Vacuum and Gravitons of Relic Gravitational Waves, and Regularization of Spectrum and Energy-Momentum Tensor

TL;DR

This paper investigates the high-frequency divergences in relic gravitational wave spectra, applying adiabatic regularization to produce finite, physically meaningful spectra and energy-momentum tensors across cosmic evolution stages.

Contribution

It systematically applies adiabatic regularization at multiple stages to remove divergences in RGW spectra and energy tensors, ensuring consistency with inflationary predictions.

Findings

Vacuum dominates at frequencies above 10^{11}Hz.

Gravitons are mainly produced during inflation-reheating transition.

Regularization schemes yield similar spectra profiles and preserve inflationary predictions.

Abstract

The spectrum of relic gravitational wave (RGW) contains high-frequency divergences, which should be removed. We present a systematic study of the issue, based on the exact RGW solution that covers the five stages, from inflation to the acceleration, each being a power law expansion. We show that the present RGW consists of vacuum dominating at $f>10^{11}$Hz and graviton dominating at $f<10^{11}$Hz, respectively. The gravitons are produced by the four cosmic transitions, mostly by the inflation-reheating one. We perform adiabatic regularization to remove vacuum divergences in three schemes: at present, at the end of inflation, and at horizon-exit, to the 2-nd adiabatic order for the spectrum, and the 4-th order for energy density and pressure. In the first scheme a cutoff is needed to remove graviton divergences. We find that all three schemes yield the spectra of a similar profile, and the primordial spectrum defined far outside horizon during inflation is practically unaffected. We also regularize the gauge-invariant perturbed inflaton and the scalar curvature perturbation by the last two schemes, and find that the scalar spectra, the tensor-to-scalar ratio, and the consistency relation remain unchanged.