Primordial Black Holes from Supercooled Phase Transitions

TL;DR

This work shows that primordial black holes can form abundantly during strongly supercooled first-order phase transitions due to late-nucleation patches that inflate and later develop large overdensities. By modeling the full past-light-cone nucleation history, the authors derive a semi-analytical expression for the PBH formation probability as a function of the phase-transition duration parameter \\beta/H and demonstrate an observable PBH population for \\beta/H \\lesssim 7, largely independent of the degree of supercooling. They also map the resulting PBH abundances onto cosmological and astrophysical constraints, identifying viable regions where PBHs could contribute to dark matter or be detectable through microlensing or gravitational waves, and highlighting the strong interplay between PBH formation and GW signals from bubble collisions. Overall, the paper provides a concrete framework linking supercooled PT dynamics to PBH production and observational consequences, establishing robust bounds on the transition rate and offering testable predictions for future searches.

Abstract

Cosmological first-order phase transitions (1stOPTs) are said to be strongly supercooled when the nucleation temperature is much smaller than the critical temperature. These are often encountered in theories that admit a nearly scale-invariant potential, for which the bounce action decreases only logarithmically with temperature. During supercooled 1stOPTs the equation of state of the universe undergoes a rapid and drastic change, transitioning from vacuum-domination to radiation-domination. The statistical variations in bubble nucleation histories imply that distinct causal patches percolate at slightly different times. Patches which percolate the latest undergo the longest vacuum-domination stage and as a consequence develop large over-densities triggering their collapse into primordial black holes (PBHs). We derive an analytical approximation for the probability of a patch to collapse into a PBH as a function of the 1stOPT duration, $β^{-1}$, and deduce the expected PBH abundance. We find that 1stOPTs which take more than $15\%$ of a Hubble time to complete ($β/H \lesssim 7$) produce observable PBHs. Their abundance is independent of the duration of the supercooling phase, in agreement with the de Sitter no hair conjecture.

Figures (11)

• Figure 1: The supercooled late-blooming mechanism: a) The nucleation of bubbles through quantum or thermal tunneling is a random process. Within certain causal patches -- such as the one delimited with a black dotted circle and labeled "late-bloomer" -- bubble nucleation can start later than the background. b) and c) In the supercooled limit, false vacuum regions in gray are vacuum-dominated while true vacuum regions in brown are energetically dominated by components which redshift like radiation (see App. \ref{['app:PT_dynamics']}). As a result, the background is rapidly redshifting while late-bloomers admit a nearly constant energy density. d) This inhomogeneity in the equation of state generates a Hubble-size over-density in the radiation fluid which, above a certain threshold, collapses into a PBH.
• Figure 2: During a supercooled first-order PT, the universe changes from a vacuum-like equation of state (EoS) with $\omega = -1$ to a radiation EoS with $\omega = 1/3$. Depending on their bubble nucleation history, distinct Hubble patches follow different EoSs. We compare the EoS of the background (blue) to the EoS of a late-blooming patch (red) inside which nucleation only starts after a critical time $t_{n_i}^{\rm \mathsmaller{PBH}}$. As a result, the radiation-like cooling phase of such a patch begins late with respect to the background, resulting in an over-density (see Fig. \ref{['fig:rho_evolution']}) which reaches a maximal density contrast at $t_{\rm max}$ which, if $\delta\rho/\rho > \delta_c$, collapses into a PBH. We use $\omega = -(\dot{a}^2+2a\ddot{a})/3\dot{a}^2$Kolb:1990vq with $a(t)$ the solution of Eq. \ref{['eq:rho_R_text']} for $t_{n_i}=t_c$ (blue) and $t_{n_i}=t_{n_i}^{\rm PBH}$ (red) so that it just passes the threshold $\delta\rho/\rho = \delta_c$ in Eq. \ref{['eq:PBHs_threshold_0']} at $t=t_{\rm max}$.
• Figure 3: An illustration depicting in chronological order the various steps leading to the eventual collapse of a Hubble patch into a PBH during a supercooled PT. The comoving Hubble horizon $r_{\rm H}(t)$ shown with red lines, shrinks when the universe is vacuum-dominated and grows when it becomes radiation-dominated. With blue lines we show the trajectory of the outermost walls which would enter the Hubble volume at $t_{\rm max}$ (see Eq. \ref{['eq:wall_traj']} in App. \ref{['sec:numerical_treatment']}) and coincide with the past light-cone of the Hubble volume at $t_{\rm max}$. A space-like slice bounded by this light-cone has the volume $V(t;t_{\rm max})$ and is shown in shaded blue (see Eq. \ref{['eq:causal_volume_text']}). The survival probability $\mathcal{P}_{\rm surv}\left(t_{n_i};t_{\rm max}\right)$ (see Eq. \ref{['eq:proba_tni_PBHs_0']}) is the probability of having no bubbles nucleated inside $V(t'; t_{\rm max})$ between $t_c< t'<t_{n_i}$.
• Figure 4: Predictions for the PBH density, $f_{\rm PBH}\equiv \rho_{\rm PBH}/\rho_{\rm DM}$, in the $\alpha^{-1/4}$--$\beta/H$ plane. As discussed in the text, $\alpha$ signifies the strength of the supercooled PT while $\beta^{-1}$ encodes its duration. For sufficiently long ($\beta\lesssim 6H$) and strong ($\alpha \gtrsim 100$) supercooled PT, significant production of PBHs is expected due to the existence of regions in which bubble nucleation is delayed, resulting in large over-densities. The gray bands show the dependence of the PBHs abundance on the critical threshold $\delta_{c}=\delta \rho/\rho$ above which gravitational collapse occurs. The solid ( dotted) lines are the numerical (semi-analytical) predictions for $f_{\rm PBH}$ assuming $\delta_c=0.50$. The PBH mass is given by the mass within the sound horizon at the time of the collapse. The temperature $T_{\rm eq}$ marks the beginning of the inflationary phase, during which the universe super-cools until bubbles nucleate at temperature $T_n$. The visible plateau associated with the asymptotic independence of the PBHs abundance on the strength of the phase transition (and hence the duration during which the universe is inflating) is a manifestation of de Sitter no hair conjecture Wald:1983ky.
• Figure 5: Constraints on strong ($\alpha>100$) supercooled PTs for which PBHs are expected to be produced, shown in the $T_{\rm eq}$--$\beta/H$ plane (with the PBH mass shown on the top x-axis). As discussed in the text, $T_{\rm eq}$ is the temperature below which an intermediate period of inflation occurs while the universe undergoes super-cooling. It is equal to the maximal reheating temperature after the phase transition up to ratio of relativistic degrees of freedom. The quantity $\beta^{-1}$ encodes the duration of the phase transition. In yellow, regions excluded by CMB either due to the expected accretion of predicted large PBHs ( left) Ali-Haimoud:2016mbvPoulin:2017bweSerpico:2020ehh or due to the evaporation of small PBHs ( right) Poulin:2016anjStocker:2018avmPoulter:2019ooo, are shown. The cyan region is excluded due to expected merging events in LIGO-Virgo DeLuca:2020qqa while the purple region is excluded from the MACHO MACHO:2000qbb, Eros EROS-2:2006ryy, OGLE Niikura:2019kqi, and HSC Niikura:2017zjd microlensing experiments. The region in which DM over-closes the universe is shown in brown, exclusion from unobserved cosmic-rays fluxes Carr:2009jmBoudaud:2018hqb is in red, and evaporation during BBN Carr:2009jm is in green. The green spot is the best fit region for contributing to the 511 keV excess DeRocco:2019fjqLaha:2019ssqKeith:2021guq. The dotted dark blue and dotted orange regions are the best-fit regions which address the anomalous microlensing events reported in HSC and OGLE data respectively Niikura:2019kqiNiikura:2017zjdSugiyama:2021xqg. Finally, in dashed and solid gray we indicate where gravitational waves from bubble collision fall within the detectability of pulsar timing arrays NANOGrav:2021flcNANOGrav:2023gorChen:2021rqpEPTA:2023fykGoncharov:2021oubReardon:2023gzhXu:2023wogAntoniadis:2022pcnInternationalPulsarTimingArray:2023mzf and exclusion of LIGO-Virgo Romero:2021kby. Additionally, in the region labeled "BBN" in gray, the reheating temperature is lower than the temperature of neutrino decoupling, $T_{\rm eq} \lesssim \text{MeV}$Rubakov:2017xzr, which is excluded. The above limits are recasted from various existing constraints shown in Fig. \ref{['fig:PBHs_constraints']}. We have fixed the PBHs threshold to $\delta_c=0.50$.