Gravitational Waves and Primordial Black Holes produced by Dark Meta Stable Vacuum Decay

TL;DR

This work investigates a dark-sector metastable vacuum that decays via bubbles nucleating at an approximately constant rate and completing by percolation, with gravity as the only SM interaction. It develops both analytic frameworks and lattice simulations to predict the resulting stochastic gravitational-wave background and primordial black-hole production, accounting for the expanding universe and the dominance of kinetic energy in ultra-relativistic bubble walls. The authors find a robust GW peak at $k_{\rm peak}=3.1\,H_{\rm PT}$ with amplitude $\Omega_{\rm GW}^{\rm peak} \simeq 1.5\,\Omega_{\gamma}\, (\Delta \rho/\rho_{\rm tot})^2$, and show the IR tail is modified to include a $k^3\log^2(k/H)$ term; PBH production is largely suppressed by $\Delta N_{\rm eff}$ constraints unless latent energy is transferred to the SM. These results imply that dark-sector phase transitions could yield detectable gravitational-wave signals while remaining consistent with cosmological radiation bounds, and they delineate conditions under which DS-origin PBHs could contribute to dark matter or be ruled out by light-element/Planck-era constraints.

Abstract

Inspired by string theory and cosmological constant problem, it is plausible that the Universe's vacuum structure is characterized by a landscape of metastable vacua. The existence of dark matter and dark energy further suggests that the dark sector may inhabit its own "dark landscape". If the dark vacuum is metastable, bubbles of lower-energy phases can nucleate at an approximately constant rate. Because the Hubble expansion rate is monotonically non-increasing with cosmic time, such nucleation can eventually lead to percolation and completion of a dark-sector phase transition. In this work, we investigate the phenomenological consequences of this transition, focusing on the resulting stochastic gravitational-wave background and the potential formation of primordial black holes. We find that the gravitational wave spectrum peaks at $k_{\mathrm{peak}}=3.1 H_{\mathrm{PT}}$, with an amplitude $Ω_{\mathrm{GW}}^{\mathrm{peak}}\simeq1.5 Ω_γ(Δρ/ρ_{\mathrm{tot}})^2$. Furthermore, the formation of primordial black holes is suppressed due to $ΔN_{\mathrm{eff}}$ constraint.

Figures (12)

• Figure 1: The lower panel displays the energy density distribution before and after the collision of two bubbles at point $c$. The upper and middle panels zoom in on the region outlined by the red rectangle, showing the detailed density profile and field configuration of a segment of the bubble wall, respectively. For this specific collision event, the boost factor is $\gamma = 10$. We observe that despite the development of multiple oscillations in the post-collision waveform, the majority of the energy remains concentrated in a thin region near the propagating wavefront of the bubble shell.
• Figure 2: Bubble collision process in three sequential stages. a) Pre-collision: The left panel depicts a kink-like configuration with false vacuum between the two bubbles. b) Collision: The middle panel shows the moment the bubble walls merge. c) Post-collision: The right panel displays the outcome, where most of the energy is carried by two outward-propagating shells. (The simulation uses a boost factor of $\gamma=80$.)
• Figure 3: The relation between $\xi$ and the energy fraction of DS. The red dotted line is the result from simulation while the blue solid line is the fitting result $\frac{\rho_{\rm DS}}{\rho_{\rm SM}}\simeq 7.3\xi^2$. The $\Delta N_{\rm eff}$ bound is shown in the black vertical line.
• Figure 4: In the left panel, the blue and red curves represent the average energy of bubble $\bar{\rho}(\eta_0)$ and the bubble radius $\bar{r}(\eta_0)$, respectively. Both quantities decrease with increasing nucleation time $\eta_0$, as bubbles nucleated later have less time to absorb energy from the false vacuum. In the right panel, the blue curve depicts the derivative $\mathrm{d}\bar{\rho}/\mathrm{d}\bar{r}$. It peaks at $(0.64, 1.95)$, indicating that bubbles carrying the largest energy fraction have a characteristic size of $r \sim 0.64 H_{\mathrm{PT}}^{-1}$.
• Figure 5: The blue and red curves depict the gradient energy distribution and the gradient energy ratio, respectively. In our simulation, two bubbles are initialized at $r=0$ and $r=2r_c$, leading to a subsequent collision at $r=r_c$. We analyze the bubble shell centered at $r=0$ to compute $E_g(r,t)$ and $\rho_g(r,t)$. The red dotted line indicates the cutoff at $E_g(r,t)=0.25$. This cutoff intersects the gradient energy ratio curve between the first and second shell peaks. As time $t$ evolves, the peak amplitude of the gradient energy decreases; however, the fractional energy contained within the first peak remains constant.