Classification of $f(R)$ Theories Of Inflation And The Uniqueness of Starobinsky Model

TL;DR

This work classifies $f(R)$ inflationary theories by exploiting a mathematical analogy between slow-roll dynamics and renormalization-group flow, introducing a running parameter $\alpha$ and a beta function $\beta_{\alpha}$ to organize models by class. By requiring $\beta_{\alpha}$ to start at least quadratically in $\alpha$ and imposing observational constraints, the authors show that only two classes remain viable, and among polynomial $f(R)$ theories the Starobinsky model $f(R)=R-\frac{1}{6M^{2}}R^{2}+2\Lambda$ is the unique slow-roll realization. Other polynomial extensions fall into a class that supports constant-roll inflation, highlighting difficulties in smooth transitions between slow-roll and constant-roll. The paper also provides a precise Jordan-to-Einstein frame map and derives the general forms of $f(R)$ for the surviving classes, enabling high-precision predictions of tensor and scalar power spectra and offering a principled, RG-inspired criterion for selecting viable primordial cosmology models.

Abstract

We classify $f(R)$ theories using a mathematical analogy between slow-roll inflation and the renormalization-group flow. We derive the power spectra and spectral indices class by class and compare them with the latest data. The framework used for the classification allows us to determine the general structure of the $f(R)$ functions that belong to each class. Our main result is that only two classes survive. Moreover, we show that the Starobinsky model is the only polynomial $f(R)$ that can realize slow-roll inflation. In fact, all other polynomials belong to a special class that can only realize constant-roll inflation, at least far enough in the past. We point out some of the issues involved in considering a smooth transition between constant-roll and slow-roll inflation in this class of models. Finally, we derive the map that transforms the results from the Jordan frame to the Einstein frame.