Disentangling the $f(R)$ - Duality

TL;DR

This paper investigates how exponential plateau-like potentials in inflation models relate to their $f(R)$-theory duals, revealing that higher order terms generally cannot be neglected and affecting CMB predictions.

Contribution

It provides a detailed analysis of the $f(R)$-formulation of inflationary potentials with broken shift symmetry, highlighting the limitations of simple $R^2$ models and exploring connections to UV theories.

Findings

$f(R)$-descriptions with $R^n$ terms where $1<n<2$ are necessary for broken shift symmetry.

Higher order $f(R)$ terms cannot be neglected and influence inflationary dynamics.

Models show power suppression at low multipoles in CMB spectra.

Abstract

Motivated by UV realisations of Starobinsky-like inflation models, we study generic exponential plateau-like potentials to understand whether an exact $f(R)$-formulation may still be obtained when the asymptotic shift-symmetry of the potential is broken for larger field values. Potentials which break the shift symmetry with rising exponentials at large field values only allow for corresponding $f(R)$-descriptions with a leading order term $R^{n}$ with $1<n<2$, regardless of whether the duality is exact or approximate. The $R^2$-term survives as part of a series expansion of the function $f(R)$ and thus cannot maintain a plateau for all field values. We further find a lean and instructive way to obtain a function $f(R)$ describing $m^2\phi^2$-inflation which breaks the shift symmetry with a monomial, and corresponds to effectively logarithmic corrections to an $R+R^2$ model. These examples emphasise that higher order terms in $f(R)$-theory may not be neglected if they are present at all. Additionally, we relate the function $f(R)$ corresponding to chaotic inflation to a more general Jordan frame set-up. In addition, we consider $f(R)$-duals of two given UV examples, both from supergravity and string theory. Finally, we outline the CMB phenomenology of these models which show effects of power suppression at low-$\ell$.