Reconstructing the inflationary $f(R)$ from observations

TL;DR

This paper explores how to reconstruct the form of $f(R)$ gravity models from observational data, showing that inflation is best described by a nearly $R^2$ theory with small quantum corrections.

Contribution

It provides a method to determine the $f(R)$ function from observational parameters, revealing that inflationary dynamics are close to a quadratic curvature model.

Findings

Inflationary $f(R)$ models are approximately $R^2$ with small deviations.

The form of $f(R)$ can be reconstructed from the scalar spectral index and tensor-to-scalar ratio.

Quantum corrections can account for slight deviations from the pure $R^2$ model.

Abstract

The BICEP2 collaboration has recently released data showing that the scalar-to-tensor ratio $r$ is much larger than expected. The immediate consequence, in the context of $f(R)$ gravity, is that the Starobinsky model of inflation is ruled out since it predicts a value of $r$ much smaller than what is observed. Of course, the BICEP2 data need verification, especially from Planck with which there is some tension, therefore any conclusion seems premature. However, it is interesting to ask what would be the functional form of $f(R)$ in the case when the value of $r$ is different from the one predicted by the Starobinsky model. In this paper, we show how to determine the form of $f(R)$, once the slow-roll parameters are known with some accuracy. The striking result is that, for given values of the scalar spectral index $n_{S}$ and $r$, the effective Lagrangian has the form $f(R)=R^{\zeta}$, where $\zeta=2-\varepsilon$ and $|\varepsilon|\ll 1$. Therefore, it appears that the inflationary phase of the Universe is best described by a $R^{2}$ theory, with a small deviation that, as we show, can be obtained by quantum corrections.