General Properties of the Gravitational Wave Spectrum from Phase Transitions

TL;DR

This paper clarifies how the gravitational-wave spectrum from post-inflationary, short-lived cosmological events is determined by the temporal and spatial structure of the underlying anisotropic stress. By deriving the general relation between the GW energy density and the diagonal of the stress spectrum, and by studying several unequal-time correlator models (incoherent, coherent, top-hat, stationary), it identifies how peak frequency and high-frequency decay depend on the source’s duration, correlation length, and time differentiability. In the specific context of bubble collisions, the authors show that previous separable models misplace the peak; a time-dependent correlation length and finite shell thickness align analytic predictions with numerical simulations, yielding a peak at $k\sim\beta$ and a mild $1/k$ tail. These insights have direct implications for forecasting GW signals from electroweak-scale phase transitions and for interpreting potential detections by future observatories.

Abstract

In this paper we discuss some general aspects of the gravitational wave background arising from post-inflationary short-lasting cosmological events such as phase transitions. We concentrate on the physics which determines the shape and the peak frequency of the gravitational wave spectrum. We then apply our general findings to the case of bubble collisions during a first order phase transition and compare different results in the recent literature.

Figures (8)

• Figure 1: The function $\beta^2k^3{\rm Re}[P_s(k,k,k)]$ for the incoherent case, as a function of $k/\beta$. The three curves correspond to $g(t)$ given by (\ref{['gconst']}), (\ref{['gdiff1']}) and (\ref{['gdiff2']}), and the velocities are $v=1.0$ (left curves) and $v=0.01$ (right curves).
• Figure 2: The qualitative behavior of the function $P_s(k, \omega, \omega)$ is shown for the totally coherent case. The diagonal, $P_s(k,k,k)$ is also plotted. In the region $\omega<\beta$ and $k<\beta/v$ we expect a white noise spectrum of the anisotropic stress. For $\omega>\beta$ and $k>\beta/v$ the spectrum is expected to decrease. Since the gravity wave spectrum only probes the diagonal $\omega=k$, we expect, in the separable case with constant $R^{-1}=\beta/v$ a first change of slope at $\omega=k=\beta$ and a second at $\omega=k=\beta/v$. Whether the first or the second is the peak frequency depends on the space and time continuity and differentiability properties of $P_s(k,\omega,\omega)$.
• Figure 3: The function $\beta^2k^3{\rm Re}[P_s(k,k,k)]$ for the coherent case, as a function of $k/\beta$: top panel, $v=1$, bottom panel, $v=0.01$. The three curves correspond to $g(t)$ given by (\ref{['gconst']}), (\ref{['gdiff1']}) and (\ref{['gdiff2']}). Notice the different peak positions for $g_1$ with respect to the other two.
• Figure 4: The function $\beta^2k^3{\rm Re}[P_s(k,k,k)]$ for the top hat case (with $x_c=1$), as a function of $k/\beta$. The three curves correspond to $g(t)$ given by (\ref{['gconst']}), (\ref{['gdiff1']}) and (\ref{['gdiff2']}) and the velocities are $v=1.0$ (left curves) and $v=0.01$ (right curves).
• Figure 5: Typical time evolution of the correlation length (corresponding to the characteristic scale of the colliding region) and therefore of the source (anisotropic stress) generating the gravitational waves calculated in numerical simulations of bubble collisions Huber:2008hg.