# Asymptotic solutions in asymptotic safety

## TL;DR

This paper develops methods to determine the asymptotic behavior of fixed point solutions in $f(R)$ quantum gravity models, revealing complex solution spaces and aiding the validation of numerical solutions.

## Contribution

It introduces techniques for analyzing asymptotic solutions in $f(R)$ approximations, mapping out fixed point solution spaces and validating numerical results.

## Key findings

- Quantum fluctuations do not decouple at large $R$
- Multiple asymptotic solutions depend on the endomorphism parameter
- The dimension of fixed point solution spaces varies with parameters

## Abstract

We explain how to find the asymptotic form of fixed point solutions in functional truncations, in particular $f(R)$ approximations. We find that quantum fluctuations do not decouple at large $R$, typically leading to elaborate asymptotic solutions containing several free parameters. By a counting argument, these can be used to map out the dimension of the fixed point solution spaces. They are also necessary to validate the numerical solution, and provide the physical part in the limit that the cutoff is removed: the fixed point equation of state. As an example we apply the techniques to a recent $f(R)$ approximation by Demmel et al, finding asymptotic matches to their numerical solution. Depending on the value of the endomorphism parameter, we find many other asymptotic solutions and fixed point solution spaces of differing dimensions, yielding several alternative scenarios for the equation of state. Asymptotic studies of other $f(R)$ approximations are needed to clarify the picture.