Loop contributions to the scalar power spectrum due to quartic order action in ultra slow roll inflation

Abstract

The investigation of the theory of inflation beyond the linearorder in perturbations is important both for theoretical consistency andpotential observables.In the contemporary literature, the calculation of modifications to theinflationary scalar power spectrum due to the loops from the higher orderinteraction terms in the Hamiltonian have led to an interesting discussionregarding the validity of perturbation theory and the robustness of itspredictions.Recently, there have been many efforts to examine the contributions to thescalar power spectrum due to the loops arising from the cubic order termsin the action describing the perturbations, specifically in inflationaryscenarios that permit an epoch of ultra slow roll (USR).A brief phase of USR during inflation is known to lead to interesting featuresin the scalar power spectrum which in turn has significant observationalconsequences, such as the copious production of primordial black holes.In this work, we consider the loop contributions to the scalar power spectrumin a scenario of USR inflation arising due to the quartic order termsin the action describing the scalar perturbations.We compute the loop contributions to the scalar power spectrum due to thedominant term in the action at the quartic order in a scenario wherein ashort phase of USR is sandwiched between two stages of slow roll (SR) inflation.We analyze the behaviour of the loop contributions in terms of the parametersthat characterize the non-trivial inflationary dynamics, viz. the onset andduration of USR, and the smoothness of transitions between the USR and SR phases.We examine three different cases of the scenario — the late, intermediate andearly epochs of USR during inflation, each of which affects the scalar powerspectrum over different ranges of wave numbers.In the inflationary scenario involving a late phase of USR, for reasonable choicesof the parameters, we show that the loop corrections are negligible for the entirerange of wave numbers.In the intermediate case, the contributions from the loops prove to be scaleinvariant over large scales and, we find that these contributions can amountto 30% of the leading order (i.e. the Gaussian) power spectrum.In the case wherein USR sets in early, we find that the loop contributions couldbe negative and can dominate the power spectrum at the leading order, whichindicates a breakdown of the validity of the perturbative expansion.We discuss the origin of the negative sign and the divergences that arise in theloop contributions to the power spectrum.We conclude with a brief summary and outlook.