TESTING GRAVITY WITH COSMOLOGY

21 IV. TESTING GRAVITY WITH COSMOLOGY We now turn to the different ways with which cosmological observations can constrain modified gravity models. We have already seen that Solar System tests provide stringent constraints on AU scales, but that modified gravity models can pass these tests through non-linear screening mechanisms while still yielding significant modifications on larger scales ( Mpc 2 10 11 AU). We can separate cosmological probes into those that measure the expansion history and geometry of the background Universe, and those that probe the evolution of structure (departures from the smooth background). The treatment of perturbations can further separated into the regime where (density) perturbations are linear, on large scales, and those in which they become non-linear, on smaller scales, to form dark matter halos, galaxies, clusters, etc. The latter regime can only rigorously be treated using simulations. The scale at which structure becomes non-linear today (redshift z = 0) is roughly 30 Mpc (depending on the definition). A. Expansion history and early Universe The expansion history of the Universe can be described through the Hubble parameter H(t). Qualitatively, the Universe transitioned from radiation-dominated (H a 2 ) to matter-dominated (H a 3/2 ) at redshift z eq 10 4. Many observations now indicate that the Universe transitioned to Dark Energy-dominated at z 1. In order from increasing to decreasing redshift, we have the following probes of H(t): Big Bang Nucleosynthesis: measurements of the abundances of light elements (in particular D and He) probe the expansion rate at t BBN 3 min after the Big Bang, through its influence on the nuclear reaction network. Since the Universe was radiation-dominated at that time, the absolute measurement of H(t BBN ) can be used together with the measurements on late-time cosmology to constrain any deviation of G N at t BBN from its value today, to the level of 5%. This puts constraints on scalar-tensor theories where G N evolves as φ 1. Recombination and cosmic microwave background (CMB): the CMB puts stringent constraints on the density of matter (both dark and baryonic) and radiation in the Universe. Together with measurements of the Hubble constant, it also provides a stringent constraint on the angular diameter distance to redshift z = 1100, and the curvature. So far, measurements are consistent with a flat Universe Ω K = 0.

22 Supernova Ia: the measurement of the apparent magnitude of Supernovae as function of observed redshift provides a measurement of the luminosity distance for redshifts z 0.1 1.7. Baryon Acoustic Oscillations: the CMB provides a accurate measurement of the acoustic sound horizon at recombination. This scale is also imprinted in the matter power spectrum and thus the power spectrum of large-scale structure tracers. A measurement of the angular scale of this feature at as given redshift provides a measurement of the angular diameter distance to that redshift. This technique has so far been applied to z 0.2 0.8. The last three measurements are used to infer the accelerated expansion. Using the Friedmann equation in GR, we can write (assuming a flat Universe) H 2 = 8πG N 3 [ ρ m + ρ DE ] = H 2 0 Ω m a 3 + Ω DE a 3(1+w), (72) parametrizing the Dark Energy through its density parameter and its equation of state w. A cosmological constant corresponds to w = 1. Current measurements are consistent with w = 1, with error bars at redshifts z < 0.5 of σ(w) 0.1. Theories such as f(r) and DGP of course modify Eq. (72); however, assuming we have only matter in the Universe, we can phrase the modification as an effective Dark Energy density ρ DE,eff. For example, in the case of self-accelerating DGP, this Dark Energy density has w eff 0.8 today, reducing to -0.5 at higher redshifts. CMB, Supernovae, and BAO are sufficient to rule out this expansion history at 4 5σ level. In the case of f(r) gravity, deviations in the Friedmann equation are of order f R, and expansion history observations rule out amplitudes of f R0 1. One of the main motivations for studying these models was as working examples to see what phenomenological signatures to look for when testing gravity. The expansion history of the Universe has one major drawback as test of gravity: we can only test gravity under the assumption of a cosmological constant or Dark Energy component. If we allow for unspecified Dark Energy, then any expansion history is possible in the GR+DE framework, rendering the expansion history a blunt probe of gravity. The growth of structure on the other hand is not directly influenced by a cosmological constant or Dark Energy 3, and thus can serve as a test of GR. 3 Unless we couple DE to matter, in which case the distrinction between DE and modified gravity becomes increasingly blurred.

23 B. Growth of structure We have no way of predicting the initial seed perturbations of the Universe; they are stochastic. However, we can obtain information on the stress-energy content of the Universe and gravity through statistical measures of the perturbations. In the commonly chosen conformal-newtonian gauge, the perturbed FRW metric can be written as (see Julien Lesgourgues lectures) ds 2 = a 2 (η) (1 + 2Ψ)dη 2 + (1 2Φ)dx 2, (73) where a(η) is the scale factor, η denotes conformal time, and Φ, Ψ are the cosmological potentials. In General Relativity (GR), Φ = Ψ in the absence of anisotropic stress, which is the case at late times in ΛCDM and most dark energy models. In addition, the potentials are conventionally re-expressed in terms of the matter overdensity δ m = (ρ ρ)/ ρ using the Poisson equation: 2 Ψ GR = 4πGa 2 ρ m δ m = 3 2 Ω mh 2 0 a 1 δ m, (74) where the derivatives on the l.h.s. are in terms of comoving coordinates (hence the factor a 2 on the r.h.s.). In case of modified gravity, both of these assumptions do not necessarily hold. In particular, we have seen that the scalar degree of freedom in scalar-tensor theories contributes differently to Φ and Ψ: Ψ = Ψ GR + αψ Φ = Φ GR αψ. (75) The dynamics of matter is governed by Ψ, while gravitational lensing measures Φ Ψ and thus is not directly influenced by the scalar field. 1. Linear Scales On large scales, the matter density contrast δ m is much less than one. This is the simplest case to consider since we can linearize all equations in terms of perturbations. In that case, Fourier modes of δ m evolve independently, and can be written in ΛCDM as δ m (k, z) = D(z)δ m (k, z i ) (76) where z i is some large initial redshift and D is called growth factor. The amplitude of δ m (k, z i ) is measured well (on a certain range of scales) by the CMB at z i 1100. If we restrict ourselves to

24 scales much smaller than the horizon (where most measurements of perturbations are done), then in GR the growth factor obeys D + 2HḊ GR = 4πG N ρ m a 2 D. (77) The r.h.s. is just 2 Ψ divided by δ m (k, z i ), and we have used Eq. (74). We have seen that scalar-tensor theories change the potential Ψ, i.e. the 00-component of the metric, by adding a contribution from the scalar field. In DGP, the brane-bending mode modifies G N G N (1 + 1/3β) (on linear scales), thus D + 2HḊ DGP = 4πG N 1 + 1 ρ m a 2 D. (78) 3β For a given expansion history and initial perturbations, DGP thus predicts a different amplitude of density perturbations, for perturbations of all (sub-horizon) scales. In f(r), forces are enhanced by 4/3 within the reach λ C = 1/m fr of the field (which in f(r) is a function of scale factor due to the evolution of the background R), so that D + 2HḊ f(r) 4 = 4πG N 3 1 m fr (a) 2 3 k 2 /a 2 + m fr (a) 2 ρ m a 2 D. (79) Here, k/a is the physical wavenumber of the perturbation, k being comoving. This tells us that the growth factor D is now a function of k, in contrast to GR, and grows towards larger k (smaller scales). The growth factor can in principle be measured through the clustering of large-scale structure tracers. However, those typically have an unknown bias parameter relating the amplitude of matter clustering to that of the tracer, which is degenerate with the amplitude of the growth factor. For this reason, other observables are more interesting: Velocities: the velocities of large-scale structure tracers are measurable statistically through the Doppler effect: the observed redshift is a sum of the cosmological redshift, which is a function of distance, and the Doppler shift, which is given by the line-of-sight component of the velocity. This can be measured through the distortion (redshift distortion) of the correlation function of tracers. Velocities of large-scale structure are determined by the gradient of the 00-component of the metric (since these motions are non-relativistic), Ψ. In the linear regime, they probe the derivative of the growth factor, dd/d ln a. Combining this with the clustering of tracers allows us to measure d ln D/d ln a, which is directly influenced by modifications to gravity.

25 Gravitational lensing: lensing denotes the deflection of light rays by spacetime perturbations; it is an inherently relativistic effect. Lensing is determined by the combination Φ + Ψ of the metric (i.e., h 00 + h ij ˆn iˆn j in our notation above). On large scales (where linear perturbation theory applies), gravitational lensing is measured through the weak statistical distortion of galaxy images. In testing gravity, our goal is to disentangle gravity from other non-standard effects which can modify the growth of structure, such as modified initial conditions, massive neutrinos, etc.. The most promising approach to disentangling these effects is to compare the two potentials Ψ and Φ + Ψ, by employing both velocity and lensing measurements. This is essentially an extension of the Solar System constraint on the PPN parameter γ, which also compares lensing (time delay and deflection) with non-relativistic dynamics (Earth s orbit). 2. Non-linear probes The bulk of cosmological information at redshifts probed by large-scale structure is in the non-linear regime (i.e., δ m is not much less than 1. In order to properly calculate predictions of interesting modified gravity models, we need to perform N-body simulations which solve the non-linear field equations together with the growth of structure. This is currently the forefront of resaearch, but several interesting results have been obtained. For example, abundance of massive galaxy clusters is very sensitive to changes in the growth factor D. Eq. (79) has been used in this way to place constraints on f(r) gravity from observations [27]. One possibility to directly test gravity is the comparison of dynamical mass estimates of objects such as clusters and galaxies with lensing mass estimates. For this, we typically average over many objects to reduce the scatter in the individual dynamics and environment of objects. We can also look for signatures of the non-linear screening mechanisms. For example, the chameleon mechanism predicts a transition in the modified gravity effects as a function of the potential well (mass) of the objects considered.

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