Gravitational radiation from a bulk flow model

TL;DR

This work introduces a computationally economical bulk-flow model to emulate the hydrodynamic evolution of a relativistic fluid during cosmological first-order phase transitions, focusing on the resulting gravitational-wave spectrum from colliding fluid shells. By combining energy-conserving shell dynamics with an analytic treatment of the azimuthal integration and a geometric data structure to track first collisions, the authors generate large ensembles of nucleation histories and parameterize the GW spectrum in terms of wall velocity $v_b$, transition duration $eta$, and latent heat $rac{ ho_{ m vac}}{ ho_{ m rad}}=rac{ ext{alpha}}{1+ ext{alpha}}$. They provide high-precision spectral fits, compare with envelope-approximation results, and explore implications relative to full hydrodynamic simulations, highlighting both qualitative agreements and key differences in spectral tails. The approach yields a flexible, scalable framework enabling broad parametric studies of gravitational waves from early-Universe phase transitions, with practical relevance for interpreting experiments like LISA.

Abstract

We perform simulations in a simple model that aims to mimic the hydrodynamic evolution of a relativistic fluid during a cosmological first-order phase transitions. The observable we are concerned with is hereby the spectrum of gravitational radiation produced by colliding fluid shells. We present simple parameterizations of our results as functions of the wall velocity, the duration of the phase transition and the latent heat. We also improve on previous results in the envelope approximation and compare with hydrodynamic simulations.

Figures (7)

• Figure 1: The plots show two sketches of the envelope (left) and bulk flow (right) approximations.
• Figure 2: The plot shows the histogram of the distribution of the bubble count for 1024 nucleation histories in a box of size $(20/\beta)^3$.
• Figure 3: A sketch of the geometry of the problem. The bubble under consideration is positioned at $x_a$ and surrounded by two bubbles at $x_b$ and $x_c$. The four points $A$, $B$, $D$ and $E$ depend on the time $\Delta t_a$ and determine which part of the shell is in the surrounding bubbles according to (\ref{['eq:def_constraint']}). The two points $A$ and $D$ rely on $\delta^b_\mu = x^\mu_b - x^\mu_a$ while $B$ and $E$ are found using $\delta^c_\mu = x^\mu_c - x^\mu_a$. The point $C$ separates the surface of the bubble at $x_a$ into two regions according to (\ref{['eq:earlier_constr']}) depending on which bubble the surface element collided first with. For example, points in the segment between $C$ and $B$ are inside both surrounding bubbles but collided with the bubble at $x_b$ before they collided with the bubble at $x_c$.
• Figure 4: The plot shows an example for the data structure that stores which is the first bubble a specific surface element collides with. Different colors correspond to different neighboring bubbles.
• Figure 5: The plot shows the energy density on the surface of the biggest bubble in a simulation for three different times ($5/\beta, 6.4/\beta, 8.2/\beta$). The energy density increases from dark to light regions. The energy density is normalized to the maximal energy density possible at the specific time, i.e. black regions denote no energy density and white regions denote uncollided regions. The plots show the envelope (left) and bulk flow (right) approximations.