Primordial Spectral Indices from Generalized Einstein Theories

TL;DR

This work examines three closely related generalized Einstein theories in which a scalar field is non-minimally coupled to gravity, and computes the primordial spectral index n_s to second order in slow-roll parameters using the potential slow-roll (PSRA) framework. By performing conformal transformations to the Einstein frame and carefully evaluating horizon-crossing conditions, the study derives ε, η, and ζ and analyzes n_s for different initial-condition regimes (new inflation vs. chaotic inflation). The results show n_s remains near unity across models, with small, ξ-dependent deviations; frame ambiguities under new inflation are acknowledged but become numerically negligible under sufficient inflation, and chaotic-inflation cases exhibit robust frame-consistent predictions. Collectively, these GETs avoid the omega problems of extended inflation and illustrate how Planck-scale physics could imprint on observable density perturbations, while offering avenues for extending the analysis to more complex scalar-tensor actions.

Abstract

Primordial spectral indices are calculated to second order in slow-roll parameters for three closely-related models of inflation, all of which contain a scalar field non-minimally coupled to the Ricci curvature scalar. In most cases, $n_s$ may be written as a function of the non-minimal curvature coupling strength $ξ$ alone, with $n_s (ξ) \leq 1$, although the constraints on $ξ$ differ greatly between \lq new inflation' and \lq chaotic inflation' initial conditions. Under \lq new inflation' initial conditions, there are discrepancies between the values of $n_s$ as calculated in the Einstein frame and the Jordan frame. The sources for these discrepancies are addressed, and shown to have negligible effects on the numerical predictions for $n_s$. No such discrepancies affect the calculations under \lq chaotic inflation' initial conditions.