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

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 ${\varphi }^n$, 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 $\alpha \to 0$. For generic inflaton potential $V(\varphi )$, it is seen that if $V^{\prime }/V\to 0$ (for some $\varphi$) 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 $\alpha \to 0$ is an ultraviolet stable fixed point of the renormalisation group trajectories.

arXiv comment1+17 pages, 2 figures. Typos corrected, new references added, additional paragraph added in conclusion. Accepted for publication in JCAP

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