Angular momentum of vacuum bubbles in a first-order phase transition

TL;DR

This work analyzes the angular momentum of primordial black holes formed from spherical false vacuum bubbles during a dark-sector first-order phase transition. By tracking cosmological perturbations in a perturbed FRW spacetime, the authors compute the RMS spin using a gauge-informed volume integral of density and velocity perturbations, incorporating transfer functions across the FOPT. They show the dimensionless spin $s=J/(G_{ m N}M^2)$ can span from $\mathcal{O}(10^{-5})$ to $\mathcal{O}(10)$ for plausible keV–GeV scales and dark-to-visible temperature ratios $r_{T,c}$ in $0.1$–$0.4$, with a clear scaling of $s_{ m rms}$ with the FOPT time scale $\beta_*/H_*$, bubble wall velocity $v_{\rm w,*}$, and $r_{T,c}$. The results reveal that the dark-sector phase transition can imprint a broad range of PBH spins through second-order perturbative effects, independent of primordial curvature enhancements, and provide a framework for predicting PBH spin distributions in this context.

Abstract

The formation of primordial black holes (PBHs) during a first-order phase transition (FOPT) in a dark sector has been of recent interest. A quantity that characterizes a black hole is its spin. We carry out the first step towards determining the spin of such PBHs, by calculating the spin of spherical false vacuum bubbles induced by cosmological perturbations. The angular momentum is given by the product of density and velocity perturbations. We carefully track the evolution of background quantities and calculate the transfer functions during the FOPT. We find that the dimensionless spin parameter $s = J/(G_{\rm N} M^2)$ of false vacuum bubbles of mass $M$ and angular momentum $J$, take a wide range of values from ${\cal{O}}(10^{-5})$ to ${\cal{O}}(10)$ for FOPTs between 10 keV and 100 GeV and a dark sector that is 0.1 to 0.4 times cooler than the visible sector. We also find a scaling relation between the root-mean-square value of the spin, the FOPT time scale, the bubble wall velocity, and the dark sector-to-visible sector temperature ratio.

Figures (14)

• Figure 1: The physical region is shaded in blue for $A = 10^{-3}, 10^{-4}$, and two values of $\lambda/\lambda_0 \simeq 1.33, 1.72$. The green curves correspond to contours of constant $T_0/T_{\rm c}$, from 0.1 to 0.9, in steps of 0.1. The dashed green contour corresponds to $T_0/T_{\rm c} = 0.99$. For all cases, $g_\rho = 4.5$.
• Figure 2: The transfer functions $T_\delta$ and $T_{\hat{\psi}}$ in a radiation dominated background cosmology, for different perturbation modes.
• Figure 3: Evolution of the dark/visible sector temperatures and FV fraction (top row), and of $c_{s\rm,D}^2$ and $w_{\rm D}$ (bottom row), for BP-2 (column a) and BP-4 (column b). The vertical black dashed lines indicate the percolation temperature in all panels, while the vertical dot-dashed and dotted lines respectively indicate $\tilde{\eta}_i$ and $\tilde{\eta}_f$ in the bottom panels.
• Figure 4: $S_3/T$ and $\beta/H$, for BP-2 (red) and BP-4 (blue). The vertical dot-dashed lines indicate the percolation temperature for each benchmark point.
• Figure 5: Contours constant of $T_*/T_{\rm c}$ (red) for $A = 10^{-3}, 10^{-4}$, for $\lambda/\lambda_0 \simeq 1.33$ and $\lambda/\lambda_0 \simeq 1.72$, and $T_{\rm c} = \unit[10]{MeV}$, $r_{T\rm,c} = 0.4$. The blue contour marks the boundary of the physical region.