Inflationary spectral tilts as a result of the dilatation symmetry breaking

Pisin Chen,Rio Saitou,Jiro Matsumoto

doi:10.1103/physrevd.100.023534

Abstract

We derive the spectral indices and their runnings of single inflation models by a new approach. We perform a dilatation transformation to the linear cosmological perturbations and derive a current (non-)conservation law. Using it, we construct a dilatation charge and a Ward-Takahashi identity for the two-point correlators, and derive two \textit{exact} expressions for the tree-level spectral indices. First, we apply the slow roll expansion to one of the exact expressions. We calculate the spectral indices and their runnings up to the second and the third order of slow roll parameters respectively, with use of the "horizon crossing formalism". By construction, our results are more rigorous and generic than the previous works. Then, we analyze another exact expression to understand how the perturbations and the slow roll parameters contribute to the spectral indices. By a numerical calculation, we confirm that only the behaviors of the slow roll parameters during a few e-folds around the horizon crossing affect significantly to the values of spectral indices. The analysis in this article indicates that if one cannot use the slow roll parameters, regardless of their values, as the expansion parameters around the horizon crossing, then one can no longer apply the slow roll expansion to the spectral indices, and it is thus necessary to apply the more generic method introduced here.

Highlights

Inflation is the phase of accelerated expansion in the early universe [1–3]
The observation of cosmic microwave background (CMB) anisotropy [4,5] indicates that the temperature fluctuations of CMB are essentially Gaussian, which is in concordance with the quantum fluctuations predicted by the inflationary theory [6–9]
We stress that we evaluate the asymptotic values of spectral indices, say, the values at the end of inflation, by using the slow roll parameters evaluated at the horizon crossing time

Summary

INTRODUCTION

Inflation is the phase of accelerated expansion in the early universe [1–3]. It can resolve the three big problems of the big bang universe and can create the seed of the large scale structure by the quantum fluctuation around the quasi–de Sitter background. In [25,26], the authors performed an uncertain Taylor expansion for the power spectra, where the convergence of the expansion is not guaranteed Their predictions rely on the slow roll expansion, so that we cannot apply them to a class of models where the slow roll parameters are not necessarily small throughout the inflation. If we perform a dilatation transformation to the two-point correlators, we should naturally obtain exact expressions of the spectral indices According to this idea, we define the dilatation transformation to the linear cosmological perturbation theory and construct the dilatation charge. By using the Ward-Takahashi (WT) identity involved with the charge, we derive two expressions of the spectral indices for the scalar and the tensor perturbations without any ansatzes nor approximations By construction, both expressions are exact and applicable to almost all wavelengths and all times for generic models. We use the unit 8πG 1⁄4 c 1⁄4 ħ 1⁄4 1 throughout this article

WARD-TAKAHASHI IDENTITY

SLOW ROLL EXPANSION

NUMERICAL ANALYSIS

Equations

Slow roll expansion vs the large NÃ expansion

Analysis of the other expression for the spectral index

SUMMARY

Power-law inflation

Ultra slow roll inflation