On the Renormalization Group perspective of $\alpha$-attractors

TL;DR

This paper explores the connection between inflationary potentials and $F(R)$ gravity, revealing that for various models, especially $oldsymbol{ extit{ extalpha}}$-attractors, the reconstructed $F(R)$ behaves as $oldsymbol{R^2}$ at large curvature, with $oldsymbol{ extalpha o0}$ being an ultraviolet fixed point.

Contribution

It introduces a method to reconstruct $F(R)$ gravity from inflationary potentials and analyzes $ extalpha$-attractors using renormalization group techniques, highlighting the $R^2$ behavior and fixed point structure.

Findings

Reconstructed $F(R)$ for simple potentials shows $F(R) o R^2$ at large $R$.

For $ extalpha$-attractors, $F(R) o R^2$ for all $R$ as $ extalpha o 0$.

$ extalpha o 0$ is identified as an ultraviolet stable fixed point.

Abstract

In this short paper we outline a recipe for the reconstruction of $F(R)$ gravity starting from single field inflationary potentials in the Einstein frame. For simple potentials one can compute the explicit form of $F(R)$, whilst for more involved examples one gets a parametric form of $F(R)$. The $F(R)$ reconstruction algorithm is used to study various examples: power-law $\phi^n$, exponential and $\alpha$-attractors. In each case it is seen that for large $R$ (corresponding to large value of inflaton field), $F(R) \sim R^2$. For the case of $\alpha$-attractors $F(R) \sim R^2$ for all values of inflaton field (for all values of $R$) as $\alpha\to0$. For generic inflaton potential $V(\phi)$, it is seen that if $V^\prime/V \to0$ (for some $\phi$) then the corresponding $F(R) \sim R^2$. We then study $\alpha$-attractors in more detail using non-perturbative renormalisation group methods to analyse the reconstructed $F(R)$. It is seen that $\alpha\to0$ is an ultraviolet stable fixed point of the renormalisation group trajectories.