Dispersion of growth of matter perturbations inf(R)gravity

Abstract

We study the growth of matter density perturbations ${\ensuremath{\delta}}_{m}$ for a number of viable $f(R)$ gravity models that satisfy both cosmological and local gravity constraints, where the Lagrangian density $f$ is a function of the Ricci scalar $R$. If the parameter $m\ensuremath{\equiv}R{f}_{,RR}/{f}_{,R}$ today is larger than the order of ${10}^{\ensuremath{-}6}$, linear perturbations relevant to the matter power spectrum evolve with a growth rate $s\ensuremath{\equiv}d\mathrm{ln}{\ensuremath{\delta}}_{m}/d\mathrm{ln}a$ ($a$ is the scale factor) that is larger than in the $\ensuremath{\Lambda}\mathrm{CDM}$ model. We find the window in the free parameter space of our models for which spatial dispersion of the growth index ${\ensuremath{\gamma}}_{0}\ensuremath{\equiv}\ensuremath{\gamma}(z=0)$ ($z$ is the redshift) appears in the range of values $0.40\ensuremath{\lesssim}{\ensuremath{\gamma}}_{0}\ensuremath{\lesssim}0.55$, as well as the region in parameter space for which there is essentially no dispersion and ${\ensuremath{\gamma}}_{0}$ converges to values around $0.40\ensuremath{\lesssim}{\ensuremath{\gamma}}_{0}\ensuremath{\lesssim}0.43$. These latter values are much lower than in the $\ensuremath{\Lambda}\mathrm{CDM}$ model. We show that these unusual dispersed or converged spectra are present in most of the viable $f(R)$ models with $m(z=0)$ larger than the order of ${10}^{\ensuremath{-}6}$. These properties will be essential in the quest for $f(R)$ modified gravity models using future high-precision observations and they confirm the possibility to distinguish clearly most of these models from the $\ensuremath{\Lambda}\mathrm{CDM}$ model.