Perturbations of Single-field Inflation in Modified Gravity Theory

TL;DR

This paper investigates single-field inflation within modified gravity theories, highlighting the differences in perturbations between Einstein and Jordan frames and fitting the models to Planck data.

Contribution

It demonstrates the non-equivalence of curvature perturbations in different frames and provides explicit calculations of the power spectrum in the Jordan frame.

Findings

Power spectrum and spectral index derived in Jordan frame

Models can fit Planck data very well

Differences between Einstein and Jordan frame perturbations shown

Abstract

In this paper, we study the case of single field inflation within the framework of modified gravity theory where the gravity part has an arbitrary form $f(R)$. Via a conformal transformation, this case can be transformed into its Einstein frame where it looks like a two-field inflation model. However, due to the existence of the isocurvature modes in such a multi-degree-of-freedom (m.d.o.f.) system, the (curvature) perturbations are not equivalent in two frames, so in despite of its convenience, it is illegal to treat the perturbations in its Einstein frame as the "real" ones as we always do for pure $f(R)$ theory or single field with nonminimal coupling. Here by pulling the results of curvature perturbations back into its original Jordan frame, we show explicitly the power spectrum and spectral index of the perturbations in the Jordan frame, as well as how it differs from the Einstein frame. We also fit our results with the newest Planck data. Since there are large parameter space in these models, we show that it is easy to fit the data very well.