Sound waves from primordial black hole formations

TL;DR

This work numerically studies primordial black hole formation from super-horizon curvature perturbations using the Misner-Sharp formalism with excision to follow post-collapse sound-wave generation. It reveals that near-critical perturbations produce a two-shell compression wave (overdense and underdense) that accompanies PBH formation, while far-from-critical perturbations yield only an outward underdense shell; softer equations of state suppress compression waves and the comoving sound-shell thickness remains nearly constant, linking the GW peak frequency to PBH mass via horizon re-entry. The authors connect nonlinear collapse to linear sound-wave propagation, show energy balance between the overdense shell, the PBH, and the compensating underdense shell, and employ a sound-shell model to estimate the resulting stochastic GW background, with the shell thickness serving as a key observable that encodes PBH mass scales. These results provide a concrete route to relate PBH properties to gravitational-wave signatures and motivate further exploration of non-spherical effects and multi-PBH configurations. The study thus advances the understanding of PBH dynamics, early-universe fluid interactions, and potential GW signals from PBH-related sound waves.

Abstract

We present a numerical investigation of primordial black hole (PBH) formation from super-horizon curvature perturbations and the subsequent generation and propagation of sound waves, which can serve as a new source of stochastic gravitational wave backgrounds (SGWBs) presented in a companion letter. Using the Misner-Sharp formalism with an excision technique, our simulations extend to significantly later times than previous work and indicate that the near-critical perturbations produce a distinct compression wave featuring both overdense and underdense shells, while significantly supercritical perturbations yield only an underdense shell. We also show that a softer equation of state suppresses the formation of compression waves. Furthermore, the comoving thickness of sound shells remains nearly constant during propagation and scales with the Hubble radius at horizon re-entry, thereby serving as a key link between the gravitational-wave peak frequency and PBH mass in the companion letter. These results offer new insights into the dynamics of PBH formation and suggest potential observational signatures of PBHs in the gravitational wave (GW) spectrum from associated sound waves.

Figures (15)

• Figure 1: Time evolution of the energy density for subcritical perturbations with $\delta_m = 0.3 \ll \delta_c$ (left panel) and $\delta_m = 0.495 \lesssim \delta_c$ (right panel). The horizontal coordinate $R/a$ indicates the comoving radius, and different coloured lines correspond to different time snapshots.
• Figure 2: Time evolution of the energy density before PBH formation for near-critical ($\delta_m = 0.5$, left panel) and far-from-critical ($\delta_m = 0.55$, right panel) perturbations. The black dashed lines indicate that a black hole has formed.
• Figure 3: Time evolution of the energy density contrast for near-critical ($\delta_m = 0.5$, left panel) and far-from-critical ($\delta_m = 0.55$, right panel) perturbations.
• Figure 4: Time evolution of the PBH mass for near-critical ($\delta_m = 0.5$, blue line) and far-from-critical ($\delta_m = 0.55$, orange line) perturbations. The red dot marks the time ($t = 50t_m$) at which the overdense shell forms.
• Figure 5: Time evolution of the energy density contrast for $\omega = 1/5$ (left panel) and $\omega = 1/10$ (right panel). In both cases, the perturbation amplitudes are chosen such that $\delta_m-\delta_c \approx 0.00226$.