Distinguishing modified gravity from dark energy

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 superhorizon scales and scale dependent on subhorizon scales. For scale-independent modifications, utilizing the conservation of the curvature scalar and a parametrized 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 parametrization for the gravitational coupling $G$ and the post-Newtonian parameter $\gamma$. These parametrizations 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.

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, parametrized by Eq. (5), 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 $\Lambda \mathrm{CDM}$ (GR) model.

Figure 3

Contour plot of ${\sigma }_8(\beta ,s)/{\sigma }_8(\Lambda \mathrm{CDM})$ 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. 1.

Figure 5

The temperature anisotropy power spectrum for different parameter choices of modified gravity.

Figure 6

Ratios of weak lensing shear correlation $C_{+}$, to its value for GR, for several sets of modified gravity parameters.

Figure 7

The ratio of the density transfer function to the predicted GR case, plotted at $a=1$ versus dimensionless wave number $k\sqrt{{\alpha }_1}$ for two values of ${\alpha }_1/{\alpha }_2$ and two values of $s$. For $a=0.5$ the transfer functions are almost identical, except that $|D/D(GR)-1|$ is reduced by approximately 13% for ${\alpha }_1/{\alpha }_2=0.5$ and 30% for ${\alpha }_1/{\alpha }_2=2$.

Figure 8

Scale dependence of the logarithmic density perturbation growth rate $\partial \ln D/\partial \ln a$ at $a=1$ for scale-dependent modified gravity models.