Chiral phase transition with primordial black holes: Distinct phase structure and catalysis

TL;DR

This paper investigates how primordial black holes modify the chiral phase transition in a curved spacetime setting using a three-flavor NJL model in a Schwarzschild background. By deriving the finite-temperature effective potential with curvature corrections and solving the bounce equation, it demonstrates a novel near-horizon phase structure where curvature can drive second-order and first-order transitions and even restore symmetry near the horizon. The PBH-catalyzed dynamics significantly enhances the inverse duration parameter $β/H$ while keeping the latent heat similar, leading to $O(1)$ shifts in the GW peak frequency and amplitude, with signals potentially detectable by LISA or PTAs depending on the energy scale. The framework connects curved-space quantum field theory, phase-transition dynamics, and gravitational-wave phenomenology, offering a versatile template for dark-sector phase transitions in the presence of PBHs across a wide range of scales.

Abstract

We study the impact of primordial black holes (PBHs) on the chiral phase transition and its associated stochastic gravitational-wave (GW) signals. Using the three-flavor Nambu-Jona-Lasinio model, we construct the chiral effective potential in a Schwarzschild spacetime background. We find that PBHs promote chiral symmetry restoration and induce a nontrivial local phase structure in the vicinity of the event horizon simultaneously. In particular, this structure exhibits a novel chiral symmetry breaking pattern involving both second- and first-order phase transitions, a feature absent in flat spacetime. We further demonstrate that PBHs act as genuine catalysts for the chiral phase transition by analyzing the bounce solution in curved spacetime. The presence of PBHs substantially enhances the inverse duration parameter $β/H$ while leaving the overall transition strength comparable to that in flat spacetime. As a consequence, even a small population of PBHs can induce $\mathcal{O}(1)$ shifts in both the peak frequency and the peak amplitude of the GW spectrum generated by the first-order dark chiral phase transition.

Figures (7)

• Figure 1: Temperature dependence of the flat-spacetime effective potential $V_{\rm eff}^{\rm flat}(\sigma, T)$ for the parameter choice $G_S\Lambda^2 = 8$, $G_A \Lambda^6 = -448$ and $m_0\Lambda^{-1} = 10^{-3}$, illustrating a first-order chiral phase transition.
• Figure 2: Radial dependence of the zero-temperature effective potential $V_R(\sigma;r)=V_{00}(\sigma)+V_{R0}(\sigma;r)$ in the Schwarzschild PBH background for the benchmark parameters in eq. \ref{['sec3:BenchMark']}. The colored curves show $V_R(\sigma;r)$ at fixed radii $r/r_s=1.2,\,1.4,\,1.7$, while the black curve corresponds to the flat-spacetime potential $V_{00}(\sigma)$. The PBH-induced curvature term lifts the broken-phase minimum and thus tends to restore chiral symmetry as $r\to r_s$.
• Figure 3: Temperature dependence of the full effective potential $V_\mathrm{eff}(\sigma;T,r)$ at $r/r_s = 1.05$ with $r_s\Lambda = 1$ and the parameters in eq. \ref{['sec3:BenchMark']}. Panels (a) and (b) respectively show the FOPT subsequent to the second-order phase transition and the symmetry restoration process.
• Figure 4: Ratio $\beta/H$ as a function of the Schwarzschild radius $r_s$ (in units of $\Lambda^{-1}$) for the benchmark $G_S \Lambda^2 = 8$, $G_A \Lambda^5 = -448$, $m_0 \Lambda^{-1} = 10^{-3}$. The orange line shows the flat–spacetime result $\beta_0/H$, while the blue curve corresponds to $\beta_{\rm PBH}/H$. The presence of a PBH enhances $\beta/H$ and hence catalyzes the phase transition, with the enhancement being larger for lighter PBHs (smaller $r_s$).
• Figure 5: Dependence of the phase transition duration $\beta/H$ on the PBH fraction $f_\mathrm{PBH}$ for the benchmark point in Eq. \ref{['sec3:BenchMark']} with $r_{s} \Lambda = 4$ and $\Lambda=1 \,\mathrm{GeV}$. The green (orange) line represents the prediction for $\beta/H$ with (without) PBHs. The blue line denotes the asymptotic value $\beta_\mathrm{PBH}/H$ in the limit $N_\mathrm{PBH}/N_\mathrm{tot} \to 1$.