The Large Scale Structure of f(R) Gravity

TL;DR

The paper develops linear perturbation theory for $f(R)$ gravity in the Jordan frame and shows that, for any given expansion history $H(a)$, there are two solution branches distinguished by $B\propto f_{RR}$. The $B<0$ branch suffers a high-curvature instability, while the stable $B>0$ branch predicts testable signatures: suppression of large-angle CMB ISW power, a modified linear matter power spectrum, and altered CMB–galaxy cross-correlations. These effects enable stringent cosmological tests of general relativity on large scales and motivate a parameterized post-Friedmann framework for describing deviations in the linear regime. The work clarifies the role of high-curvature stability in viable $f(R)$ models and provides concrete predictions for current and upcoming cosmological data sets.

Abstract

We study the evolution of linear cosmological perturbations in f(R) models of accelerated expansion in the physical frame where the gravitational dynamics are fourth order and the matter is minimally coupled. These models predict a rich and testable set of linear phenomena. For each expansion history, fixed empirically by cosmological distance measures, there exists two branches of f(R) solutions that are parameterized by B propto d^2 f/dR^2. For B<0, which include most of the models previously considered, there is a short-timescale instability at high curvature that spoils agreement with high redshift cosmological observables. For the stable B>0 branch, f(R) models can reduce the large-angle CMB anisotropy, alter the shape of the linear matter power spectrum, and qualitatively change the correlations between the CMB and galaxy surveys. All of these phenomena are accessible with current and future data and provide stringent tests of general relativity on cosmological scales.

Figures (6)

• Figure 1: Every expansion history that can be parameterized by a dark energy model with $\rho_{\rm DE}(\ln a)$ can be reproduced by a one parameter family of $f(R)$ models, indexed by $B_0 \propto f_{RR}/(1+f_R)$ at the present epoch (left end point of curves), that approaches the Einstein-Hilbert action in the high curvature limit. (a) $\Lambda$CDM expansion history ($w=-1$, $\Omega_{\rm DE}=0.76$, $h=0.73$). (b) Dynamical dark energy expansion history ($w=-0.9$, $\Omega_{\rm DE}=0.73$, $h=0.69$).
• Figure 2: Evolution of metric fluctuations $\Phi$ (upper panel) and $\Phi_{-}$ (lower panel) for $B_{0}=1$ and a $\Lambda$CDM expansion history. The different closure relations on super and sub-horizon scales for $\Psi$, Eqs. (\ref{['eq:phipsiclosure']}) and (\ref{['eq:Sclosure']}), lead to qualitatively different evolution for the two limits with a transition region in between. $\Phi_{-}$, which controls effects in the CMB and enters directly into the Poisson equation, has a scale-dependent growth that makes it increasingly larger than the $\Lambda$CDM prediction at high $k$. Results for other values of $B_{0}$ can be scaled from this figure by noting that the transition occurs when $k/aH \approx B^{-1/2}$.
• Figure 3: CMB quadrupole power $6 C_2/2\pi$ contributed by the modified ISW effect (dashed curve) and total (solid curve) as a function of $B_0$ in the $\Lambda$CDM expansion history. For reference, the $\Lambda$CDM total quadrupole is also shown (horizontal line). The change in the growth of the potential causes a near nulling of the ISW effect at $B_0 \approx 3/2$ and a substantial reduction of power between $0.2 \lesssim B_0 \lesssim 2.5$.
• Figure 4: CMB angular power spectra for the $\Lambda$CDM expansion history for $B_0=0$ ($\Lambda$CDM), 1/2, 3/2. Power in the low multipoles is lowered by the reduction in the ISW effect. Power at the high multipoles of the acoustic peaks is left unchanged.
• Figure 5: Linear matter power spectrum for several values of $B_0$ in the $\Lambda$CDM expansion history. The change in the amplitude of the power at high $B_0 \gtrsim 0.1$ is nearly degenerate with galaxy bias. Smaller values of $0.001 \lesssim B_0 \lesssim 0.1$ change the shape of the linear power spectrum at a potentially observable level. All spectra are normalized to the WMAP anisotropy from recombination.