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 lna (a is the scale factor) that is larger than in the Lambda CDM 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 less than or similar to gamma(0) less than or similar to 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 less than or similar to gamma(0) less than or similar to 0.43. These latter values are much lower than in the Lambda 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(-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 Lambda CDM model.