Frame-Covariant Formulation of Inflation in Scalar-Curvature Theories

TL;DR

This paper introduces a frame-covariant approach to inflation in scalar-curvature theories, providing generalized slow-roll parameters and demonstrating frame invariance of cosmological observables, applicable to various inflation models.

Contribution

It develops a new frame-covariant formalism for inflation that generalizes slow-roll parameters and ensures frame-invariant cosmological predictions in scalar-curvature theories.

Findings

Frame-invariant expressions for power spectrum and spectral indices.

Application to induced gravity, Higgs inflation, and $F(R)$ models.

Consistent lowest-order results with existing literature.

Abstract

We develop a frame-covariant formulation of inflation in the slow-roll approximation by generalizing the inflationary attractor solution for scalar-curvature theories. Our formulation gives rise to new generalized forms for the potential slow-roll parameters, which enable us to examine the effect of conformal transformations and inflaton reparameterizations in scalar-curvature theories. We find that cosmological observables, such as the power spectrum, the spectral indices and their runnings, can be expressed in a concise manner in terms of the generalized potential slow-roll parameters which depend on the scalar-curvature coupling function, the inflaton wavefunction, and the inflaton potential. We show how the cosmological observables of inflation are frame-invariant in this generalized potential slow-roll formalism, as long as the end-of-inflation condition is appropriately extended to become frame-invariant as well. We then apply our formalism to specific scenarios, such as the induced gravity inflation, Higgs inflation and $F(R)$ models of inflation, and obtain more accurate results, without making additional approximations to the potential. Our results are shown to be consistent to lowest order with those presented in the literature. Finally, we outline how our frame-covariant formalism can be naturally extended beyond the tree-level approximation, within the framework of the Vilkovisky--DeWitt effective action.