$R^2$ Inflation Revisited and Dark Energy Corrections

TL;DR

This paper revisits the $R^2$ inflation model, revealing it is a deformed version with extra terms, and explores its late-time dark energy implications, showing it can match observational constraints and differ from standard $ ext{ extLambda}$CDM.

Contribution

It demonstrates that the actual inflationary $f(R)$ gravity is a deformed $R^2$ model with additional terms and proposes a dark energy correction compatible with observations.

Findings

The deformed $R^2$ model closely matches pure $R^2$ inflation predictions.

Adding a dark energy term allows late-time acceleration consistent with Planck data.

The model differs from $ ext{ extLambda}$CDM but exhibits similar late-time behavior.

Abstract

The vacuum $R^2$ model is known to generate a quasi-de Sitter evolution for inflation, using solely the slow-roll assumptions. Using standard reconstruction techniques, we demonstrate that the $f(R)$ gravity which actually realizes the quasi-de Sitter evolution is not simply the $R^2$ model but a deformed $R^2$ model which contains extra terms in addition to the $R^2$ model. We analyze in detail the inflationary dynamics of the deformed $R^2$ model and we demonstrate that the predictions are quite close to the ones of the pure $R^2$ model, regardless the values of the free parameters. Basically the deformed $R^2$ model is also a single parameter inflationary model, exactly like the ordinary $R^2$ model. In contrast to the early-time era, where the deformed $R^2$ model is quite similar to the $R^2$ model, at late times, the phenomenological picture is different. The deformed $R^2$ model describes stronger gravity compared to the ordinary $R^2$ with an effectively smaller effective Planck mass. We propose the addition of an early dark energy term which does not affect at all the inflationary era, but takes over the control of the late-time dynamics at late times. We study in some detail the predicted dark energy era evolution, and we demonstrate that the dark energy corrected deformed $R^2$ model can describe a viable dark energy era, compatible with the Planck constraints on cosmological parameters. Furthermore the model is distinct from the $\Lambda$-Cold-Dark-Matter model, but shows a qualitatively similar behavior.