Large scale structure of $f(R)$ gravity

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 parametrized by $B\propto d^{2}f/d{R}^2$. 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.

Figure 1

Every expansion history that can be parametrized by a dark energy model with ${\rho }_{\mathrm{DE}}(\ln a)$ 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) $\Lambda \mathrm{CDM}$ expansion history ($w=-1$, ${\Omega }_{\mathrm{DE}}=0.76$, $h=0.73$). (b) Dynamical dark energy expansion history ($w=-0.9$, ${\Omega }_{\mathrm{DE}}=0.73$, $h=0.69$).

Figure 2

Evolution of metric fluctuations $\Phi$ (upper panel) and ${\Phi }_{-}$ (lower panel) for $B_0=1$ and a $\Lambda \mathrm{CDM}$ expansion history. The different closure relations on super and subhorizon scales for $\Psi$, Eqs. (42, 46), 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 $\Lambda \mathrm{CDM}$ 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 $6C_2/2\pi$ contributed by the modified ISW effect (dashed curve) and total (solid curve) as a function of $B_0$ in the $\Lambda \mathrm{CDM}$ expansion history. For reference, the $\Lambda \mathrm{CDM}$ 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 $\Lambda \mathrm{CDM}$ expansion history for $B_0=0$ ($\Lambda \mathrm{CDM}$), $1/2$, $3/2$. 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 $\Lambda \mathrm{CDM}$ 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.

Figure 6

Cross correlation coefficient between the CMB and galaxies in the $\Lambda CDM$ expansion history. Shown are two representative redshift bins centered around $\overline{z}=0.2$ and 0.6 with $B_0=0$ ($\Lambda \mathrm{CDM}$), $1/2$, 1, $3/2$. The cross correlation is substantially reduced for $1/2\lesssim B_0\lesssim 1$ and becomes negative for $B_0\gtrsim 3/2$.