Freeze-out and spectral running of primordial gravitational waves in viscous cosmology

TL;DR

This paper studies how shear viscosity in the early universe affects primordial gravitational waves, leading to modifications in their spectrum and potential observational signatures.

Contribution

It provides a general analytical framework to quantify the impact of shear viscosity on primordial gravitational wave spectra in various post-inflationary scenarios.

Findings

Shear viscosity introduces additional damping, modifying the transfer function.

A constant shear viscosity-to-Hubble ratio causes an extra red tilt in the spectrum.

Viscosity of the electron-photon-baryon plasma induces a $k$-dependent blue tilt.

Abstract

We investigate the impact of shear viscosity on the propagation of primordial gravitational waves (pGW) after inflation. Without assuming a specific inflationary scenario we focus on the evolution of pGWs after they re-enter the horizon during a cosmological epoch characterized by the presence of shear viscosity. We show that shear viscosity introduces an additional damping term in the tensor equation, modifying both the transfer function and the energy density power spectrum. For a constant shear viscosity-to-Hubble ratio the transfer function acquires an extra red tilt, while a time-dependent viscosity leads to a running spectral index $\Omega_\text{GW}\sim k^{n_\text{eff}(k)}$ controlled by the time evolution of the mean free path of the viscous fluid. Our analysis provides a general framework to analytically quantify how shear viscosity can alter the primordial gravitational wave background in standard and non-standard post-inflationary scenarios. As a case study we evaluate the effect of viscosity of the electron-photon-baryon plasma, on both the transfer function and the normalized energy density, finding a $k$-dependent blue tilt due to gravitational wave freeze-out from the viscous phase. This effect corresponds to a fractional difference of order $10^{-3}$.