Running of the Spectral Index and Inflationary Dynamics of $F(R)$ Gravity

TL;DR

This paper provides a model-independent analysis of the spectral index running in vacuum $F(R)$ gravity, classifies models based on their predictions, and compares them with inflationary scalar field models, finding certain universal behaviors.

Contribution

It introduces a general $n_s-r$ relation for vacuum $F(R)$ gravity and classifies models based on their spectral index running predictions, including $R^2$-attractor models.

Findings

$R^2$-attractor models predict $a_s$ in $-10^{-3}$ to $-10^{-4}$ range.

Some $F(R)$ models predict $a_s$ outside the $-10^{-3}$ to $-10^{-4}$ range, sometimes positive.

No models found with $r<10^{-4}$ and $a_s<-10^{-3}$.

Abstract

In this work we shall provide a model-independent general calculation of the running of the spectral index for vacuum $F(R)$ gravities. We shall exploit the functional form of the spectral index and of the tensor-to-scalar ratio in order to present a general $n_s-r$ relation for vacuum $F(R)$ gravity theories. As we show, viable $F(R)$ gravity theories can be classified to two classes of models regarding their prediction for the running spectral index. The $R^2$-attractor models predict a running of the spectral index in the range $-10^{-3}<a_s<-10^{-4}$, which classifies them in the same universality class that most inflationary scalar field models belong to. We provide three models of this sort, for which we verify our claims in detail. However there exist viable $F(R)$ gravity models with running of spectral index outside the range $-10^{-3}<a_s<-10^{-4}$ and in some cases it can be positive. We also present an $R^2$-corrected scalar field model, which also predicts a running of the spectral index in the range $-10^{-3}<a_s<-10^{-4}$. For all the cases we studied, we found no evidence for the most phenomenologically interesting scenario of having $r<10^{-4}$ and a running $a_s<-10^{-3}$, which in principle could be realized.