Numerical simulations of density perturbation and gravitational wave production from cosmological first-order phase transition

TL;DR

Cosmological first-order phase transitions (FOPTs) can imprint large density perturbations on small scales and produce a stochastic gravitational wave background, potentially seeding primordial black holes (PBHs). The authors perform 3D lattice simulations of bubble nucleation, expansion, and collisions to quantify density perturbations and GW spectra as functions of $\alpha$ and $\beta/H$. They identify two principal sources of density perturbations: delayed false-vacuum decay (dominant for $\alpha<1$) and forward bubble-wall motion (dominant for $\alpha>1$), with density spectra $\mathcal{P}_\delta(k)$ following $k^3$ at small $k$ and $k^{-1.5}$ at large $k$, and GW spectra with slopes $k^3$ and $k^{-2}$ in the respective regimes. They find PBH formation is favored for slow transitions (smaller $\beta/H$) and can occur down to $\beta/H\sim 6$, with PBH abundance increasing with $\alpha$ but more strongly suppressed by $\beta/H$, yielding testable predictions for future GW detectors and curvature-perturbation constraints.

Abstract

We conducted three-dimensional lattice simulations to study the density perturbation and gravitational waves (GWs) during first-order phase transition (FOPT). We find that for phase transition strength $α> 1$, the forward motion of bubble walls becomes the primary source, whereas for $α< 1$, the dominant contribution to the density perturbation comes from the delay of vacuum decay. Additionally, the power spectrum of density perturbations generated by the phase transition exhibits a slope of $k^3$ at small wavenumbers and $k^{-1.5}$ at large wavenumbers. Furthermore, we calculated the GW power spectra, which exhibit the slope of $k^3$ at small wavenumbers and $k^{-2}$ at large wavenumbers. Our numerical simulations confirm that slow PTs can produce PBHs and provide predictions for the GW power spectrum, offering theoretical support for GW detection.

Figures (12)

• Figure 1: Equation of state (EOS) evolution for weak phase transition strength (top, solid line $\alpha$=0.5, dash-dotted line $\alpha$=1) and strong phase transition strength (bottom, solid line $\alpha$=5, dash-dotted line $\alpha$=10). Four colors represent different $\beta/H$ values. In all cases, $\omega$ transitions from $\omega \to -1$ (vacuum energy domination) to $\omega = 1/3$. A smaller $\alpha$ results in a longer transition time to reach $\omega = 1/3$.
• Figure 2: The PBH abundance variation with respect to the PT strength parameter $\alpha$ and the inverse duration $\beta/H$.
• Figure 3: Evolution of $\sigma_\delta$ for weak phase transition strength (top, solid line $\alpha$=0.5, dash-dotted line $\alpha$=1) and strong phase transition strength (bottom, solid line $\alpha$=5, dash-dotted line $\alpha$=10).
• Figure 4: Variation of the standard deviation $\sigma_\delta$ of accumulated overdensity perturbations with respect to the PT parameters $\alpha$ and $\beta/H$.
• Figure 5: The power spectra of density perturbations for different $\beta/H$ values during the late stage of the phase transition. Top panel: solid line $\alpha$=0.5, dash-dotted line $\alpha$=1; Bottom panel: solid line $\alpha$=5, dash-dotted line $\alpha$=10.