 0.1◦), liesabout mid-way between the GA and GR predictions (with a preference for GA, but only atthe level of ∆χ2 ' 1). Nevertheless, the best fit for GN/GR is inconsistent with both GA andGR at 2.7 and 2.9σ, respectively. The latter is a manifestation of the well-known tensionbetween the Planck upper limit on tensor modes, and the reported detection by BICEP2 (at684.4. Discussionleast for standard ΛCDM cosmology with a power-law primordial power spectrum).Let us now try to qualitatively understand what might be responsible for the differenttrends that we observe when fitting different data sets, as we turn up the GGA parameter.The first step is to obtain the gross structure of the CMB CTT` power spectrum peaks byfixing θ, the ratio of the sound horizon at last scattering, to the distance to the last-scatteringsurface. For any value of GN/GR, this can be done by picking appropriate values of Ωm andh, which explains the degeneracy directions in Fig. 4.7 for these parameters.The next step is to recognize the effect of free steaming on the damping tail of the CMBpower spectrum. Similar to the effect of free streaming of additional neutrinos, boostingthe gravitational effect of neutrinos leads to additional suppression of power at small scales,or high `, in the CMB power spectrum, as we can see in Fig. 4.5. This can be partiallycompensated for by increasing the spectral index of the scalar perturbations, leading to abluer primordial spectrum. In fact, we see that combinations of data sets that prefer largerGR (in Tables 4.1–4.2) prefer a near scale-invariant power spectrum, ns ' 1 (which is upfrom the value ns ' 0.96 in GR+ΛCDM).Finally, a bluer scalar spectral index tends to suppress scalar power for ` . 100, whichthen relaxes the upper bound on tensors from the Planck temperature power spectrum.This allows a higher value of r than the limit (r < 0.11 [1]) found from the temperatureanisotropies in ΛCDM.Of course, none of these degeneracies are perfect. In particular, the additional dampingdue to free-streaming is much steeper than a power law, which is why even the best-fit GAmodel underpredicts CMB power for ` & 1000 in Fig. 4.5. This is also why adding higherresolution CMB observations (from ACT and SPT), pushes the best fit away from GA. It ispossible that adding a positive running for the spectral index might be able to partially can-cel the effect, at least for the observable range of multipoles. However, a significant positiverunning would be hard to justify in simple models of inflation, and may also exacerbate theobservational tensions with structure formation on small scales in ΛCDM.A more stringent constraint on GA (and thus anomalous pressure coupling) comes from694.4. Discussionthe degeneracy with the Hubble constant, which can also be seen in Fig. 4.6. Additionalgravitational coupling to pressure, of the sort required in GA, requires H0 > 80 km s−1 Mpc−1,which is larger than even the highest measurements in the current literature (see e.g., fig-ure 16 in Ref. [1]). In particular, BAO geometric constraints place tight bounds of H0 ' 68–72 km s−1 Mpc−1, which is why the inclusion of these data substantially cuts off the smallervalues of GN/GR. However, we should note that this inference is based on a simple cosmo-logical constant model for dark energy at low redshifts; more complex descriptions of darkenergy, as suggested by some recent BAO [135] or Supernovae Ia [136] studies, could relaxthese H0 constraints.There are certainly hints of possible systematics among the different data sets that couldexplain some of these tensions. For example, the power spectrum of WMAP-9 appears to beabout 2.5% higher than Planck [1, 137], independent of scale. Additionally, the first 30 orso multipoles appear low (in both WMAP and Planck data), which, coupled with calibration,can affect the best fit in the damping tail. A perhaps related issue is that the best-fit lensingamplitude in Planck and Planck+HighL spectra, appears to be around 20% higher thanexpected in the ΛCDM model [1, 138]. Since lensing moves power from small `s to high`s, this could also have an indirect effect on the shape of the high-` power spectrum.Finally, there are legitimate questions about whether BICEP2 analysis [134] has un-derestimated the effect of instrumental systematics or Galactic foregrounds (e.g., [139]).Decreasing the primordial amplitude of B-modes would reduce the tension with Planck,and thus relax the need for anomalous pressure coupling (i.e., GN < GR).On balance it seems premature to claim that ζ4 > 0 is required by the current cosmolog-ical data. The simple GA theory (with ζ4 = 1/3) certainly appears disfavoured by the data.However, as the quality of the data continue to improve, it is worth bearing in mind that theGGA picture provides a particular degree of freedom. This should be considered in futurefits, particularly with the upcoming release of the Planck polarization data.704.5. Conclusions, and open questions4.5 Conclusions, and open questionsIn this paper, we have closely examined the question of anomalous pressure coupling togravity in cosmology. This was done in the context of the Generalized Gravitational Aetherframework, which allows for an anomalous sourcing of gravity by pressure (ζ4 in the PPNframework), while not affecting other precision tests of gravity. The idea would mean thatthe gravitational constant during the radiation era, when p = 13ρ, is boosted to GR = (1 +ζ4)GN, compared to the gravitational constant for non-relativistic matter GN. In particular,the case with ζ4 = 1/3 or GR = 4GN/3, can be used to decouple vacuum energy fromgravity, and thus solve the (old) cosmological constant problem.We have implemented cosmological linear perturbations for this theory into the codeCAMB, and explored the models that best fit different combinations of cosmological data.The effects are qualitatively similar to introducing additional neutrinos (Neff), or dark radi-ation. Our constraints are summarized in Tables 4.1–4.2 and Figs. 4.6–4.7.There is clearly some mild tension between different data combinations, but ζ4 = 1/3is inconsistent with current observations at around the 2.6-−5σ level, depending on thecombination used. CMB B-mode observations (from BICEP2) push for larger ζ4, whilehigh resolution CMB or baryonic acoustic oscillations, go in the opposite direction. Thebest fit is in the range 0.04 . ζ4 . 0.15, or 0.87 . GN/GR . 0.96, with statistical errorsof a half to third of this range. It may be interesting to notice that even GR (ζ4 = 0) isdisfavoured at 3σ when we combine lower resolution CMB observations.To bring some statistical perspective, we should note that even if the gravitational aethersolution to the cosmological constant problem is ruled out at 5σ, the standard GR+ΛCDMparadigm, with no fine-tuning, is ruled out at > 1060 σ! Therefore, while the first attemptat solving the problem might not have been entirely successful (compared to a model thattakes the liberty of fine-tuning the vacuum energy), we argue, that it may be a step in theright direction. So, other than working to improve the quality and consistency of observa-tional data, what can we do to tackle this problem, that quantum fluctuations appear not togravitate?714.5. Conclusions, and open questionsFrom the theoretical standpoint, there are several clear avenues that we have alreadyalluded to:1. As we discussed in the Introduction, gravitational aether is a classical theory for aneffective low energy description of gravity. Therefore, like all effective theories, it hasan energy cut-off above which it will not be valid. In fact, the length-scale λc (inverseenergy scale) associated with this cut-off should be λc ∼ 0.1 mm, since a smallerλc would not fully solve the cosmological constant problem, while larger λc couldhave been seen in torsion balance tests of gravity (although it is not entirely clearwhat the signature would be). It is worth noting that the number density of baryonsat CMB last scattering is 0.33 mm−3, implying that to calculate Tαα in Eq. (4.1), itmight be necessary to use a microscopic description of atoms interacting with aether,as opposed to the usual mean fluid density picture.13 If this is the case, then eachmicroscopic particle would carry an aether halo of size about ∼ λc; this would appearlike a renormalization of particle mass for all macroscopic gravitational effects, butotherwise (like for other vacuum tests of gravity), the theory would be indistinguish-able from GR. Nevertheless, in lieu of a quantum theory of gravitational aether, it isnot clear how much progress can be made in this direction.2. Another possibility is to modify the simple ansatz in Eq. (4.2) for the energy-momentumtensor of the gravitational aether, e.g., by introducing a density, ρ′. This might be areasonable approach if one is also attempting to connect gravitational aether to darkenergy (which does have both density and pressure at late times). However, Eq. (4.3)will no longer be sufficient to predict the evolution of the aether, and thus we wouldneed another equation to fix the aether equation of state.3. In solving for the evolution of aether with respect to dark matter, ω, we have assumedthat the two substances were originally comoving, i.e.,ω = 0 at early times. However,depending on the process that generates primordial scalar fluctuations in this picture,13 The density of dark matter particles is much more model dependent, but is expected to be even less thanthis baryon value for conventional WIMP models724.5. Conclusions, and open questionsω could have also been sourced in the early Universe. So, even though its amplitudedecays as a−1 on super-horizon scales, depending on its amplitude and spectrum,it can impact CMB observations. This would be akin to introducing isocurvaturemodes, but for aether perturbations. Although, since ω decays exponentially on sub-horizon scales, this could only affect the CMB at ` . 100.4. Finally, we have not included the effect of neutrino mass in our GGA treatment. Mas-sive neutrinos will be qualitatively different from other components, as they start asradiation, which does not couple to aether, but then gradually start sourcing aether asthey become non-relativistic. However, this happens relatively late in cosmic history,long after CMB last-scattering, and when neutrinos make up only a small fraction ofcosmic density. Therefore, although this would be a useful direction to pursue, we donot expect a significant change from the analyses presented here.In contrast to unfalsifiable approaches for solving the cosmological constant problem,such as landscape/multiverse ideas with anthropic arguments, the gravitational aether con-cept has the very distinct advantage of being predictive and hence it can be falsified. Here,we have demonstrated this explicitly, since the basic picture does not appear to fit the cur-rent cosmological data. However, like elsewhere in physics, the logical next step wouldbe to learn from this process and propose better physical models (rather than relying onmetaphysics). We believe that the GGA approach yields a useful parameterization of a par-ticular degree of freedom in models of modified gravity, and that this idea is worth pursuingfurther.73Chapter 5Planck 2015 Results. XIV. DarkEnergy and Modified Gravity5.1 IntroductionThe cosmic microwave background (CMB) is a key probe of our cosmological model [2],providing information on the primordial Universe and its physics, including inflationarymodels [71] and constraints on primordial non-Gaussianities [140]. In this chapter we usethe 2015 data release from Planck14 [42] to perform an analysis of f (R) theories and effec-tive field theory approaches to modified gravity.Observations have long shown that only a small fraction of the total energy density inthe Universe (around 5 %) is in the form of baryonic matter, with the dark matter neededfor structure formation accounting for about another 26 %. In one scenario the dominantcomponent, generically referred to as dark energy (hereafter DE), brings the total close tothe critical density and is responsible for the recent phase of accelerated expansion. Inanother scenario the accelerated expansion arises, partly or fully, due to a modification ofthe theory of gravity on cosmological scales. Elucidating the nature of this DE and testingGeneral Relativity (GR) on cosmological scales are major challenges for contemporarycosmology, both on the theoretical and experimental sides [e.g., 141–145].In preparation for future experimental investigations of DE and modified gravity (here-14 Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instru-ments provided by two scientific consortia funded by ESA member states and led by Principal Investigatorsfrom France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific con-sortium led and funded by Denmark, and additional contributions from NASA (USA).745.1. Introductionafter MG), it is important to determine what we already know about these models at differentepochs in redshift and different length scales. CMB anisotropies fix the cosmology at earlytimes, while additional cosmological data sets further constrain on how DE or MG evolveat lower redshifts.The simplest model for DE is a cosmological constant, Λ, first introduced by Einstein[146] in order to keep the Universe static, but soon dismissed when the Universe was foundto be expanding [147, 148]. This constant has been reintroduced several times over theyears in attempts to explain several astrophysical phenomena, including most recently theflat spatial geometry implied by the CMB and supernova observations of a recent phaseof accelerated expansion [149, 150]. A cosmological constant is described by a single pa-rameter, the inclusion of which brings the model (ΛCDM) into excellent agreement withthe data. ΛCDM still represents a good fit to a wide range of observations, more than 20years after it was introduced. Nonetheless, theoretical estimates for the vacuum density aremany orders of magnitude larger than its observed value, as is more thoroughly discussedin Chapter. 4. In addition, ΩΛ and Ωm are of the same order of magnitude only at present,which marks our epoch as a special time in the evolution of the Universe (sometimes re-ferred to as the “coincidence problem”). This lack of a clear theoretical understanding hasmotivated the development of a wide variety of alternative models. Those models whichare close to ΛCDM are in broad agreement with current constraints on the background cos-mology, but the perturbations may still evolve differently, and hence it is important to testtheir predictions against CMB data.There are at least three difficulties in studying modified gravity models. First, there ap-pears to be a vast array of possibilities in the literature and no agreement yet in the scientificcommunity on a comprehensive framework for discussing the landscape of models. A sec-ond complication is that robust constraints come from a combination of different data setsworking in concert. Hence we have to be careful in the choice of the data sets so that we donot find apparent hints for non-standard models that are in fact due to systematic errors. Athird area of concern is the fact that numerical codes available at present for DE and MG are755.1. Introductionnot as well tested in these scenarios as for ΛCDM, especially given the accuracy reachedby the data. Furthermore, in some cases, we need to rely on stability routines that deservefurther investigation to assure that they are not excluding more models than required.Among all the possible options, we consider the effective field theory (EFT) for MG[e.g. 151], which has a clear theoretical motivation, since it includes all theories derivedwhen accounting for all symmetry operators in the Lagrangian, written in unitary gauge,i.e. in terms of metric perturbations only. This is a very general classification that has theadvantage of providing a broad overview of (at least) all universally coupled DE models.One advantage of the currently available EFT numerical codes is the theoretical prior itputs on the MG models, as we will discuss further later in this chapter. However, thesestability routines are not fully tested and may discard more models than necessary. Anotherclear disadvantage of this framework is that the number of free parameters is large and theconstraints are consequently weak.As a complementary approach, we focus on f (R) models, since they have already beenthoroughly discussed in the literature and are better understood theoretically. This partof the study can partly be considered as applications of the previous case where the CMBconstraints are more informative, because there is less freedom in any particular theory thanin a generic parameterization.The CMB is the cleanest probe of large scales, which are of particular interest for mod-ifications to gravity. We will investigate the constraints coming from Planck data in com-bination with other data sets, addressing strengths and potential weaknesses of differentanalyses. Before describing in detail the models and data sets that correspond to our re-quirements, in Sect. 5.2 we first address the main question that motivates this chapter, dis-cussing why CMB is relevant for DE. We then present the specific model parameterizationsin Sect. 5.3. We present results in Sect. 5.6 and discuss conclusions in Sect. 5.7.765.2. Why is the CMB relevant for modified gravity?5.2 Why is the CMB relevant for modified gravity?The CMB anisotropies are largely generated at the last-scattering epoch, and hence can beused to pin down the theory at early times. In fact many forecasts of future DE or MGexperiments are for new data plus constraints from Planck. However, there are also severaleffects that DE and MG models can have on the CMB, some of which are to:1. change the expansion history and hence distance to the last scattering surface, with ashift in the peaks, sometimes referred to as a geometrical projection effect [152];2. cause the decay of gravitational potentials at late times, affecting the low-multipoleCMB anisotropies through the integrated Sachs-Wolfe (or ISW) effect [153, 154];3. enhance the cross-correlation between the CMB and large-scale structure, throughthe ISW effect [155];4. change the lensing potential, through additional DE perturbations or modifications ofGR [156, 157];5. change the growth of structure [158, 159] leading to a mismatch between the CMB-inferred amplitude of the fluctuations As and late-time measurements of σ8 [160,161];6. impact small scales, modifying the damping tail in CTT` , giving a measurement of theabundance of DE at different redshifts [162, 163];7. affect the ratio between odd and even peaks if modifications of gravity treat baryonsand cold dark matter differently [164];8. modify the lensing B-mode contribution, through changes in the lensing potential[165];9. modify the primordial B-mode amplitude and scale dependence, by changing thesound speed of gravitational waves [165, 166].775.3. Models and parameterizationsIn this chapter we restrict our analysis to scalar perturbations. The dominant effects onthe temperature power spectrum are then due to lensing and the ISW effect, as was discussedin length in chapter 3.5.3 Models and parameterizationsWe now provide an overview of the models addressed in this chapter. Details on the specificparameterization will be discussed in Sect. 5.6, where we also present the results for eachspecific choice of parameters.We start by noticing that one can generally follow two different approaches: (1) givena theoretical set up, one can specify the action (or Lagrangian) of the theory and derivebackground and perturbation equations in that framework; or (2) more phenomenologically,one can construct functions that map closely onto cosmological observables, probing thegeometry of spacetime and the growth of perturbations. We discussed this latter option inchapter 3, and will consider the first option in here.Assuming spatial flatness for simplicity, the geometry is given by the expansion rate Hand perturbations to the metric. If we consider only scalar-type components the metric per-turbations can be written in terms of the gravitational potentials Φ and Ψ (or equivalentlyby any two independent combinations of these potentials). Cosmological observations thusconstrain one “background” function of time H(a) and two “perturbation” functions of scaleand time Φ(k, a) and Ψ(k, a) (as was discussed earlier and also e.g. in [167]). These func-tions fix the metric, and thus the Einstein tensor Gµν. Einstein’s equations link this tensorto the energy-momentum tensor Tµν, which in turn can be related to DE or MG properties.Throughout this chapter we will adopt the metric given by the line elementds2 = a2[−(1 + 2Ψ)dτ2 + (1 − 2Φ)dx2]. (5.1)The gauge invariant potentials Φ and Ψ are related to the Bardeen [36] potentials ΦA andΦH and to the Kodama and Sasaki [35] potentials ΨKS and ΦKS in the following way:785.4. Modified gravity and effective field theoryΨ = ΦA = ΨKS and Φ = −ΦH = −ΦKS. We use a metric signature (−,+ + +) and followthe notation of Ma and Bertschinger [168]; the speed of light is set to c = 1, except whereexplicitly stated otherwise.We define the equation of state p¯(a) = w(a)ρ¯(a), where p¯ and ρ¯ are the average pressureand energy density. The sound speed cs is defined in the fluid rest frame in terms of pressureand density perturbations as δp(k, a) = c2s (k, a)δρ(k, a). The anisotropic stress σ(k, a) isthe scalar part of the off-diagonal space-space stress energy tensor perturbation. The setof functions {H,Φ,Ψ} describing the metric is formally equivalent to the set of functions{w, c2s , σ} [169].Specific theories typically cover only subsets of this function space and thus make spe-cific predictions for their form. In the following sections we will discuss the particulartheories that we consider in this chapter.5.4 Modified gravity and effective field theoryModified gravity models (in which gravity is modified with respect to GR) in general af-fect both the background and the perturbation equations. In this section we lay out thefoundations of the EFT theory approach to modified gravity.This approach starts from a Lagrangian, derived from an effective field theory (EFT)expansion [170], discussed in [151] in the context of MG. Specifically, EFT describes thespace of (universally coupled) scalar field theories, with a Lagrangian written in unitarygauge that preserves isotropy and homogeneity at the background level, assumes the weakequivalence principle, and has only one extra dynamical field besides the matter fields con-795.5. Dataventionally considered in cosmology. The action reads:S =∫d4x√−gm202 [1 + Ω(τ)] R + Λ(τ) − a2c(τ)δg00 + M42(τ)2 (a2δg00)2−M¯31(τ)2a2δg00δKµµ −M¯22(τ)2(δKµµ)2 − M¯23(τ)2δKµν δKνµ +a2Mˆ2(τ)2δg00δR(3)+ m22(τ)(gµν + nµnν)∂µ(a2g00)∂ν(a2g00)  + S m[χi, gµν]. (5.2)Here R is the Ricci scalar, δR(3) is its spatial perturbation, Kµν is the extrinsic curvature,and m0 is the bare (reduced) Planck mass. The matter part of the action, S m, includesall fluid components except dark energy, i.e., baryons, cold dark matter, radiation, andneutrinos. The action in Eq. (5.2) depends on nine time-dependent functions [171], here{Ω, c,Λ, M¯31 , M¯42 , M¯23 ,M42 , Mˆ2,m22}, whose choice specifies the theory. In this way, EFTprovides a direct link to any scalar field theory. A particular subset of EFT theories are theHorndeski [68] models, which include (almost) all stable scalar-tensor theories, universallycoupled to gravity, with second-order equations of motion in the fields, and depend on fivefunctions of time [172, 173].Although the EFT approach has the advantage of being very versatile, in practice it isnecessary to choose suitable parameterizations for the free functions listed above, in orderto compare the action with the data. We will describe our specific choices, together withresults for each of them, in the next section.5.5 DataWe now discuss the data sets we use, both from Planck and in combination with otherexperiments. We should notice that if we combine many different data sets (not all of whichwill be equally reliable) and take them all at face value, we risk attributing systematicproblems between data sets to genuine physical effects in modified gravity models. On theother hand, we need to avoid bias in confirming ΛCDM, and remain open to the possibilitythat some tensions may be providing hints that point towards a genuine deviation from the805.5. Datamodel. While discussing results in Sect. 5.6, we will try to assess the impact of additionaldata sets, separating them from the Planck baseline choice, keeping in mind caveats thatmight appear when considering some of them.5.5.1 Planck data setsPlanck low-` dataThis data set consists of the foreground-cleaned Planck LFI 70 GHz polarization mapswhich are processed together with the temperature map. This likelihood is pixel-based,extends up to multipoles ` = 29 and masks the polarization maps with a specific polar-ization mask, which uses 46 % of the sky. Use of this likelihood is denoted as “lowP”hereafter.Planck high-` dataFollowing [174], the high-` part of the likelihood (30 < ` < 2500) uses a Gaussian approx-imation,−logL(Cˆ|C(θ)) = 12(Cˆ −C(θ))T · C−1 · (Cˆ −C(θ)) + const. , (5.3)with Cˆ the data vector, C(θ) the model with parameters θ and C the covariance matrix. Thedata vector consists of the temperature power spectra of the best CMB frequencies of theHFI instrument. Specifically, as discussed in [175], we use 100 GHz, 143 GHz and 217 GHzhalf-mission cross-spectra, measured on the cleanest part of the sky, avoiding the Galacticplane.Planck CMB lensingGravitational lensing by large-scale structure introduces dependencies in CMB observableson the late-time geometry and clustering, which otherwise would be degenerate in the pri-mary anisotropies [176, 177]. This provides some sensitivity to dark energy and late-timemodifications of gravity from the CMB alone. The source plane for CMB lensing is the815.5. Datalast-scattering surface, so the peak sensitivity is to lenses at z ≈ 2 (i.e., half-way to the last-scattering surface) with typical sizes of order 102 Mpc. Although this peak lensing redshiftis rather high for constraining simple late-time dark energy models, CMB lensing deflec-tions at angular multipoles ` . 60 have sources extending to low enough redshift that DEbecomes dynamically important (e.g., Pan et al. 178).The main observable effects of CMB lensing are a smoothing of the acoustic peaksand troughs in the temperature and polarization power spectra, the generation of significantnon-Gaussianity in the form of a non-zero connected 4-point function, and the conversionof E-mode to B-mode polarization. The smoothing effect on the power spectra is includedroutinely in all results in this paper. We additionally include measurements of the powerspectrum Cφφ`of the CMB lensing potential φ, which are extracted from the Planck temper-ature and polarization 4-point functions, as presented in Planck Collaboration XV [179].The construction of the CMB lensing likelihood we use in this chapter is described fullyin Planck Collaboration XV [179]; see also Planck Collaboration XIII [2]. It is a simpleGaussian approximation in the estimated Cφφ`bandpowers, covering the multipole range40≤ `≤ 400. The Cφφ`are estimated from the full-mission temperature and polarization 4-point functions, using the the component-separated maps [180] over approximately 70 % ofthe sky.Planck CMB polarizationThe T E and EE likelihood follows the same principle as the TT likelihood described inSect. 5.5.1. The data vector is extended to contain the T E and EE cross-half-mission powerspectra of the same 100 GHz, 143 GHz, and 217 GHz frequency maps. Following [181], theregions where the dust intensity is important are masked, and 70 %, 50 %, and 41 % of thesky is retained for the three frequencies.825.5. Data5.5.2 Background data combinationWe identify a first basic combination of data sets that we mostly rely on, for which we havea high confidence that systematics are under control. Throughout this paper, we indicatefor simplicity with “BSH” the combination BAO + SN-Ia + H0, which we now discuss indetail.Baryon acoustic oscillationsThe BAO data can be used to measure both the angular diameter distance DA(z), and theexpansion rate of the Universe H(z) either separately or through the combinationDV(z) =[(1 + z)2D2A(z)czH(z)]1/3. (5.4)In this chapter we use the SDSS Main Galaxy Sample at zeff = 0.15 [182]; the BaryonOscillation Spectroscopic Survey (BOSS) “LOWZ” sample at zeff = 0.32 [183]; the BOSSCMASS (i.e. “constant mass” sample) at zeff = 0.57 of Anderson et al. [183]; and the six-degree-Field Galaxy survey (6dFGS) at zeff = 0.106 [184]. The first two measurements arebased on peculiar velocity field reconstructions to sharpen the BAO feature and reduce theerrors on the quantity DV/rs; the analysis in Anderson et al. [183] provides constraints onboth DA(zeff) and H(zeff). In all cases considered here the BAO observations are modelled asdistance ratios, and therefore provide no direct measurement of H0. However, they providea link between the expansion rate at low redshift and the constraints placed by Planck atz ≈ 1100.SupernovaeType-Ia supernovae (SNe) are among the most important probes of expansion and histori-cally led to the general acceptance that a DE component is needed [150, 185]. Supernovaeare considered as “standardizable candles” and so provide a measurement of the luminositydistance as a function of redshift. However, the absolute luminosity of SNe is considered835.5. Datauncertain and is marginalized out, which also removes any constraints on H0.Consistently with [2], we use here the analysis by Betoule et al. [186] of the “JointLight-curve Analysis” (JLA) sample. JLA is constructed from the SNLS and SDSS SNedata, together with several samples of low redshift SNe. Cosmological constraints from theJLA sample are discussed by Betoule et al. [187], and as mentioned in [2] the constraintsare consistent with the 2013 and 2104 Planck values for standard ΛCDM.The Hubble constantThe CMB measures mostly physics at the epoch of recombination, and so provides onlyweak direct constraints about low-redshift quantities through the integrated Sachs-Wolfeeffect and CMB lensing. The CMB-inferred constraints on the local expansion rate H0 aremodel dependent, and this makes the comparison to direct measurements interesting, sinceany mismatch could be evidence of new physics.Here, we rely on the re-analysis of the Riess et al. [22] Cepheid data made by Efstathiou[23]. By using a revised geometric maser distance to NGC 4258 from Humphreys et al.[188], Ref. [23] obtains the following value for the Hubble constant:H0 = (70.6 ± 3.3) km s−1 Mpc−1, (5.5)which is within 1σ of the Planck TT+lowP estimate.5.5.3 Perturbation data setsThe additional freedom present in MG models can be calibrated using external data that testperturbations in particular. In the following we describe other available data sets that weincluded in the grid of runs for this chapter.845.5. DataRedshift space distortions (RSD)Observations of the anisotropic clustering of galaxies in redshift space permit the mea-surement of their peculiar velocities, which are related to the Newtonian potential Ψ viathe Euler equation. This, in turn, allows us to break a degeneracy with gravitational lens-ing that is sensitive to the combination Φ + Ψ. Galaxy redshift surveys now provide veryprecise constraints on redshift-space clustering. The difficulty in using these data is thatmuch of the signal currently comes from scales where nonlinear effects and galaxy bias aresignificant and must be accurately modelled [see, e.g., the discussions in 189, 190].In linear theory, anisotropic clustering along the line of sight and in the transverse di-rections measures the combination f (z)σ8(z), where the growth rate is defined byf (z) =d lnσ8d ln a. (5.6)Anisotropic clustering also contains geometric information from the Alcock-Paczynski ef-fect [191], which is sensitive toFAP(z) = (1 + z)DA(z)H(z) . (5.7)In addition, fits which constrain RSD frequently also measure the BAO scale, DV (z)/rs,where rs is the comoving sound horizon at the drag epoch, and DV is given in Eq. (5.4).The Baryon Oscillation Spectroscopic Survey (BOSS) collaboration have measured thepower spectrum of their CMASS galaxy sample [192] in the range k = 0.01–0.20 h Mpc−1.Samushia et al. [51] have estimated the multipole moments of the redshift-space correlationfunction of CMASS galaxies on scales > 25 h−1Mpc. Both papers provide tight constraintson the quantity fσ8, and the constraints are consistent. The Samushia et al. [51] resultwas shown to behave marginally better in terms of small-scale bias compared to mocksimulations, so we choose to adopt this as our baseline result.855.6. ResultsGalaxy weak lensingThe distortion of the shapes of distant galaxies by large-scale structure along the line ofsight (weak gravitational lensing or cosmic shear) is particularly important for constrainingDE and MG, due to its dependence on the growth of fluctuations and the two scalar metricpotentials.Currently the largest weak lensing (WL) survey is the Canada France Hawaii TelescopeLensing Survey (CFHTLenS), and we make use of this data set in our analysis.5.6 ResultsWe now proceed by illustrating in more detail the model and parameterization described inSect. 5.3, through presenting results for a specific subset of them. In Sect. 5.6.1 we studythe constraints on the presence of non-negligible dark energy perturbations, in the contextof general modified gravity models described through effective field theories. The last part,Sect. 5.6.3, illustrates results for a range of particular examples often considered in theliterature.5.6.1 Perturbation parameterizationsGeneral modifications of gravity change both the background and the perturbation equa-tions, allowing for contribution to clustering and anisotropic stress. Here we discuss resultsfor EFT cosmologies, with a “top-down” approach that starts from the most general ac-tion allowed by symmetry and then selects from there interesting classes belonging to theso-called “Horndeski models”, which, as mentioned in Sect. 5.4, include almost all stablescalar-tensor theories, universally coupled, with second-order equations of motion in thefields.The full background and perturbation equations of the action 5.2 have been derivedin [193]. The implementation of the equations in the publicly available Boltzmann code865.6. ResultsEFTCAMB is also discussed in [194]15. We will briefly explain the role of the nine pa-rameters that appear in action 5.2 in this section. From the set of the nine parameters,{Ω, c,Λ, M¯31 , M¯42 , M¯23 ,M42 , Mˆ2,m22}, three are in charge of the background evolution. Sincethe EFT Lagrangian preserves homogeneity and isotropy at the background level, the onlyextra physical degree of freedom at the background is the sound speed of an extra fluid.Therefore, from the three functions {Ω, c,Λ} that are in charge of the background evolu-tion, only one is an independent function. Given an expansion history (which we fix tobe ΛCDM, i.e., effectively w = − 1) and an EFT function Ω(a), EFTCAMB computes c andΛ from the Friedmann equations and the assumption of spatial flatness [195]. In addition,EFTCAMB uses a set of stability criteria in order to specify whether a given model is stableand ghost-free, i.e. without negative energy density for the new degrees of freedom. Thiswill automatically place a theoretical prior on the parameter space while performing theMCMC analysis.The remaining six functions, M¯31 , M¯42 , M¯23 , M42 , Mˆ2, m22, are internally redefined in termsof the dimensionless parameters αi with i running from 1 to 6:α41 =M42m20H20, α32 =M¯31m20H0, α23 =M¯22m20,α24 =M¯23m20, α25 =Mˆ2m20, α26 =m22m20.We should notice that unlike the PPF formalism discussed in chapter 3, all of these ninefunctions are 1-dimensional functions of the scale factor, as opposed to the 2-dimensionalfunctions of PPF.15 http://www.lorentz.leidenuniv.nl/˜hu/codes/, version 1.1, Oct. 2014.875.6. Results5.6.2 Modified gravity: EFT and Horndeski modelsWe will always demand thatm22 = 0 (or equivalently α26 = 0), (5.8)M¯23 = −M¯22 (or equivalently α24 = −α23), (5.9)which eliminates models containing higher-order spatial derivatives [172]. In this case thenine functions of time discussed above reduce to a minimal set of five functions of timethat can be labelled {αM, αK, αB, αT, αH}, in addition to the Planck mass M2∗ (the evolutionof which is determined by H and αM), and an additional function of time describing thebackground evolution, e.g., H(a). The former are related to the EFT functions via thefollowing relations [172]:M2∗ = m20Ω + M¯22 ; (5.10)M2∗HαM = m20Ω˙ + ˙¯M22 ; (5.11)M2∗H2αK = 2c + 4M42 ; (5.12)M2∗HαB = −m20Ω˙ − M¯31 ; (5.13)M2∗αT = −M¯22 ; (5.14)M2∗αH = 2Mˆ2 − M¯22 . (5.15)These five α functions are closer to a physical description of the theories under in-vestigation. For example: αT enters in the equation for gravitational waves, affectingtheir speed and the position of the primordial peak in B-mode polarization; αM affectsthe lensing potential, but also the amplitude of the primordial polarization peak in B-modes[165, 166, 196]. It is then possible to relate the desired choice for the Horndeski variables885.6. Resultsto an appropriate choice of the EFT functions,∂τ(M2∗ ) = HM2∗αM, (5.16)m20(Ω + 1) = (1 + αT)M2∗ , (5.17)M¯22 = −αTM2∗ , (5.18)4M42 = M2∗H2αK − 2c, (5.19)M¯31 = −M2∗HαB + m20Ω˙, (5.20)2Mˆ2 = M2∗ (αH − αT), (5.21)where H is the conformal Hubble function, m0 the bare Planck mass and M∗ the effectivePlank mass. Fixing αM corresponds to fixing M∗ through Eq. (5.16). Once αT has beenchosen, Ω is obtained from Eq. (5.17). Finally, αB determines M¯31 via Eq. (5.20), whilethe choice of αH fixes Mˆ2 via Eq. (5.21). In this way, our choice of the EFT functions canbe guided by the selection of different “physical” scenarios, corresponding to turning ondifferent Horndeski functions.To avoid possible consistency issues with higher derivatives, we set16 M¯23 = M¯22 = 0in order to satisfy Eq. (5.9). From Eq. (5.18) and Eq. (5.10) this implies αT = 0, so thattensor waves move with the speed of light. In addition, we set αH = 0 so as to remain in theoriginal class of Horndeski theories, avoiding operators that may give rise to higher-ordertime derivatives [197]. As a consequence, Mˆ2 = 0 from Eq. (5.21) and M2∗ = m20(1 + Ω)from Eq. (5.10). For simplicity we also turn off all other higher-order EFT operators andset M¯31 = M42 = 0. Comparing Eq. (5.11) and Eq. (5.20), this implies αB = −αM.In summary, in the following we consider Horndeski models in which αM = −αB, αKis fixed by Eq. (5.13), with M2 = 0 as a function of c and αT = αH = 0.The only free function in this case is αM, which is linked to Ω through:αM =aΩ + 1dΩda. (5.22)16 Because of the way EFTCAMB currently implements these equations internally, it is not possible to satisfyEq. (5.9) otherwise.895.6. ResultsBy choosing a non-zero αM (and therefore a time evolving Ω) we introduce a non-minimalcoupling in the action (see Eq. 5.2), which will lead to non-zero anisotropic stress and tomodifications of the lensing potential, typical signatures of MG models. Here we will usea scaling ansatz, αM = αM0aβ, where αM0 is the value of αM today, and β > 0 determineshow quickly the modification of gravity decreases in the past.Integrating Eq. (5.22) we obtainΩ(a) = exp{αM0βaβ}− 1, (5.23)which coincides with the built-in exponential model of EFTCAMB for Ω0 = αM0/β. Themarginalized posterior distributions for the two parameters Ω0 and β are plotted in Fig. 5.1for different combinations of data. For αM0 = 0 we recover ΛCDM. For small values of Ω0and for β = 1, the exponential reduces to the built-in linear evolution in EFTCAMB,Ω(a) = Ω0 a . (5.24)The results of the MCMC analysis are shown in Table 5.1. For both the exponential and thelinear model we use a flat prior Ω0 ∈ [0, 1]. For the scaling exponent β of the exponentialmodel we use a flat prior β ∈ (0, 3]. For β → 0 the MG parameter αM remains constantand does not go to zero in the early Universe, while for β = 3 the scaling would correspondto M functions in the action (5.2) which are of the same order as the relative energy den-sity between DE and the dark matter background, similar to the suggestion in [172]. Animportant feature visible in Fig. 5.1 is the sharp cutoff at β ≈ 1.5. This cutoff is due to“viability conditions” that are enforced by EFTCAMB and that reject models due to a set oftheoretical criteria (see [195] for a full list of theoretical priors implemented in EFTCAMB).Disabling some of these conditions allows to extend the acceptable model space to largerβ, and we find that the constraints on αM0 continue to weaken as β grows further, extendingFig. 5.1 in the obvious way. We prefer however to use here the current public EFTCAMBversion without modifications. A better understanding of whether all stability conditions905.6. Resultsimplemented in the code are really necessary or exclude a larger region than necessary inparameter space will have to be addressed in the future. The posterior distribution of thelinear evolution for Ω is shown in Fig. 5.2 and is compatible with ΛCDM. Finally, it isinteresting to note that in both the exponential and the linear expansion, the inclusion ofgalaxy weak lensing (WL) data set weakens constraints with respect to Planck alone. Thisis due to the fact that in these EFT theories, WL and Planck are in tension with each other,WL preferring higher values of the expansion rate with respect to Planck.0.00 0.04 0.08 0.12 0.16αM00.00.40.81.21.6βPlanckPlanck+ BSHPlanck+ WLPlanck+ BAO/RSDPlanck+ BAO/RSD + WLFigure 5.1. Marginalized posterior distributions at 68 % and 95 % C.L. for the two parame-ters αM0 and β of the exponential evolution, Ω(a) = exp(Ω0 aβ) − 1.0, see Sect. 5.6.2. HereαM0 is defined as Ω0β and the background is fixed to ΛCDM. αM0 = 0 corresponds to theΛCDM model also at perturbation level. Note that Planck means Planck TT+lowP. AddingWL to the data sets results in broader contours, as a consequence of the slight tension be-tween the Planck and WL data sets.915.6. Results0.00 0.04 0.08 0.12αM00.00.20.40.60.81.0P/PmaxPlanckPlanck + BSHPlanck + WLPlanck + BAO/RSDPlanck + BAO/RSD + WLFigure 5.2. Marginalized posterior distribution of the linear EFT model background pa-rameter, Ω, with Ω parameterized as a linear function of the scale factor, i.e., Ω(a) = αM0 a,see Sect. 5.6.2. The equation of state parameter wde is fixed to −1, and therefore, Ω0 = 0will correspond to the ΛCDM model. Here Planck means Planck TT+lowP. Adding CMBlensing to the data sets does not change the results significantly; high-` polarization tightensthe constraints by a few percent, as shown in Tab. 5.1.5.6.3 Further examples of particular modelsQuite generally, DE and MG theories deal with at least one extra degree of freedom thatcan usually be associated with a scalar field. For ‘standard’ DE theories the scalar fieldcouples minimally to gravity, while in MG theories the field can be seen as the mediator ofa fifth force in addition to standard interactions. This happens in scalar-tensor theories (in-cluding f (R) cosmologies), massive gravity, and all coupled DE models, both when matteris involved or when neutrino evolution is affected. Interactions and fifth forces are there-925.6. ResultsParameter TT+lowP+BSH TT+lowP+WL TT+lowP+BAO/RSD TT+lowP+BAO/RSD+WL TT,TE,EELinear EFT . . . . . . . . .αM0 . . . . . . . . . . . . . . < 0.052(95 %CL) < 0.072(95 %CL) < 0.057(95 %CL) < 0.074(95 %CL) < 0.050(95 %CL)H0 . . . . . . . . . . . . . . . 67.69 ± 0.55 67.75 ± 0.95 67.63 ± 0.63 67.89 ± 0.62 67.17 ± 0.66σ8 . . . . . . . . . . . . . . . 0.826 ± 0.015 0.818 ± 0.014 0.822 ± 0.014 0.814 ± 0.014 0.830 ± 0.013Exponential EFT . . . . .αM0 . . . . . . . . . . . . . . < 0.071(95 %CL) < 0.098(95 %CL) < 0.071(95 %CL) < 0.094(95 %CL) < 0.056(95 %CL)β . . . . . . . . . . . . . . . . 0.90+0.55−0.25 0.91+0.55−0.25 0.88+0.55−0.27 0.89+0.55−0.27 0.88+0.56−0.27H0 . . . . . . . . . . . . . . . 67.70 ± 0.56 67.78 ± 0.96 67.60 ± 0.62 67.87 ± 0.63 67.15 ± 0.65σ8 . . . . . . . . . . . . . . . 0.826 ± 0.015 0.817 ± 0.014 0.821 ± 0.014 0.814 ± 0.014 0.830 ± 0.013Table 5.1. Marginalized mean values and 68 % CL intervals for the EFT parameters, bothin the linear model, αM0, and in the exponential one, {αM0, β} (see Sect. 5.6.2) Adding CMBlensing does not improve the constraints, while small-scale polarization can more stronglyconstraint αM0.fore a common characteristic of many proposed models, the difference being whether theinteraction is universal (i.e., affecting all species with the same coupling, as in scalar-tensortheories) or is different for each species (as in coupled DE, [198, 199] or growing neutrinomodels, [200, 201]). In the following we will test one of the well known examples of thesemodels, namely f (R) theories.Universal couplings: f (R) cosmologiesA well-investigated class of MG models is constituted by the f (R) theories that modifythe Einstein-Hilbert action by substituting the Ricci scalar with a more general function ofitself:S =12κ2∫d4x√−g(R + f (R)) +∫d4xLM(χi, gµν), (5.25)where κ2 = 8piG. f (R) cosmologies can be mapped to a subclass of scalar-tensor theories,where the coupling of the scalar field to the matter fields is universal.For a fixed background, the Friedmann equation provides a second-order differentialequation for f (R(a)) [see e.g., 202, 203]:H2 − (HH ′) fR + 16 f a2 +H2 fRRR′ = 8 piGρ a23. (5.26)935.6. ResultsHere a prime denotes a derivative with respect to ln(a). One of the initial conditions isusually set by requiringlimR→∞f (R)R= 0, (5.27)and the other initial (or boundary condition), usually called B0, is the present value ofB(z) =fRR1 + fRHR˙H˙ − H2 . (5.28)Here, fR and fRR are the first and second derivatives of f (R), and a dot means a derivativewith respect to conformal time. B(z) = 0 corresponds to f (R) ∝ (R+ const.), i.e. GR plus acosmological constant, after a rescaling of the gravitational constant. Higher values of B0hint towards a GR modification. They tend to suppress power at large scales in the CMBpower spectrum, due to a modified ISW effect, and increase the amplitude of the CMBlensing potential, resulting in slightly smoother peaks at higher `s [204–207].It is possible to restrict EFTcamb to describe f (R)-cosmologies. Given an evolutionhistory for the scale factor and the value of B0, EFTcamb effectively solves the Friedmannequation for f (R). It then uses this function at the perturbation level to evolve the metricpotentials and matter fields. The merit of EFTcamb over the other available similar codes isthat it checks the model against some stability criteria and does not assume the quasi-staticregime, where the scales of interest are still linear but smaller than the horizon and the timederivatives are ignored.As shown in Fig. 5.3, there is a degeneracy between the optical depth, τ, and the f (R)parameter, B0. Adding any structure formation probe, such as WL, redshift space (RSD) orCMB lensing, breaks the degeneracy. Figure 5.4 shows the likelihood of the B0 parameterusing EFTcamb, where a ΛCDM background evolution is assumed, i.e., wDE = −1.As the different data sets provide constraints on B0 that vary by more than four ordersof magnitude, we show plots for log10 B0; to make these figures we use a uniform prior inlog10 B0 to avoid distorting the posterior due to prior effects. However, for the limits quotedin the tables we use B0 (without log) as the fundamental quantity and quote 95 % limits945.6. Resultsbased on B0. In this way the upper limit on B0 is effectively given by the location of thedrop in probability visible in the figures, but not influenced by the choice of a lower limitof log10 B0. Overall this appears to be the best compromise to present the constraints on theB0 parameter. In the plots, the GR value (B0 = 0) is reached by a plateau stretching towardsminus infinity.−7.5 −6.0 −4.5 −3.0 −1.5 0.0log(B0)0.020.040.060.080.100.12τPlanckPlanck+BSHPlanck+WLPlanck+BAO/RSDPlanck+BAO/RSD+WLFigure 5.3. 68 % and 95 % contour plots for the two parameters, {Log10(B0), τ} (see Sect.5.6.3). There is a degeneracy between the two parameters for Planck TT+lowP+BSH.Adding lensing will break the degeneracy between the two. Here Planck indicates PlanckTT+lowP.955.6. Results−7.5 −6.0 −4.5 −3.0 −1.5 0.0log(B0)0.00.20.40.60.81.0P/PmaxPlanck+lensingPlanck+lensing+BSHPlanck+lensing+WLPlanck+lensing+BAO/RSDPlanck+lensing+BAO/RSD+WLFigure 5.4. Likelihood plots of the f (R) theory parameter, B0 (see Sect. 5.6.3). CMBlensing breaks the degeneracy between B0 and the optical depth, τ, resulting in lower upperbounds.Finally, we note that f (R) models can be studied also with the MGcamb parameteriza-tion, assuming the quasi-static limit. We find that for the allowed range of the B0 parameter,the results with and without the quasi-static approximation are the same within the uncer-tainties. The 95 % confidence intervals are reported in Table 5.2. These values show animprovement over the WMAP analysis made with MGcamb (B0 < 1 (95 % C.L.) in [202])and are similar to the limits obtained in [208] with MGcamb.965.7. Conclusionsf (R) models Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP+BSH +WL +BAO/RSD +WL+BAO/RSDB0 . . . . . . . . . . . . < 0.79 (95 % CL) < 0.67 (95 % CL) < 0.082 (95 % CL) < 0.88 × 10−4 (95 % CL) < 0.10 × 10−3 (95 % CL)H0 . . . . . . . . . . . . 68.9 ± 1.0 68.37 ± 0.58 69.1+1.0−1.2 67.64 ± 0.65 67.92 ± 0.63σ8 . . . . . . . . . . . . 1.150+0.043−0.031 1.142+0.044−0.030 0.992+0.080−0.056 0.842+0.017−0.020 0.837+0.017−0.019B0 (+lensing) . . . . < 0.12 (95 % CL) < 0.07 (95 % CL) < 0.04 (95 % CL) < 0.95 × 10−4 (95 % CL) < 0.83 × 10−4(95 % CL)H0 . . . . . . . . . . . . 69.0 ± 1.1 68.27 ± 0.57 69.1 ± 1.1 67.75 ± 0.61 67.91 ± 0.60σ8 . . . . . . . . . . . . 1.017+0.046−0.030 1.002+0.044−0.028 0.959 ± 0.053 0.836+0.014−0.019 0.833+0.014−0.018Table 5.2. 95 % CL intervals for the f (R) parameter, B0 (see Sect. 5.6.3). While the plotsare produced for log10 B0, the numbers in this table are produced via an analysis on B0 sincethe GR best fit value (B0 = 0) lies out of the bounds in a log10 B0 analysis and its estimatewould be prior dependent.5.7 ConclusionsThe quest for Dark Energy and Modified Gravity is far from over. A variety of differenttheoretical scenarios have been proposed in literature and need to be carefully comparedwith the data. This effort is still in its early stages, given the variety of theories and param-eterizations that have been suggested, together with a lack of well tested numerical codesthat allow us to make detailed predictions for the desired range of parameters. In this chap-ter, we considered a yet different approach to modified gravity in cosmology. Even thoughmost of the weight in the Planck data lies at high redshift, Planck can still provide tightconstraints on DE and MG, especially when used in combination with other probes. Ourfocus has been on the scales where linear theory is applicable, since these are the mosttheoretically robust. Overall, the constraints that we find are consistent with the simplestscenario, ΛCDM, with constraints on MG models (including effective field theory, and f (R)models) that are significantly improved with respect to past analyses.We discussed how to restrict EFT theories for MG, which include (almost) all univer-sally coupled models in MG via nine generic functions of time, to Horndeski theories, de-scribed in terms of five free functions of time. Using the publicly available code EFTcamb,we have then varied three of these functions, in the limits allowed by the code, which cor-respond to a non-minimally coupled K-essence model (i.e. αB, αM, and αK are varying975.7. Conclusionsfunctions of the scale factor). We have found limits on the present value αM0 < 0.052 at95% C.L. (in the linear EFT approximation), in agreement with ΛCDM. Constraints de-pend on the stability routines included in the code, which will need to be further tested inthe future, together with allowing for a larger set of choices for the Horndeski functions,not available in the present version of the numerical code.We also considered f (R) models, written in terms of B(z), conventionally related tothe first and second derivatives of f (R) with respect to R. The results of which show aremarkable agreement with GR.98Chapter 6ConclusionIn this thesis, we discussed modified theories of gravity in general, and explored three dif-ferent methods for studying these models in cosmology. We introduced a phenomenologicalmethod in chapter 3, studied a specific model of modified gravity (i.e. gravitational aether)in chapter 4, and considered approaches to modifying the Lagrangian in chapter 5. We dis-cussed a variety of models, data sets, and numerical codes, but there are of course scenariosnot included in this thesis that deserve future attention. Additional cosmologies within theEFT (and Horndeski) framework, massive gravity models (see [209] for a recent review),general violations of Lorentz invariance as a way to modify GR [210], and non-local grav-ity [211], have all been considered in recent years for a variety of different reasons, andcosmology can provide useful tests for them.Cosmology has proven to be unique in testing gravity, thanks to its time-dependentbackground metric, and the diverse range of environmental conditions in the Universe sincethe big bang. The radiation-domination, matter-domination, and Λ-domination phases ofcosmology provide unique tests for alternative models of gravity. Up to now, the tightestconstraints have come from CMB temperature and polarization anisotropies, which exploreearly stages in the evolution of the Universe (together with BBN). However, there are otherproposed and ongoing experiments that aim at studying the later epochs [212–215], andfurther promising constraints may come from a more precise measurement of the matterpower spectrum, gravitational waves, and the reionization epoch. As of now, GR has passedall of the observational tests, and ΛCDM has proven to be the simplest model for describingthe Universe. However, this is not considered desirable by many in theoretical physics.This situation can (and hopefully will) change in the future with access to more data from a99Chapter 6. Conclusionvariety of different experiments. With access to all of these data sets, it may be temping tocombine the results of all the different experiments and find the tightest constraints on themodified gravity parameters. However, we want to argue against this simplistic approachfor the following two reasons.The first reason deals with the issue of “false positives”. We define false positives assigns of modified gravity that originate from inconsistencies between two data sets (giventhe ΛCDM model). These signs are considered “false”, since the inconsistencies are eitherdue to systematic noise, or cosmic variance fluctuations. Clearly, one needs to be careful toavoid being misled into claiming detection of a violation of GR that comes from systematiceffects between data sets. Examples of this case have been discussed in Ref. [19].The second reason is the “true negatives”, i.e. real deviations from the standard modelthat are ignored due to compiling independent data sets. Let us explain this situation with amade-up example from the history of science. Let us imagine a case where the phenomenonof gravitational lensing was discovered prior to introducing GR (together with the alreadyknown empirical excess in precession of Mercury). The precession of Mercury can beexplained in Newtonian mechanics by adding an A/r2 term to the Newtonian gravitationalpotential. The amplitude A, is a free-parameter here, and lacking a more complete theory,is just a number that comes out of the fitting process. Fitting for this amplitude parameterwould result in a non-zero value and hence would stand out as a hint for “modified gravity”(i.e. deviations from Newtonian gravity). On the other hand, gravitational lensing, whichis also deviant from Newtonian gravity, does not prefer a 1/r2 term, but a gravitationalconstant that is twice as strong for photons. Combining the lensing experiment with theprecession of Mercury, would therefore result in null results for deviations from Newtonianmechanics, even though these are both classical examples of such a deviation. One can keepcombining more data (such as projectile trajectories, planetary orbits other than Mercury,gyroscope experiments, etc.) and increasing confidence on the A = 0 value by loweringthe upper 95% confidence limit on A. We made up this example based of the fact that boththe gravitational lensing, and the precession of Mercury are linear first order deviations100Chapter 6. Conclusionfrom Newtonian mechanics, while the underlying theory (i.e. GR) produces totally differentcorrection terms for them. This means that if GR is not the ultimate theory of gravity, it isplausible to believe that a more complete (and fundamentally different) theory would resultin totally different correction functions for different observables. Using the language ofchapter 3 to parameterize deviations from GR via two functions of scale and time, A andB, this would mean that combining different data sets could result in constraints that areconsistent with the GR value of these functions, i.e. A = B = 0, no matter if that is actuallythe case. It is therefore possible to ignore genuine GR deviations by combining enoughindependent data sets in the wrong theoretical framework.Based on these observations, our suggestion for future experiments that are aimed attesting gravity is to keep an open mind and follow up any suggested (even statisticallyinsignificant) deviations from GR in the existing data, using improved data sets. One suchdeviation is an oscillatory residual in Planck data at high multipoles in the temperatureanisotropies [175]. This oscillatory residual is not statistically significant at the moment,due to the noise in data, and it is not clear whether it is an attribute of the Universe or isdue to systematic noise (or just a statistical fluctuation). A future more precise experimentwould be able to clear this up.Besides detecting deviations from GR, there is also a need for building a theory thatcan explain the deviations. The above example also shows that a “reconstruction process”(i.e. using experimental data to build theories) could easily result in unnecessary compli-cations if the new theory uses the same fundamentals as the existing theory. For example,one such approach might be a modified theory of Newtonian gravity where the gravitationalconstant smoothly becomes twice as strong when the mass of the particle goes to zero (witha parameter that controls this transition),17 and a 1/r2 term is added to the potential suchthat it only couples with the angular momentum of an object in orbital motion. 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