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

TL;DR

This work studies how shear viscosity in the post-inflationary cosmological medium damps primordial gravitational waves (pGWs) by introducing a friction term $\delta(τ)$ in the tensor evolution equation $h_k'' + 2\mathcal{H}(1+δ) h_k' + k^2 h_k = 0$, with $δ(τ) = H_V(τ)/H(τ)$ and $H_V = λ_p^2 η$. An exact solution is obtained for constant $δ$, and a controlled perturbative expansion is developed for evolving $δ$, revealing a viscous freeze-out imprint on sub-horizon modes and, for time-dependent $δ$, a running spectral index $n_{\mathrm{eff}}(k)$ governed by the mean free path evolution. Applying the framework to the tightly coupled photon–baryon–electron plasma shows a small, scale-dependent blue tilt and a fractional suppression of the gravitational-wave energy density, $Ω_{\rm GW}$, at the level of ~10^{-3} in the CMB–LSS window, while the standard pGW signal remains inaccessible to near-future detectors under ΛCDM. The analytical approach provides a general link between microphysical viscosity and the spectral shape of the stochastic GW background, and is readily extendable to non-standard dissipative scenarios such as warm inflation or hidden sectors, potentially yielding observable signatures for future space-based detectors and pulsar timing arrays.

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 $Ω_\text{GW}\sim k^{n_\text{eff}(k)}$ 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 $10^{-3}$.

Figures (4)

• Figure 1: Relative difference of the analytic and numeric transfer function and $\Omega_{\text{GW}}$ in a viscous radiation epoch evaluated at $\tau=\tau_{\text{max}}$, for $\alpha=1$, $\delta_{\text{max}}=10^{-2}$ and $\lambda_{\text{mfp}}=0$. In gray the relative difference, in orange the $k$ average.
• Figure 2: Relative difference of the analytic and numeric transfer function and $\Omega_{\text{GW}}$ in a viscous matter epoch evaluated at $\tau=\tau_{\text{max}}$, for $\alpha=1$ and $\delta_{\text{max}}=10^{-2}$. In gray the relative difference, in orange the $k$ average.
• Figure 3: Evolution of the comoving viscous cutoff scale $k_{\rm vis}(\tau)$ (thick red curve). Modes below the curve ($k < k_{\rm vis}$) propagate within the hydrodynamic, shear-viscous regime of the photon-baryon-electron plasma and undergo dissipative damping, represented illustratively by the oscillatory wiggly mode. At the viscous freeze-out time $\tau_{\rm exit}(k)$ (black point), the mode exits the viscous regime and subsequently propagates in a non-viscous background. Horizontal and vertical markers correspond respectively to characteristic scales $k \simeq \tau_{\rm eq}^{-1}, \tau_{\rm rec}^{-1}$ and to matter-radiation equality ($\tau_{\text{eq}}$) and recombination ($\tau_{\text{rec}}$). The colored regions classify modes according to their re-entry time into the Hubble horizon and viscous freeze-out time.
• Figure 4: In light gray, the relative difference between the first order analytic approximation viscous $\Omega_{\text{GW}}$ and the analytic non-viscous case. In orange the expressions Eqs. \ref{['visoutOrad']}, \ref{['visoutOmat']} and \ref{['matdamp2']}. The $\sim k^2$ for $k\to0$ is added to account for the viscous super-horizon suppression.