Dynamics and collapse of collisionless self-gravitating systems is described by the coupled collisionless Boltzmann and Poisson equations derived from $f(R)$ gravity in the weak field approximation. Specifically, we describe a system at equilibrium by a time-independent distribution function $f_0(x,v)$ and two potentials ${\Phi }_0(x)$ and ${\Psi }_0(x)$ solutions of the modified Poisson and collisionless Boltzmann equations. Considering a small perturbation from the equilibrium and linearizing the field equations, it can be obtained a dispersion relation. A dispersion equation is achieved for neutral dust-particle systems where a generalized Jeans wave number is obtained. This analysis gives rise to unstable modes not present in the standard Jeans analysis (derived assuming Newtonian gravity as weak filed limit of $f(R)=R$). In this perspective, we discuss several self-gravitating astrophysical systems whose dynamics could be fully addressed in the framework of $f(R)$ gravity.

• Received 4 December 2011

DOI:https://doi.org/10.1103/PhysRevD.85.044022