How large are curvature perturbations from slow first-order phase transitions? A gauge-invariant analysis

TL;DR

The paper addresses the generation of super-horizon curvature perturbations from slow, strongly supercooled FOPTs and the gauge ambiguities in prior analyses. It develops a gauge-invariant multi-fluid framework and uses DeltaPT-based simulations to compute the comoving curvature perturbation $\\mathcal{R}$, deriving a fitting formula for $P_{\\mathcal{R}}(k)$ as a function of the transition strength $\\alpha$, reheating temperature $T_{reh}$, and nucleation parameter $\\beta/H_n$. The results show that PBHs are unlikely to form from this mechanism and that scalar-induced gravitational waves are subdominant, while the $P_{\\mathcal{R}}(k)$ template enables robust constraints from SKA, PTA, and UCMH observations. Overall, the work provides a practical, gauge-consistent framework to translate FOPT parameters into observable signatures, reducing gauge-related uncertainties and enabling stringent tests of slow, supercooled phase transitions in the early universe.

Abstract

When strongly supercooled cosmological first-order phase transitions (FOPTs) are sufficiently slow, super-horizon inhomogeneities can be generated. We compute these super-horizon curvature perturbations by employing a gauge-invariant, multi-fluid formalism. By resolving the gauge ambiguities inherent in conventional separate-universe simulations, we demonstrate that Primordial Black Holes are unlikely to be produced by these super-horizon inhomogeneities. We also derive a fitting formula for the resulting curvature perturbations and discuss potential observational constraints on FOPTs imposed by limits on primordial curvature perturbations and associated scalar-induced gravitational waves.

Figures (6)

• Figure 1: Time evolution of the absolute value of the false vacuum fraction $F(t)$, the comoving curvature perturbation $\mathcal{R}$, and the density contrast $\delta_{\rm NG}$ of an independently evolving volume $4\pi k^{-3}/3$, where $k=0.9 k_{\rm max}$ and $\beta / H_n =7$. The value of $9/4\; \delta_{\rm NG}$ is given for reference. The dashed line represents the time when the $k=0.9 k_{\rm max}$ mode re-enters the horizon.
• Figure 2: Variance of the comoving curvature perturbations $\sigma^2_{\mathcal{R}}$ as a function of $\beta$ and $k/k_{\rm max}$. The dashed curves represent the corresponding fits, while the dashed black lines indicate the $\beta^{-5}$ and $k^3$ reference power laws.
• Figure 3: Distributions of density contrast of independently evolving volumes $4\pi k^{-3}/3$, where $k=0.9 k_{\rm max}$. The blue and orange envelopes represent $\delta_C$, the density contrast in the comoving-gauge. The black envelope represents the density contrast computed from separate universe simulations ($\delta_{\rm NG}$). The PBH formation threshold $\delta_{\rm crit} \approx 0.45$ is marked as a vertical dashed line.
• Figure 4: Various benchmark spectra $P_{\mathcal{R}}(k)$ for $\alpha=10$ with varying $\beta/H_{n}$ and $T_{\rm reh}$. Also displayed is a compilation of existing constraints on the primordial power spectrum $P_{\mathcal{R}}(k)$ (solid-outlined filled regions), sensitivity forecasts for next-generation observatories (dotted-outlined filled regions), and UCMH-based constraints (dashed lines). The non-UCMH constraints are derived from CMB anisotropies Planck:2018vyg (light blue), Lyman-$\alpha$ observations Bird:2010mp (orange), CMB spectral distortions Chluba:2012we (green), Pulsar Timing Arrays Byrnes:2018txb (red), as well as SKA and LISA forecasts Inomata:2018epa (pink and purple). The UCMH-based constraints, which assume a monochromatic power-spectrum enhancement and that WIMP dark matter annihilates into into $b\bar{b}$ with a mass $m_{\rm DM}=1$ TeV and $s$-wave thermal relic cross section $\langle \sigma v \rangle_0=3 \times 10^{-26} \mathrm{cm}^{3}\mathrm{s}^{-1}$, are obtained from diffuse $\gamma$-ray searches Delos:2018ueo (black dashed line) and limits on extra energy injection into the CMB FrancoAbellan:2023sby (gray dashed line).
• Figure 5: GWs spectra from PT-BUBBLE and PT-SOUND models as the primary GW sources together with SIGWs from curvature perturbations.