Using Gravitational Wave Signals to Disentangle Early Matter Dominated Epochs

TL;DR

This work investigates the stochastic background of gravitational waves produced at second order by curvature perturbations during an early matter-dominated epoch (eMD) and how the end of that epoch encodes the underlying physics. It develops a fully time-dependent decay-rate formalism for physically motivated matter components—primordial black holes (PBHs) and four Q-ball variants (thin-wall, thick-wall, delayed)—and applies it to both monochromatic and log-normal mass distributions. A key finding is that the resonant poltergeist-type peak near k η_{eq,2} ~ 10^3 differs in amplitude and shape across PBHs and the various Q-ball scenarios, enabling discrimination in the monochromatic limit; broadened mass distributions suppress the peak in distinct ways, with broader spectra approaching the slow-transition limit. However, when realistic mass distributions and spectral tilt are included, the resonant signal is substantially suppressed, making near-term detection unlikely except in narrow parameter regions. The paper thus provides a robust framework to extract information about the duration and end of eMD epochs from future GW data, while highlighting challenges from non-linear evolution, non-Gaussianities, and damping effects that warrant future study.

Abstract

Curvature perturbations induce gravitational waves (GWs) at second order, contributing to the stochastic gravitational wave background. The resulting gravitational wave spectrum is sensitive to the evolutionary history of the universe and can be substantially enhanced by early matter-dominated (eMD) epochs, particularly if they end rapidly. Such epochs can be caused by primordial black holes (PBHs) and non-topological solitons (Q-balls), for example. Prior analysis approximated the end of the eMD epoch as instantaneous or used a Gaussian smoothing. In this work, we present a complete analysis fully incorporating their time-evolving decay rates. We demonstrate that the resulting signal spectra from PBH, thin wall Q-ball, thick wall Q-ball, and delayed Q-ball eMD epochs are distinguishable for monochromatic distributions. We then consider log-normal mass distributions and discuss the distinguishability of the various GW spectra. Importantly we find that the change in the spectrum from a finite mass width is qualitatively different from the change arising from a slower transition to radiation domination.

Figures (9)

• Figure 1: A comparison of the GW power spectrum originating from a eMD epoch dominated by PBH (solid-blue curve, with $n=1/3$), delayed Q-ball (orange-dashed curve, with $n=3/5$), thick-walled Q-ball (dotted-green curve, with $n=1$) and thin-walled Q-ball (dot-dashed-red curve, with $n=3$). For all situations considered, we have set the length of the eMD epoch to be $\eta_{\rm eq,2}/\eta_{\rm eq,1}=500$.
• Figure 2: GW power spectrum induced from a scale invariant power spectrum ($n_s=1$) during a (a) PBH epoch, (b) delayed Q-ball epoch, (c) thick-walled Q-ball epoch and (d) a thin-walled Q-ball epoch. The different curves correspond to eMD eras of different length, denoted by $\eta_{\rm eq,2}/\eta_{\rm eq,1}$.
• Figure 3: A comparison of the GW signal induced from a long eMD epoch dominated by delayed Q-balls, with $\eta_{\rm eq,2}/\eta_{\rm eq,1}=500$, to a short eMD epoch dominated by PBHs, with $\eta_{\rm eq,2}/\eta_{\rm eq,1}=75$.
• Figure 4: GW power spectrum induced from a scale invariant power spectrum ($n_s=1$) during a (a) PBH epoch, (b) delayed Q-ball epoch, (c) thick-walled Q-ball epoch and (d) a thin-walled Q-ball epoch. In all cases the length of the eMD epoch is fixed to $\eta_{eq,2}/\eta_{eq,1}=500$ by adjusting the value of $t_{\rm eva,0}$. The different curves correspond to different widths $\sigma$ for the initial mass distributions in eq. \ref{['eq:initial_distribution']}.
• Figure 5: GW power spectrum induced from a scale invariant power spectrum ($n_s=1$) during a (a) PBH epoch, (b) delayed Q-ball epoch, (c) thick-walled Q-ball epoch and (d) a thin-walled Q-ball epoch. In all cases the length of the eMD epoch is fixed to $\eta_{eq,2}/\eta_{eq,1}=100$ by adjusting the value of $t_{\rm eva,0}$. The different curves correspond to different widths $\sigma$ for the initial mass distributions in eq. \ref{['eq:initial_distribution']}.