The dispersion of growth of matter perturbations in f(R) gravity

We study the growth of matter density perturbations 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=Rf_{,RR}/f_{,R} today is larger than the order of 10^{-6}, linear perturbations relevant to the matter power spectrum evolve with a growth rate s=d (ln delta_m)/d (ln a) (a is the scale factor) that is larger than in the LCDM model. We find the window in the free parameter space of our models for which spatial dispersion of the growth index gamma_0= gamma(z=0) (z is the redshift) appears in the range of values 0.40< gamma_0<0.55, as well as the region in parameter space for which there is essentially no dispersion and gamma_0 converges to values around 0.40<gamma_0<0.43. These latter values are much lower than in the LCDM 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^{-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 LCDM model.