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 $\alpha > 1$, the forward motion of bubble walls becomes the primary source, whereas for $\alpha < 1$, 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.
Categories: hep-ph astro-ph.CO hep-th