Parameterizing Theories of Gravity on Large and Small Scales in Cosmology

Parameterizing theories of gravity on large and small scales in cosmology

Timothy Clifton1 and Viraj A. A. Sanghai2 1School of Physics & Astronomy, Queen Mary University of London, UK. 2School of Mathematics & Statistics, Dalhousie University, Canada. We present a link between parameterizations of alternative theories of gravity on large and small scales in cosmology. This relationship is established using theoretical consistency conditions only. We find that in both limits the “slip” and “effective Newton’s constant” can be written in terms of a set of four functions of time, two of which are direct generalizations of the α and γ parameters from post-Newtonian physics. This generalizes previous work that has constructed frameworks for testing gravity on small scales, and is to the best of our knowledge the first time that a link between parameterizations of gravity on such very different scales has been established. We expect our result to facilitate the imposition of observational constraints, by drastically reducing the number of functional degress of freedom required to consistently test gravity on multiple scales in cosmology.

Introduction – There has recently been a lot of inter- bation to the energy density, and where dots denote d/dτ est in testing the validity of Einstein’s theory of General and H =a/a ˙ . This follows the same logic as the PPN Relativity (GR) using cosmological observables [1]. To formalism, where the consequences of additional gravita- fully exploit such tests requires us to understand the pre- tional degrees of freedom are encoded in the parameters dictions that alternative theories of gravity could make µ and ζ, rather than being including as extra terms that for observations made over a wide range of scales, and have no counterparts in Einstein’s theory. to develop suitable frameworks for parameterizing these If µ = 1 and ζ = 0 then Eqs. (1)-(2) can be seen to phenomena. This is a challenging prospect, as the physics reduce to those of GR. In the quasi-static limit they also involved in horizon-sized cosmological perturbations is reduce to well-known “effective gravitational constant” quite different to that which occurs on smaller scales, and “slip” parameters, defined via [4, 5] where galaxies and clusters of galaxies are present. ∇2Ψ ∇2Φ We show that it is possible to relate the functions µ = − and ζ = 1 − . 4πG δρa2 ∇2Ψ that parameterize gravity on non-linear scales (k & 10−2 Mpc−1) to those that parameterize it on very large Alternative theories of gravity generically result in µ 6= 1 −4 −1 scales (k . 10 Mpc ), where k is the wavenumber of and ζ 6= 0. It is therefore of direct physical interest to perturbations. Our approach to this problem is to lean constrain these parameters with observations, to learn heavily on the parameterized post-Newtonian (PPN) for- about how gravity behaves on cosmological scales. malism [2], and to directly extend it to cosmology. Part of The purpose of the current paper is to relate the small the reason for this is that the PPN approach has proven and large-scale behaviour of these parameters to each to be highly successfully in parameterising gravitational other, and to the background expansion. This will be physics on solar system and astrophysical scales. An- achieved through the application of the following three other is that the PPN formalism is known to encompass consistency criteria: a wide array of alternative theories. Our final results give (i) On small scales, the gravitational field of non-linear a parameterization of gravity in cosmology that, at lead- structures should be describable using the parame- ing order in perturbations, depends on only four func- terized post-Newtonian formalism. tions of time. We call this approach parameterized post- (ii) The global rate of expansion of the Universe should Newtonian cosmology (PPNC) [3]. be compatible with gravitational fields of the non- We will consider perturbations about an FLRW back- linear structures within it. ground, which in longitudinal gauge has the scalar part (iii) On very large scales, the evolution of perturba- " 2 2 2 # 2 2 2 (dx + dy + dz ) tions in any theory should be consistent with the ds = a − (1 − 2Φ)dτ + (1 + 2Ψ) 2 1 + 1 κr2 expansion rate of the Friedmann solutions of that 4 theory. where a = a(τ) is the scale factor, τ = R a−1dt is con- We expect these criteria to be applicable to a wide range formal time, κ is the spatial curvature parameter, and of metric theories of gravity, which are in a suitable r2 = x2 + y2 + z2. We then suppose that the linearized sense “close” to GR, and that both admit perturbed field equations that govern the metric potentials above FLRW solutions and fit into the PPN framework (see can be written as Refs. [2, 3] for explicit examples). We will consider the 1 2 2 ˙ 4πG 2 3 ∇ Ψ − H Φ − HΨ + κΨ = − 3 µ δρa (1) implications of each of these three conditions in turn, before returning to show that together they give the 1 ∇2Φ + 2H˙ Φ + HΦ˙ + Ψ¨ + HΨ˙ = − 4πG µ(1 − ζ) δρa2 (2) 3 3 values of µ and ζ, on both large and small scales, in where µ = µ(τ, x) and ζ = ζ(τ, x) are yet to be deter- terms of a set of just four unspecified functions of time: mined functions of time and scale, where δρ is the pertur- {α, γ, αc, γc}. The first two of these are, in fact, exactly 2 the same as the α and γ parameters that occur in the The functions α, γ, αc and γc that are used here PPN formalism (now allowed to vary over cosmic time), should in general be expected to be functions of time while the second two are intrinsic to cosmology. We only, if inhomogeneous gravitational fields are expected will refer to the full set of four functions as the PPNC to be sourced by matter fields (see Ref. [3] for details). parameters. We set c = 1 throughout this paper. Greek The reader will note that the PPN parameter α has been letters are used to denote spatial indices, and Latin made explicit in the expression for h00. This is usually letters to denote space-time indices. set to unity when the PPN formalism is presented, so that the standard Newton-Poisson equation is recovered Condition (i): A viable weak field – The indus- in the appropriate limit. Here we have made this pa- try standard for investigating and constraining theories rameter explicit as we only require the Newton-Poisson of gravity in the slow motion and weak field regime is equation with the experimentally recovered value of G the PPN formalism [2]. This formalism is built on the to be applicable at the present moment of time. Earlier post-Newtonian expansion of gravitational and matter (or later) cosmological epochs may have different values fields, and is expected to be valid in the presence of all of Newton’s constant in alternative theories of gravity non-linear structures down to the scale of neutron stars [9]. The second PPN parameter, γ, affects Shapiro [6]. All reasonable gravitational potentials are then pos- time delay and deflection of light rays, among other tulated, and (constant) parameters are included before things, and is currently most strongly constrained from each of them in the metric, in order to create a general observations of radio signals from the Cassini spacecraft geometry that can be both constrained by observations to be γ = 1 + (2.1 ± 2.3) × 10−5 [10]. For the case of and compared to the predictions of individual theories. GR, α = γ = 1 and α = −2γ = Λ for all time. It seems reasonable to assume that the gravitational c c fields of galaxies and clusters of galaxies should also be Condition (ii): A compatible Hubble rate – The described by post-Newtonian expansions, as long as these Hubble rate of the Universe’s expansion is not indepen- systems are small enough that the Hubble flow across dent of the gravitational fields of the objects within it. them is still much smaller than the speed of light (i.e. On the contrary, one can build explicit constructions in that they are sub-horizon sized). If this is the case, then which the global expansion emerges from the Newtonian- we can directly apply the PPN formalism to describe the level gravitational fields of the structures it contains gravitational fields within their vicinity. Such a descrip- [7], with post-Newtonian fields giving small corrections tion can be shown, by direct transformation, to be iso- [8, 11, 12]. Such models make clear that the large-scale metric to a region of perturbed FLRW space-time [7, 8]. expansion of a cosmological model should be able to be In this case the coordinate transformations t˜ → t + parameterized with the same set of functions as weak 1 2 µ µ 1 2 2 2 aHr andx ˜ → a x (1 + 4 H r ) allow the scalar grav- gravitational fields on small scales. This fact was used itational potentials in the perturbed FLRW geometry to in [3] to extract the large-scale expansion of a Universe be written in terms of those in the isometric perturbed that is governed by the PPN metric on small scales. Here Minkowski space as [7] we generalize this construction to universes constructed from regions of space with periodic boundary conditions, 1 1 2 Φ = h00 − H˙ r (3) rather than the more restrictive reflection symmetry that 2 2 was previously assumed. 1 µν 1 2 2 Ψ = δ hµν + H + κ r , (4) First we take two spatial derivatives of Eqs. (3) and 6 4 (4), and average the result over a large volume of space, where the post-Newtonian metric is given by gab = ηab + to find hab. If the post-Newtonian gravitational fields are then 8πG 2γ a2 H2 = γ hρi a2 − c − κ (5) taken to be given by the form they take in the PPN 3 3 formalism, we have [2, 3] 4πG α a2 H˙ = − α hρi a2 + c , (6) 1 3 3 h = 2αU + α r˜2 and δµν h = 6γU + γ r˜2 , 00 3 c µν c where angular brackets denote the spatial average of the quantity within, such that h· · · i = R ··· d3x/ R d3x. In where U is the Newtonian gravitational potential that deriving these equations we have used the result h∇2Φi = satisfies ∇˜ 2U = −4πGρ, where ρ is the energy density, 0 = h∇2Ψi for a region of space with periodic boundary and where tildes denote coordinates in the Minkowski conditions, and we have assumed that the PPNC param- space. The two additional terms, proportional to α and c eters {α, γ, α , γ } are not functions of space. Failure of γ in these equations, are introduced in order to include c c c these results to hold would indicate the presence of large the effects of Λ, and any of the other slowly-varying (or cosmological back-reaction [13], and would invalidate the constant) additional fields that are often introduced in use of a global FLRW background, which is not the sce- alternative theories. This gives a fully specified form for nario that we wish to consider here. the cosmological scalar gravitational potentials on small Subtracting the averaged Eqs. (5)-(6) from the original scales, where non-linear structures are present, in terms Eqs. (3)-(4) then gives of a set of four familiar quantities (see [3] for further explanation). ∇2Φ = −4πG α δρa2 and ∇2Ψ = −4πG γ δρa2, (7) 3

∂ δ ∂ ln a Ψ where δρ = ρ − hρi. It can be seen that the PPN param- Φ + Ψ = χ 1 − 2 − + δ , (10) eters α and γ parameterize both the matter terms in the ∂τ H ∂ ln κ H χ Friedmann Eqs. (5)-(6) and the small-scale cosmological where we have eliminated the dependence on δτ, and perturbations in Eqs. (7). On the other hand, the cosmo- where Eq. (9) tells us δχ must be a constant. Using hρi ∝ logical parameters αc and γc occur only in the Friedmann a−3, we can then Taylor expand δ to obtain δ = −3(Ψ − equations, and take the place of the dark energy terms. δ ), and hence δ˙ = −3Ψ˙ (as expected from perturbing The reader may note that in addition to the equations χ the continuity equation on very large scales). Using this above, we have an additional integrability condition on result to replace δ in Eq. (10), and performing further {α, γ, α , γ } given by χ c c manipulations, one finds 4πGhρi = (α + 2γ + γ0 ) (α − γ + γ0) , (8) δ c c c −H2Φ − HΨ˙ + κΨ = − H2 − H˙ + κ (11) 3 0 ! where = d/d ln a. The derivation of this equation δ 2HH˙ − H¨ uses the result hρi ∝ a−3, which can be obtained from 2H˙ Φ + HΦ˙ + Ψ¨ + HΨ˙ = . (12) 3 H averaging the energy-momentum conservation equations, as well as the standard assumption that we do not have Finally, we can simplify the right-hand sides of these any interactions between matter fields at the level of the equations by using the parameterized Friedmann Eqs. background cosmology. Explicit expressions for these (5)-(6), and the constraint in Eq. (8), to get PPNC parameters, for scalar-tensor and vector-tensor 2 theories of gravity, were derived in [3]. The following − H Φ − HΨ˙ + κΨ sections extend the domain of validity of this work to 4πG 1 1 = − δρ a2 γ − γ0 + γ0 (13) horizon-sized scales, and so give information on the scale 3 3 12πGhρi c dependence of modified gravity parameters. and Condition (iii): Large-scale perturbations – On ˙ ˙ ¨ ˙ sufficiently large scales the spatial gradients of pertur- 2HΦ + HΦ + Ψ + HΨ bations are expected to be become much smaller than 4πG 1 1 = − δρ a2 α − α0 + α0 . (14) their time derivatives. Neglecting their contribution to 3 3 12πGhρi c the field equations then results in a situation where only the time dependence of the gravitational potentials is of These results can be compared to Eqs. (1)-(2). They significance for their evolution. It therefore becomes pos- constitute a parameterization of super-horizon-sized sible to model these perturbations as Friedmann solu- fluctuations in terms of the PPNC parameters and their tions with perturbed energy density and spatial curva- time derivatives only. The reader may note that nowhere ture, as well as space and time coordinates. As noted by in this derivation have we assumed anything about the Bertschinger; the evolution of such perturbations must field equations of gravity, except that they should result then be specified entirely by the Friedmann equations, in the parameterized Friedmann Eqs. (5)-(6), which and can be deduced without knowing any further infor- themselves are a direct result of treating gravity on mation about the field equations of gravity [14]. small scales as being governed by the PPN metric. In the present situation, this result is especially useful Results – We can now bring together the results above to as our parameterized Friedmann Eqs. (5)-(6) can now be demonstrate the link between parameterizations of grav- used to determine the form of very-large-scale perturba- ity on large and small scales in cosmology. Comparing tions. The result must then also be dependent only on the Eqs. (7) with (1)-(2) allows the effective gravitational PPNC parameters {α, γ, αc, γc}. The first step in doing constant parameter, µ, and the gravitational slip param- this is to perturb the FLRW background in spherical po- eter, ζ, to be related to the α and γ parameters via [3] lar coordinates such that the conformal time τ → τ + δτ lim µ = γ (15) and radial coordinate χ → χ(1+δχ), and to perturb spa- k→∞ tial curvature and energy density such that κ → κ(1+δ ) α κ lim ζ = 1 − , (16) and hρi → hρi(1 +δ), where δτ, δχ, δκ and δ are all small k→∞ γ quantities. Furthermore, the quantity δκ must also be a where k → ∞ corresponds to the small-scale limit. constant, as it is a perturbation of the (constant) spatial We expect this parameterization to be valid for k & curvature. The χ coordinate here is such that the con- 0.01 Mpc−1, where non-linear structures exist, and post- formal part of the spatial line-element can be written as 2 2 2 2 Newtonian expansions are expected to be valid. ds¯(3) = dχ +r ˆ (χ, κ)dΩ , where dΩ is the solid angle. Similarly, comparing Eqs. (13)-(14) with (1)-(2) allows Writing the large-scale perturbation in terms of the us to read off transformed Friedmann solution then means that the fol- 1 0 1 0 lowing must be true [14]: lim µ = γ − γ + γc (17) k→0 3 12πGhρi 0 0 1 α − α /3 + αc/12πGhρi δχ = − δκ (9) lim ζ = 1 − 0 0 , (18) 2 k→0 γ − γ /3 + γc/12πGhρi 4

PPNC observational PPNC observational that if {α, γ, αc, γc} are all independent of time then the parameter constraint derivative constraint large and small-scale limits of the slip and effective New- α 1 α0 0 ± 0.01 [9] ton’s constant are identical. This is not a surprise, as in- γ 1 ± 10−5 [10] γ0 0.0 ± 0.1 [17] dependence from spatial scale should be expected to lead 2 0 2 αc/H0 2.07 ± 0.03 [18] αc/H0 0.12 ± 0.25 [18] to independence from time in metric theories of gravity. 2 0 2 These results reduce to those expected from GR, when γc/H0 −1.04 ± 0.02 [18] γc/H0 −0.06 ± 0.12 [18] α = γ = 1 and αc = −2γc = Λ, as expected [15, 16]. TABLE I. Observational constraints on the present-day values Eqs. (15)-(18) represent the main results of this of the PPNC parameters {α, γ, αc, γc}, and their derivative paper. They extend parameterised gravity from small with respect to ln a. Here we have taken α = 1 (by defini- non-linear scales, to large horizon-sized scales. In order tion), and have assumed the Λs that appear in both Fried- to show the power of this approach, we have considered mann equations are identical, with ΩΛ = 0.69 ± 0.01 [18]. We −1 −1 indicative observational constraints on the individual have used H0 = 68 ± 1 km s Mpc [18] to find the relevant 0 0 PPNC parameters in Table I. These are then subse- derivatives, and derived the constraints on αc and γc using quently used in Fig. 1 to show how current bounds on the result ω + 1 = −0.02 ± 0.04 [18]. The α0 and γ0 results Λ the values of {α, γ, α , γ } can be used to derive bounds come from adapting constraints on G/G˙ and Σ from Refs. [9] c c and [17]. A more detailed study of observational constraints on ζ and µ on both small and large scales. It is the on these parameters will be presented in Ref. [19]. possible time-variation of the PPNC parameters that is the reason for the difference between the value of ζ and µ − 1 over this range of k. The constant value of ζ and −4 −1 −2 −1 where primes again represent d/d ln a, and where the spa- µ in the regions k . 10 Mpc and k & 10 Mpc tial gradient terms in Eqs. (1)-(2) are expected to be is due to the scale-independence of {α, γ, αc, γc}, as negligible. Here the limit k → 0 indicates that this pa- discussed above. Due to the different values of ζ and rameterization should be expected to be valid on super- µ in these two regions, we require scale dependence in −1 the region 10−4 Mpc−1 k 10−2 Mpc−1. We have horizon scales, such that k . 0.0001 Mpc . The reader . . may note that the terms involving hρi can be replaced by achieved this with an example interpolating curve in our parameters using Eq. (8), if required. Fig. 1. The first thing that can be noted about Eqs. (15)-(18) Conclusions – We have obtained a direct link between is that the large and small-scale parameterizations can horizon-sized cosmological perturbations, and those on both be written exclusively in terms of the PPNC param- smaller non-linear scales, in terms of a set of just four eters, {α, γ, α , γ }. These parameters also occur in the c c functions of time: {α, γ, αc, γc}. This set of functions background Friedmann Eqs. (5)-(6), and therefore pro- parameterize a wide array of minimal modifications to vide a complete set (in addition to the usual cosmological GR, of the type that has been long used to test gravity parameters) that can parameterize gravitational physics in the Solar System and binary pulsars. Here we have over a wide range of scales. Further, the reader may note found that they can be used to describe gravitational physics in the Solar System, through astrophysical and small cosmological scales, all the way up to the super- horizon scales involved in considering the entire observ- 0.06 able universe. 0.04 To the best of our knowledge, this is the first time that 0.02 theories of gravity have been parameterized consistently 1 on such a large range of scales, using such a compact 0.00 μ - set of parameters. These results can be contrasted with , ζ -0.02 the parameterized post-Friedmannian (PPF) [20–25] and effective field theory (EFT) [26–35] approaches, -0.04 which can contain larger numbers of unknown functions, -0.06 and that often require the degrees of freedom in the theory to be specified from the outset. We expect 10-5 10-4 0.001 0.010 0.100 our parameterization to be useful for testing minimal k/ Mpc -1 deviations from GR with future large-scale surveys [36–38], and in particular for future precision constraints −1 on gravity in cosmology. Future work will consider the FIG. 1. The small-scale (k & 0.01 Mpc ) and large-scale −4 −1 (k . 10 Mpc ) limits of the ζ (red) and µ (blue) parame- effect of screening mechanisms and fifth forces in this ters at the present time, connected by an interpolating tanh approach [39]. function (dotted). We have used the values of {α, γ, αc, γc} and their associated errors and derivatives from Table I. The Acknowledgements: We are grateful to P. Bull and A. shaded areas show the 1σ confidence regions, where errors Pourtsidou for helpful discussions. VAAS acknowledges have been assumed to be independent. the support of AARMS, and TC of the STFC. 5

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