Dynamics of Linear Perturbations in f(R) Gravity

TL;DR

This work develops a comprehensive, gauge-free framework for linear perturbations in $f(R)$ gravity and maps between the Jordan and Einstein frames to study structure formation. By deriving full perturbation equations in the Jordan frame and translating to the Einstein frame, the authors reveal that late-time modifications generically suppress large-scale density fluctuations, creating tension with galaxy power spectra and CMB observations. An explicit analysis of two representative models shows that an early onset of acceleration leads to excessive large-scale suppression, effectively ruling out a wide class of $f(R)$ theories as viable alternatives to dark energy. The results underscore the importance of using full, frame-consistent perturbation theory rather than quasi-static approximations when testing modified gravity against cosmological data.

Abstract

We consider predictions for structure formation from modifications to general relativity in which the Einstein-Hilbert action is replaced by a general function of the Ricci scalar. We work without fixing a gauge, as well as in explicit popular coordinate choices, appropriate for the modification of existing cosmological code. We present the framework in a comprehensive and practical form that can be directly compared to standard perturbation analyses. By considering the full evolution equations, we resolve perceived instabilities previously suggested, and instead find a suppression of perturbations. This result presents significant challenges for agreement with current cosmological structure formation observations. The findings apply to a broad range of forms of f(R) for which the modification becomes important at low curvatures, disfavoring them in comparison with the LCDM scenario. As such, these results provide a powerful method to rule out a wide class of modified gravity models aimed at providing an alternative explanation to the dark energy problem.

Figures (2)

• Figure 1: Perturbation evolution for [left panel] the $f(R)=- \mu^{4}/R$ and [right panel] the $f(R)=- \mu_{1}H_{0}^{2}\exp(-R/\mu_{2}H_{0}^{2})$ model with $\Omega_{m}^{eff} = \rho_{m}^{0}/3H_{0}^{2} =0.3$ and $\mu^{4}$, and $\{ \mu_{1}, \mu_{2}\}$ chosen in each case to give $H_{0} = 70 kms^{-1}Mpc^{-1}$. [Top panels] The density fluctuations for the f(R) theory are compared to that for the equivalent $\Lambda$CDM scenario for two comoving scales $k=10^{-1}Mpc^{-1}$ (left) and $k=10^{-3}Mpc^{-1}$ (right). The diamond shows the analytic value of $x_{eq}$ in the limit of no suppression from $k\tau_{eq}>1$, as described in (\ref{['eq1']}). [Lower panels] The power law evolution of the density fluctuations $d\delta /d\ln a$ for the different density components. The main scaling solutions are shown by dotted lines, respectively $d\ln\delta /d\ln a=3$ and $d\ln\delta /d\ln a=2$ for the scalar field and matter components in the radiation era and $d\ln\delta /d\ln a=1.5$ and $d\ln\delta /d\ln a=0$ for the matter perturbations in the matter dominated and accelerated eras. In the figure $(E)$ and $(J)$ denote the Einstein and Jordan frame quantities respectively.
• Figure 2: The matter power spectrum for $\Lambda$CDM (full black) and [left panel] the $f(R)= -\mu^{4}/R$ model and [right panel] the $f(R)= -\mu_{1}H_{0}^{2}\exp(-R/\mu_{2}H_{0}^{2})$ model for the same normalization (red dashed) and for normalization to give small scale agreement between the two models (red dot-dashed) are shown against the SDSS matter power spectrum data Tegmark:2003uf. One can see that the f(R) model cannot simultaneously give small scale agreement with galaxy matter power spectrum and large scale agreement with the CMB.