Gravitional radiation from first-order phase transitions in the presence of a fluid

TL;DR

The paper investigates gravitational radiation from cosmological first-order phase transitions in a system where a scalar field is coupled to a relativistic fluid. Using high-resolution 3D simulations, it maps how the final GW spectrum depends on the field–fluid energy ratio β and the coupling ξ, across thick-wall and thin-wall regimes parameterized by α. It finds that while the field commonly dominates the spectrum, the fluid can dominate in certain parameter regions, and the spectrum shows multiple characteristic peaks from bubble dynamics and fluid turbulence. The work provides a robust framework for predicting Ω_GW today from such transitions and highlights the interplay between field and fluid sources in shaping the observable signal.

Abstract

First-order phase transitions are a source of stochastic gravitational radiation. Precision calculations of the gravitational waves emitted during these processes, sourced by both the degrees of freedom undergoing the transition and anisotropic stress of the coupled, ambient constituents, have reached an age of maturity. Here we present high-resolution numerical simulations of a scalar field coupled to a fluid and parameterize the final gravitational wave spectrum as a function of the ratio of the energies of the two sectors and the coupling between the two sectors for a set of models that represent different types of first-order phase transitions. In most cases, the field sector is the dominant source of gravitational radiation, but it is possible in certain scenarios for the fluid to have the most important contribution.

Figures (9)

• Figure 1: Here we plot the size of bubbles in the fully non-linear simulations (dashed line) as in Giblin:2013kea compared to the hyperbolic approximation (thin solid lines). The consistency of the hyperbolic approximation justifies the use of Eq. \ref{['rad_vs_time']}. This is a sample for $\alpha = 0.45$, $\beta=0.028$ and varying couplings $\xi$. From top to bottom the couplings are $0.1$, $0.2$, $0.4$, $0.8$, $1.6$, $3.2$, $6.4$.
• Figure 2: A time-depedent representation of our simulations. The vertical axis, $y$, is a one-dimensional slice through the box, and the horizontal axis is a space-like hyper surface, $x+ c_f t$, for some $c_f<1$ that spans the course of a simulation. The upper panel shows a scaled version of the field value, roughly $ln|\psi-\psi_{-}|$, with $\psi_{-}$ the field value in the true minimum. The lower panel shows the logarithm of the energy density of the fluid $\ln (\epsilon)$. Lighter regions are at higher field values/energy density. The scale is exaggerated to show definition. The reader should notice that the fluid is being pushed outward by the field after the initial nucleation of bubbles and continues to "slosh" around after the field has homogenized about the true minimum.
• Figure 3: Agreement of spectrum position and amplitude for $\alpha = 0.45$ for various resolutions and box sizes. All spectra have different initial conditions with $l_* = 5.5$, $\beta = 0.1$, and $\xi=3.2$. The solid red line is $N^3 = 256^3$ and $L=36 R_0$. The dot-dashed blue lines are $N^3 = 256^3$, $L=72 R_0$; and $N^3 = 128^3$, $L=32 R_0$. The dashed green lines are $N^3 = 512^3$, $L=36 R_0$; and $N^3 = 256^3$, $L=18 R_0$. Spectra are plotted during the coalescence phase when $R(t) = l_*$, and the spectra are scaled so that the solid red line has $L = 2 H_*^-1$. Disagreement begins around $500 H_*/m$. The bump at very high frequencies is a numerical artifact.
• Figure 4: Equivalence of gravitational wave spectra with the transverse-traceless projection taken at different points in the code. The dashed line (green) is a result of making no projections at all, and using the full $T_{ij}$ as the source in Eq. \ref{['evolutioneq']}. The dot-dashed (blue) line is a result of sourcing using $T_{ij}^{\rm T}$ (Eq. \ref{['pseudosrc']}) and performing no further projections. There are three solid lines, indicating different projected versions of the spectra: using the projected anisotropic stress tensor $S^{\rm TT}_{ij}$ as a source and not projecting the metric perturbations (dark red), using the full $T_{ij}$ as the source and projecting the metric perturbations $\bar{b}_{ij}$, (light red), and finally using $T_{ij}^{\rm T}$ and projecting the metric perturbations $\bar{b}_{ij}$ (light purple). The projected lines are virtually indistinguishable. This was for a simulation in which $\alpha = 0.45$, $\beta = 0.028$, $L = 36$, $N^3 = 128^3$, $\xi = 3.2$, and $l_* = 6.3$, where the spectra are evaluated at a Hubble time, $t = H_*^{-1}$.
• Figure 5: A spectrum run with parameters similar to that of Hindmarsh:2013xza, for a bubble spacing $l_* \sim 7 R_0$, coupling $\xi = \eta/m = 3.2$, $\beta = \alpha_{T}/4 = 0.028$, and $\alpha = 0.45$. We run in a box with $N^3 = 256^3$, at two scales, $L = 72 R_0$ (dot-dashed, blue) and $L = 18 R_0$ (dashed, green). The amplitudes are scaled so that the latter is half of a Hubble volume. The spectra are plotted at a time equal to the largest amplitude spectrum shown in Hindmarsh:2013xza, around $t = 1400 T_c^{-1}$ (in the units of Hindmarsh:2013xza), or approximately $18.6 R_0 ~ 2.8 l_*$.