Non-linear evolution in $f(R)$ gravity: perturbative modelling of the Chameleon mechanism

Abstract

We investigate the non-linear evolution of matter perturbations in $f(R)$ models with the Chameleon screening mechanism. The novel feature of our investigation is an iterative solution for the non-linear equation for the scalar field $\chi = \Phi - \Psi$, where $\Phi$ and $\Psi$ are the potentials that characterise scalar perturbations of the metric. We demonstrate the scheme on spherical perturbations - smooth, compensated top-hats of varying length scales. We find that the effect of the Chameleon mechanism is seen most prominently on scales where the size of the top-hat is comparable to the Compton scale of the background. There is a density enhancement near the outer edge of the top-hat and the top-hat does not retain its shape. We explain this well-known observation in the context of the spatio-temporal evolution of the Compton scale. Additionally, we find a slight enhancement of the density near the origin, a feature not reported previously in the literature. On scales much smaller or much larger than the background Compton length, including the Chameleon screening has no appreciable effect on the perturbations. In the former, the growth is enhanced as compared to GR and is almost the same as GR in the latter. Finally, we examine the non-linear density velocity divergence (DVDR) relation and find that for evolution affected by Chameleon screening, the DVDR is no longer one-to-one even for a single profile. The relation between density and velocity depends on the location within the perturbation.