General Expressions for $n_{\rm s}$ and $r$ for Several Minimally Coupled Single Scalar Field Models

TL;DR

This paper derives generalized formulas for the scalar spectral index and tensor-to-scalar ratio in various minimally coupled single scalar field inflation models, extending previous results to second order in slow-roll parameters.

Contribution

It provides a unified framework to calculate inflationary observables for multiple models using generalized expressions for the pump field and sound speed.

Findings

Derived second-order expressions for $n_s$ and $r$ in single scalar models.

Unified approach applicable to various minimally coupled inflation models.

Enhanced precision in inflationary parameter predictions.

Abstract

We derive a generalized expressions for the scalar power spectrum and the observational parameters of inflation (the scalar spectral index $n_{\rm s}$ and the tensor-to-scalar ratio $r$) for several single field scalar models minimally coupled with gravity. We work in the framework where the perturbed Einstein equations, at linear order, are of the same form to those obtained in the standard \textit{k}-inflation, with generalized form of the pump field $z$. By generalizing the expressions for $z$ and the speed of sound squared $c_{\rm s}^2$, using the slow-roll approximation, we calculate $n_{\rm s}$ and $r$ up to the second order in the Hubble flow parameters.