Reconstructing $f(R)$ gravity from the spectral index

TL;DR

This paper reconstructs $f(R)$ gravity models from the spectral index $n_s$, showing that for typical slow-roll inflation, the models asymptote to $R^2$ gravity at large curvature.

Contribution

It provides a method to reconstruct $f(R)$ gravity from the spectral index, extending previous potential-based reconstructions to modified gravity frameworks.

Findings

$f(R)$ asymptotes to $R^2$ for large $R$.

Reconstruction applies to both Einstein and Jordan frames.

Results are valid for $n_s-1=-p/N$ with $p$ as a parameter.

Abstract

Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1\simeq -2/N$, where $N$ is the number of e-foldings. In the previous work, the reconstruction of the inflaton potential for a given $n_s$ was studied, and it was found that for $n_s-1=-2/N$, the potential takes the form of either $\alpha$-attractor model or chaotic inflation model with $\phi^2$ to the leading order in the slow-roll approximation. Here we consider the reconstruction of $f(R)$ gravity model for a given $n_s$ both in the Einstein frame and in the Jordan frame. We find that for $n_s-1=-2/N$ (or more general $n_s-1=-p/N$), $f(R)$ is given parametrically and is found to asymptote to $R^2$ for large $R$. This behavior is generic as long as the scalar potential is of slow-roll type.