Renormalization Group Equation for $f(R)$ gravity on hyperbolic spaces

TL;DR

This paper derives an exact renormalization group flow equation for $f(R)$ gravity on hyperbolic spaces, revealing differences from compact space cases and identifying quantum corrections near an asymptotically free fixed point.

Contribution

It provides the first exact trace evaluation for $f(R)$ gravity on hyperbolic spaces using the optimized cutoff, highlighting spectral gaps and quantum corrections.

Findings

Exact flow equation for $f(R)$ gravity on hyperbolic spaces.

Poorer convergence of polynomial solutions compared to compact spaces.

Identification of an $R^2 ext{log} R^2$ quantum correction near the fixed point.

Abstract

We derive the flow equation for the gravitational effective average action in an $f(R)$ truncation on hyperbolic spacetimes using the exponential parametrization of the metric. In contrast to previous works on compact spaces, we are able to evaluate traces exactly using the optimised cutoff. This reveals in particular that all modes can be integrated out for a finite value of the cutoff due to a gap in the spectrum of the Laplacian, leading to the effective action. Studying polynomial solutions, we find poorer convergence than has been found on compact spacetimes even though at small curvature the equations only differ in the treatment of certain modes. In the vicinity of an asymptotically free fixed point, we find the universal beta function for the $R^2$ coupling and compute the corresponding effective action which involves an $R^2 \log R^2$ quantum correction.