Generalized virial theorem in Palatini $f(R)$ gravity

TL;DR

This paper derives a generalized virial theorem within Palatini $f(R)$ gravity, linking geometric terms to galaxy cluster mass discrepancies and providing a potential observational test for these modified gravity theories.

Contribution

It introduces a new virial theorem in Palatini $f(R)$ gravity that relates geometric contributions to galaxy cluster masses and observational parameters.

Findings

The virial mass is proportional to geometric terms in the field equations.

Derived velocity dispersion relations for galaxy clusters.

Expressed metric components and $f(R)$ Lagrangian in terms of observational data.

Abstract

We use the collision-free Boltzmann equation in Palatini $f({\mathcal{R}})$ gravity to derive the virial theorem within the context of the Palatini approach. It is shown that the virial mass is proportional to certain geometrical terms appearing in the Einstein field equations which contribute to gravitational energy and that such geometric mass can be attributed to the virial mass discrepancy in cluster of galaxies. We then derive the velocity dispersion relation for clusters followed by the metric tensor components inside the cluster as well as the $f({\mathcal{R}})$ lagrangian in terms of the observational parameters. Since these quantities may also be obtained experimentally, the $f({\mathcal{R}})$ virial theorem is a convenient tool to test the viability of $f({\mathcal{R}})$ theories in different models. Finally, we discuss the limitations of our approach in the light of the cosmological averaging used and questions that have been raised in the literature against such averaging procedures in the context of the present work.