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Cosmological Tests of General Relativity

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Cosmological Tests of General Relativity Cosmological Tests of General Relativity Rachel Bean (Cornell University) Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 The challenge: to connect theory to observation 2 • While GR is a remarkably beautiful, and comparatively Category Theory References Scalar-Tensor theory [21, 22] minimal metric theory, it is not (incl. Brans-Dicke) the only one. We should test it! f(R)gravity [23, 24] f(G)theories [25–27] • Rich and complex theory Covariant Galileons [28–30] Horndeski Theories The Fab Four [31–34] space of cosmological K-inflation and K-essence [35, 36] relevant modifications to GR Generalized G-inflation [37, 38] Kinetic Gravity Braiding [39, 40] Quintessence (incl. • Motivated by cosmic [41–44] universally coupled models) acceleration, but implications Effective dark fluid [45] from horizon to solar system Einstein-Aether theory [46–49] Lorentz-Violating theories scales Ho˘rava-Lifschitz theory [50, 51] DGP (4D effective theory) [52, 53] > 2newdegreesoffreedom EBI gravity [54–58] Aim: look for and confidently TeVeS [59–61] extract out signatures from Baker et al 2011 TABLE I: A non-exhaustive list of theories that are suitable for PPF parameterization. We will not treat all of these explicitly cosmological observations? 2 µν µνρσ in the present paper. G = R − 4Rµν R + RµνρσR is the Gauss-Bonnet term. Rachel Bean, Tes-ng Gravity, Vancouver, January 2015 derlying our formalism are stated in Table II. PPN and formalism. PPF are highly complementary in their coverage of dif- ferent accessible gravitational regimes. PPN is restricted ii) A reader with a particular interest in one of the ex- to weak-field regimes on scales sufficiently small that lin- ample theories listed in Table I may wish to addition- ally read II B-II D to understand how the mapping ear perturbation theory about the Minkowski metric is § an accurate description of the spacetime. Unlike PPN, into the PPF format is performed, and the most rel- evant example(s) of III. PPF is valid for arbitrary background metrics (such as § the FRW metric) provided that perturbations to the cur- iii) A reader concerned with the concept of parameter- vature scalar remain small. PPF also assumes the valid- ized modified gravity in and of itself may also find ity of linear perturbation theory, so it is applicable to II F and IV useful for explaining how the approach large length-scales on which matter perturbations have presented§ § here can be concretely implemented (for not yet crossed the nonlinear threshold (indicated by example, in numerical codes). IV also discusses the δM (knl) 1); note that this boundary evolves with red- § ∼ connection of PPF to other parameterizations in the shift. present literature. Perturbative expansions like PPN and PPF cannot be used in the nonlinear, strong-field regime inhabited Our conclusions are summarized in V. §−2 by compact objects. However, this regime can still We will use the notation κ = MP =8πG and set be subjected to parameterized tests of gravity via elec- c = 1 unless stated otherwise. Our convention for the metric signature is ( , +, +, +). Dots will be used to in- tromagnetic observations [62, 63] and the Parameter- − ized Post-Einsteinian framework (PPE) for gravitational dicate differentiation with respect to conformal time and waveforms [64, 65]. Note that despite the similarity in hatted variables indicate gauge-invariant combinations, nomenclature, PPE is somewhat different to PPN and which are formed by adding appropriate metric fluctua- PPF, being a parameterization of observables rather than tions to a perturbed quantity (see II D). Note that this meansχ ˆ = χ. § theories themselves. ̸ The purpose of this paper is to present the formalism that will be used for our future results [66] and demon- strate its use through a number of worked examples. We would like to politely suggest three strategies for guiding II. THE PPF FORMALISM busy readers to the most relevant sections: i) The casually-interested reader is recommended to as- A. Basic Principles similate the basic concepts and structure of the pa- rameterization from II A and II E, and glance at As stated in the introduction, the PPF framework sys- Table I to see some example§ theories§ covered by this tematically accounts for allowable extensions to the Ein- for both the LSST DESC and MS-DESI. Modifications to GR include the presence of extra degrees of freedom (e.g. Carroll et al. (2004)), massive gravitons (e.g. Hinterbichler (2012)), gravity pervading extra dimensions (e.g. Dvali et al. (2000)), and those which attempt to resolve the fine tuning cosmological constant problem through degravitation (Dvali et al. 2007; de Rham et al. 2008). In stark contrast to ⇤ and quintessence, modifications to gravity can have marked effects on both the growth for both the LSST DESC and MS-DESI.of large scale Modifications structure and to the GR background include the expansion. presence It of is extra common degrees that ofmodels that modify GR are able to freedom (e.g. Carroll et al. (2004)),reproduce massive the gravitons distance measurements (e.g. Hinterbichler but alter (2012)), the growth gravity of pervading large scale extra structure, opening up the possibility dimensions (e.g. Dvali et al. (2000)),of testing and and those discriminating which attempt between to resolve the different the fine theories. tuning cosmological Generically, the Poisson equation relating constant problem through degravitationover-densities (Dvali to et al.gravitational 2007; de Rham potentials et al. is 2008). altered and the potential that determines geodesics of relativistic (Φ+ )/2 In stark contrast to ⇤ and quintessence,particles, modifications in terms of the to gravity Newtonian can have gauge marked potentials effects on both, differsthe growth from that that determines the motion of non-relativistic particles, . Creating Φ/ =1during an accelerative era is extraordinarily difficult in of large scale structure and the background expansion. It is common that models6 that modify GR are able to reproduce the distance measurementsfluid models but alter of the dark growth energy of (Hularge 1998). scale structure, Measuring opening it, therefore, up the could possibility be a smoking gun of a deviation from GR. In Zhang et al. (2007b) we proposed a way to constrain Φ/ , by contrasting the motions of galaxies of testingTime-space and (Synchronous) discriminating between the different theories. Generically, the Poisson equation relating m2⌦ 2k2⌘˜˙ k (h˙ +6⌘˜˙) =(¯⇢ + P¯ )kv + m2⌦˙ k2⇡˙ +[⇢ + P ] k2⇡ (D.14) Time-space0 (Synchronous)− 0 m andm the0 lensingQ Q distortions of light from distant objects that LSST and MS-DESI data will be idea for. over-densities toh gravitationali potentials is altered and the potential that determines geodesics of relativistic Space-space trace2 (Synchronous)2 ˙ ˙ ˙ ¯ 2 ˙ 2 2 m0⌦ 2k ⌘˜ k0(h +6⌘˜) =(¯⇢m + Pm)kv + m0⌦k ⇡˙ +[⇢Q + PQ] k ⇡ (D.14) ¨ 2 − Time-space (Synchronous)particles,2 in termsh ˙ of2kh the2k0 Newtoniani ˙ gauge potentials (Φ+ )/2, differs from that that determines the motion mSpace-space⌦ hH trace+ (Synchronous)⌘˜ ⌘˜ = δP + PQ⇡Bean+(⇢Q + P andQ)˙⇡ her group have(D.15) developed software based on the publicly available CAMB and CosmoMC codes 0 − 3 − 3a2 − a2 2 2 ˙ ˙ " ˙ ¯ #2 ˙ 2 2 m0⌦ 2k ⌘˜ k0(h +6⌘˜) =(¯¨ ⇢m + Pm)kv2 + m0⌦k ⇡˙ +[⇢Q + PQ] k ⇡ (D.14) − 2 h 2k 2k0 Φ/2 ˙=1 of non-relativisticm ⌦ hH˙ particles,+ ⌘˜ ⌘˜ = δ.P + Creating(Lewis⌦¨P˙ ⇡ +(⇢ + etP 2)˙ al.k⇡ 2000;h during Lewisk0 &(D.15) an Bridle accelerative 2002) era to perform is extraordinarily likelihood difficult analyses in and forecasting for generic pho- Space-space traceh (Synchronous)0 i 3 3a2 a2 + m2⌦˙ ⇡˙Q+¨⇡ +3QH⇡˙ +Q ⇡ + + ⇡ 3H2 + "− − − # 0 ˙ 3 a2 36 a2 "⌦ ✓ ◆# ¨ fluid2 models of dark energy (Hu 1998).¨ Measuring2 ˙ it, therefore, could be a smoking gun of a deviation from 2 h ˙ 2k Space-space2k0 traceless˙ (Synchronous) tometric2 ⌦ and2 k spectroscopich 2 k0 surveys (Bean & Tangmatitham 2010; Laszlo et al. 2011; Kirk et al. 2011; m0⌦ hH + ⌘˜ ⌘˜ = δP + PQ⇡ +(⇢Q + PQ)˙⇡ + m ⌦˙ ⇡˙ +¨⇡ +3H(D.15)⇡˙ + ⇡ + + ⇡ 3H + − 3 − 3a2 − a2 0 ⌦˙ 3 a2 3 a2 " Frameworks#m2⌦ 2k2 to tie" observationsk2 1 ✓ to theory◆# GR. In Zhang0 et⌘˜ al.3H (2007b)(h˙ +6⌘˜˙) (h¨ +6 we⌘˜¨) = proposedP¯ ⇧ + m2⌦˙ ⇡ a+ wayh˙ +6⌘˜˙ to constrain(D.16) Φ/ , by contrasting the motions of galaxies 2 ¨ 2 Mueller˙ m 0 &2 Bean 2013). This includes peculiar velocity, weak lensing and galaxy clustering correlations Space-space2 a 2 traceless˙ −⌦ (Synchronous)− 2 k h 2 ak0 2 + m0⌦ ⇡˙ +¨⇡ +3H⇡˙ + ⇡ + + ⇡ 3H + ⇣ ⌘ 2 2⌦˙ 3 a2 3 a2 2 and the lensingWe can translatem0⌦ distortions2k" these into Newtonian˙ ˙ of gauge. light¨ and¨ The from required¯✓ cross-correlations distant gauge2 ˙ ◆#k transformations objects1 ˙ ˙ are detailed thatwith LSST the CMB. and MS-DESI The code data models will dark be idea energy for. and modified gravity using a variety ⌘˜ 3H(h +6⌘˜) (h +6⌘˜) = Pm⇧ + m0⌦ ⇡ + h +6⌘˜ (D.16) Space-space traceless (Synchronous)in Appendix• Modify B.2 a2 −homogeneous− and perturbeda2 2 Einstein’s equations ⇣ ⌘ 2 2 Time-time (Newtonian) 2 of phenomenological parameterizations, including the equation of state w(z), the Hubble expansion rate m0⌦ 2k We can translate these into2 Newtoniank 1 gauge. The required gauge transformations are detailed ⌘˜ 3H(h˙ +6⌘˜˙) (h¨ +6⌘˜¨) 2= P¯ ⇧ + m ⌦˙ ⇡ + h˙ +6⌘˜˙ (D.16) Bean2 andin Appendix her group B.2k 6km have0 developed2 software based on the publicly available CAMB and CosmoMC codes 2 a − −m2⌦(t) − 0 φN +6H(φ˙aN + H 2 N ) 0 a2 ⇣H(z⌘), the logarithmic growth factor, f (z), and its exponent, γ(z), and a parameterization directly related Time-time (Newtonian) g We can translate(Lewis these into et Newtonian al. 2000;N gauge.N The Lewis requiredN & gaugeN Bridle transformations 2002) are detailed to perform likelihood analyses and forecasting for generic pho- = ⇢ ⇢˙Q⇡ 2 2c(t)(⇡ ˙ ) in Appendix B.
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