Observational Tests of Modified Gravity

TL;DR

The paper develops a model-independent framework to test modified gravity (MG) against dark energy (DE) using large-scale structure observables. By parameterizing MG with two key functions, $\tilde{G}_{\rm eff}(k,t)$ and $\eta(k,t)$, and by relating the four perturbation variables $\phi$, $\psi$, $\delta$, and $\theta$ to observables like gravitational lensing, galaxy dynamics, clustering, and cluster abundances, it shows how combined probes can distinguish MG from DE. It demonstrates potential degeneracies when only a subset of observables is available, and argues that incorporating three or more observables breaks these degeneracies, with concrete analysis of the DGP model showing no DE analogue can reproduce all MG signatures. The work also discusses quasilinear effects and three-point statistics as additional MG signatures, emphasizing the role of cross-correlations and future surveys in delivering robust tests of gravity on cosmological scales.

Abstract

Modified gravity theories have richer observational consequences for large-scale structure than conventional dark energy models, in that different observables are not described by a single growth factor even in the linear regime. We examine the relationships between perturbations in the metric potentials, density and velocity fields, and discuss strategies for measuring them using gravitational lensing, galaxy cluster abundances, galaxy clustering/dynamics and the ISW effect. We show how a broad class of gravity theories can be tested by combining these probes. A robust way to interpret observations is by constraining two key functions: the ratio of the two metric potentials, and the ratio of the Gravitational ``constant'' in the Poisson equation to Newton's constant. We also discuss quasilinear effects that carry signatures of gravity, such as through induced three-point correlations. Clustering of dark energy can mimic features of modified gravity theories and thus confuse the search for distinct signatures of such theories. It can produce pressure perturbations and anisotropic stresses, which breaks the equality between the two metric potentials even in general relativity. With these two extra degrees of freedom, can a clustered dark energy model mimic modified gravity models in all observational tests? We show with specific examples that observational constraints on both the metric potentials and density perturbations can in principle distinguish modifications of gravity from dark energy models. We compare our result with other recent studies that have slightly different assumptions (and apparently contradictory conclusions).

Figures (3)

• Figure 1: Upper panel: We plot the normalized distance $d(z)$ (solid curves, almost coincident) and the linear growth rate $D(z)/a(z)$ (dotted curves) for $0<z<3$ for two dark energy models (in black and red) and a DGP model (in blue). The distances and growth rates are normalized to give 1 at high redshift (z=1100). Lower panel: The fractional deviations in distance (solid curves) and linear growth (dotted curves) from the fiducial $\Lambda$CDM model are shown for a dark energy model (black) and DGP (blue) (see text for details). Note that the DGP growth curve is sensitive to the parameters chosen to fit the distance redshift relation: for a distance curve that matches $\Lambda$CDM better, the growth curve would also have smaller deviation.
• Figure 2: First consistency condition for at least one DE model to mimic $\phi$, $\psi$ and $\delta$ in a flat DGP model. The dashed line given by Eqn. \ref{['eqn:eta_Geff']} represents the required condition, while the solid curve is the actual relation in flat DGP. When $a\rightarrow 0$, $\eta\rightarrow 1$ and for $a\rightarrow \infty$, $\eta\rightarrow 1/2$. The points on the curve with $a=0.5$ and $a=1$ are indicated. (For flat DGP lines with different $\Omega_m$ lie on top of each other.) The disagreement between the two curves shows that DGP is a modified gravity model that can not be mimicked by any dark energy model.
• Figure 3: The second necessary condition for at least one DE model to mimic $\phi$, $\psi$ and $\delta$ in the given DGP model. The condition $\eta^{-1}\simeq w\beta+1+3w$, given by Eqn. \ref{['eqn:eta']} implies the variable on the y-axis should be zero for all $a$. For flat DGP with $\Omega_m=0.2$, this condition is also severely violated for $a>0$.