One-loop divergences for $f(R)$ gravity

TL;DR

This paper computes the one-loop divergences in $f(R)$ gravity on arbitrary backgrounds, extending previous results limited to constant curvature spaces, and discusses implications for quantum equivalence and cosmology.

Contribution

It provides a generalized calculation of quantum corrections in $f(R)$ gravity on arbitrary backgrounds, including new technical insights into degenerate operators.

Findings

Derived the divergent part of the one-loop effective action for $f(R)$ gravity on arbitrary backgrounds.

Extended previous results from constant curvature spaces to more general backgrounds.

Discussed applications in cosmology and the quantum equivalence between $f(R)$ and scalar-tensor theories.

Abstract

We calculate the divergent part of the one-loop effective action for $f(R)$ gravity on an arbitrary background manifold. Our result generalizes previous results for quantum corrections in $f(R)$ gravity, which have been limited to spaces of constant curvature. We discuss a new technical aspect connected to operators with degenerate principal symbol. Our result has important applications in cosmology and allows to study the quantum equivalence between $f(R)$ theories and scalar-tensor theories.