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

TL;DR

This paper classifies $f(R)$ inflationary theories using a mathematical analogy to RG flow, showing only two classes are viable and identifying the Starobinsky model as the unique polynomial $f(R)$ for slow-roll inflation.

Contribution

It introduces a classification framework for $f(R)$ theories, demonstrating the uniqueness of the Starobinsky model among polynomial $f(R)$ functions for slow-roll inflation.

Findings

Only two classes of $f(R)$ theories survive the classification.

The Starobinsky model is the only polynomial $f(R)$ capable of slow-roll inflation.

Other polynomial $f(R)$ models can only realize constant-roll inflation.

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.