Newton law in covariant unimodular $F(R)$ gravity

TL;DR

This paper introduces a covariant, ghost-free unimodular $F(R)$ gravity theory with a three-form field, showing that Newton's law remains unchanged from General Relativity and that cosmological solutions are equivalent in covariant and non-covariant formulations.

Contribution

It develops a covariant unimodular $F(R)$ gravity model with a three-form field and demonstrates its Newtonian limit and cosmological equivalence to non-covariant formulations.

Findings

Newton's law is unchanged from General Relativity in covariant unimodular $F(R)$ gravity.

Cosmological solutions are equivalent in covariant and non-covariant unimodular $F(R)$ theories.

The theory is ghost-free and analogous to a charged particle in a magnetic field.

Abstract

We propose a covariant ghost-free unimodular $F(R)$ gravity theory, which contains a three-form field and study its structure using the analogy of the proposed theory with a quantum system which describes a charged particle in uniform magnetic field. Newton's law in non-covariant unimodular $F(R)$ gravity as well as in unimodular Einstein gravity is derived and it is shown to be just the same as in General Relativity. The derivation of Newton's law in covariant unimodular $F(R)$ gravity shows that it is modified precisely in the same way as in the ordinary $F(R)$ theory. We also demonstrate that the cosmology of a Friedmann-Robertson-Walker background, is equivalent in the non-covariant and covariant formulations of unimodular $F(R)$ theory.