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. r2Ψ r2Φ We show that it is possible to relate the functions µ = − and ζ = 1 − : 4πG δρa2 r2Ψ 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 r 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 r2Φ + 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- fα; γ; αc; γcg. 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.