Distinguishing Modified Gravity from Dark Energy

TL;DR

The paper addresses whether cosmic acceleration is due to dark energy or to modifications of gravity on cosmological scales. It develops a minimal, infrared-consistent framework with scale-independent gravity parameterized by $\gamma(a)$ and, for scale-dependent cases inspired by $f(R)$ theories, by $G_\Phi(k,t)$ and $\gamma(k,t)$, deriving observable predictions for structure growth, the CMB via the ISW effect, and weak lensing. It finds that scale-independent models with $\gamma(a)=1+\beta a^s$ can modify the growth and lensing without altering the transfer-function shape, but that a dark-energy fluid with entropy and shear can mimic these effects, complicating discrimination; scale-dependent models imprint a scale-dependent transfer function and growth, offering a potential route to test gravity when combined with multiple probes, though degeneracies persist without Lagrangian priors. The work emphasizes the need for joint observational constraints on $G_\Phi(k,t)$ and $\gamma(k,t)$ and suggests nonlinear simulations with variable $G_\Phi$ to exploit beyond-linear information, presenting compact parameterizations as practical tools for data-driven tests of gravity on cosmological scales.

Abstract

The acceleration of the universe can be explained either through dark energy or through the modification of gravity on large scales. In this paper we investigate modified gravity models and compare their observable predictions with dark energy models. Modifications of general relativity are expected to be scale-independent on super-horizon scales and scale-dependent on sub-horizon scales. For scale-independent modifications, utilizing the conservation of the curvature scalar and a parameterized post-Newtonian formulation of cosmological perturbations, we derive results for large scale structure growth, weak gravitational lensing, and cosmic microwave background anisotropy. For scale-dependent modifications, inspired by recent $f(R)$ theories we introduce a parameterization for the gravitational coupling $G$ and the post-Newtonian parameter $γ$. These parameterizations provide a convenient formalism for testing general relativity. However, we find that if dark energy is generalized to include both entropy and shear stress perturbations, and the dynamics of dark energy is unknown a priori, then modified gravity cannot in general be distinguished from dark energy using cosmological linear perturbations.

Figures (8)

• Figure 1: Evolution of the scalar potentials $\Phi$ and $\Psi$ in the long wavelength, scale-independent limit, assuming a $\zeta=1$ normalization. Modified gravity effects, parameterized by eq. (\ref{['gamma']}), arise later for larger $s$. The GR model assumes $\gamma=1$ and a cosmological constant. For $\gamma<1$ the Newtonian potential grows, unlike general relativity with a cosmological constant.
• Figure 2: Evolution of the logarithmic density perturbation growth rate $d\ln D/d\ln a$. Models with $\gamma<1$ have enhanced growth relative to the $\Lambda$CDM (GR) model.
• Figure 3: Contour plot of $\sigma_8(\beta,s)/\sigma_8(\Lambda {\rm CDM})$ with $\beta$ running along the horizontal axis and $s$ running along the vertical axis.
• Figure 4: Evolution of $G_{\Phi}$ and $G_{\Psi}$. The behavior is dominated by the potentials and exhibits the same features shown in Fig. \ref{['fig:F']}.
• Figure 5: The temperature anisotropy power spectrum for different parameter choices of modified gravity.