Asymptotically Safe $f(R)$-Gravity Coupled to Matter II: Global Solutions

TL;DR

This paper investigates the existence and stability of ultraviolet fixed points in $f(R)$-gravity coupled with matter fields using the functional renormalisation group, providing numerical solutions and analyzing the impact of matter on asymptotic behavior.

Contribution

It extends previous work by finding global solutions for $f(R)$-gravity with matter, analyzing their asymptotic behavior, and assessing matter fields' effects on fixed point stability.

Findings

Global solutions exist for discrete parameter sets.

Matter fields do not necessarily destabilize the fixed point.

Different classes of fixed point functions are identified.

Abstract

Ultraviolet fixed point functions of the functional renormalisation group equation for $f(R)$-gravity coupled to matter fields are discussed. The metric is split via the exponential parameterisation into a background and a fluctuating metric, the former is chosen to be the one of a four-sphere. Also when scalar, fermion and vector fields are included global quadratic solutions exist as in the pure gravity case for discrete sets of values for some endomorphism parameters defining the coarse-graining scheme. The asymptotic, large-curvature behaviour of the fixed point functions is analysed for generic values of these parameters. Examples for global numerical solutions are provided. A special focus is given to the question whether matter fields might destabilise the ultraviolet fixed point function. Similar to a previous analysis of a polynomial, small-curvature approximation to the fixed point functions different classes for such functions are found.