Birth of Primordial Black Holes from Tensor Perturbations during Phase Transitions ✦
Context
The nature of dark matter, first hinted at by the observations of Vera Rubin, remains one of physics' great mysteries. Primordial black holes (PBHs), formed in the early universe, are natural candidates for its composition. Their emergence usually requires large curvature fluctuations during inflation, but an alternative path—using gravitational waves from first-order phase transitions and domain walls—not only avoids fine-tuning but also opens new avenues for testing theories beyond the Standard Model. A key role here is played by the expansion of the universe, which determines when perturbations enter the horizon and their subsequent evolution.
Methods
Within second-order perturbation theory, analytical expressions were derived for the spectrum of scalar inhomogeneities induced by the gravitational-wave background. Using peak formalism and critical collapse, the fraction of the horizon collapsing into PBHs and their mass distribution were calculated. Phenomenological spectra, consistent with lattice simulations, described gravitational wave emission from the phase transition and domain wall annihilation. Symbolic regression yielded compact semi-analytical formulas directly linking phase transition parameters to PBH abundance. Accounting for the evaporation of black holes, predicted by Stephen Hawking, provides independent constraints on allowed masses.
Results
It was found that primordial black holes of asteroid-scale masses (around 10⁻¹⁵ M⊙) can account for all dark matter for phase transition temperatures T⋆ between 4×10² and 10⁴ GeV and ratio β/H ≃ 6, provided the transition strength parameter α exceeds unity. After accounting for cosmological redshift, the peak of the corresponding gravitational wave spectrum reaches an amplitude Ωp GW h² ∼ 10⁻⁸ at frequencies 10⁻⁵–10⁻² Hz, within the sensitivity range of LISA and SKA. For domain walls, all dark matter is achieved for tension σ^(1/3) ∈ [10⁶, 10⁸] TeV and bias V_bias^(1/4) ∈ [10⁷, 10¹⁰] MeV; the wave peak Ωp GW h² ∼ 10⁻⁹ (with α_ann ∼ 10⁻²) at frequencies 4×10⁻⁴–10⁻¹ Hz, accessible to LISA and ET. Comparison with Hawking evaporation data and cosmic microwave background constraints confirms the scenario's viability.
Implications
The proposed mechanism links the observed fraction of PBHs in dark matter to the characteristics of cosmological phase transitions, providing model-independent constraints on physics beyond the Standard Model. Combining data from nucleosynthesis, the cosmic microwave background, and future gravitational-wave projects will test theories with extended Higgs sectors, additional gauge groups, and discrete symmetries. This approach also resolves the cosmological domain wall problem, which would otherwise lead to early universe domination.
Future development
Future work will incorporate non-Gaussianity of primordial perturbations, nonlinear effects in curved spacetime, and the influence of black hole spin. Improved numerical methods and lattice simulations will refine the shape of gravitational wave spectra and induced scalar perturbations. Comparisons with upcoming data from LISA, ET, SKA, and gravitational lensing surveys will determine which phase transition scenario nature chose and the contribution of PBHs to dark matter.
Impact
The developed formalism will impact interpretations of the stochastic gravitational wave background, constraints on dark matter parameters, and the construction of ultraviolet completions of the Standard Model. The pioneering work of Rainer Weiss in gravitational wave detection laid the foundation for this interdisciplinary synthesis.
Next steps
In the near term, it is essential to include nonlinear evolution effects, a realistic equation of state for the early universe, and comparisons with new gravitational lensing data (including future observations of light bending in the NGRST project) and black hole mergers.
Key open problems
The study connects the nature of dark matter, the origin of the stochastic gravitational wave background, the generation of baryon asymmetry (which phase transitions may also explain), and the hierarchy problem in general relativity. It also suggests a path to solving the mystery of cosmological domain walls, which would otherwise quickly overrun the universe.
🎯 Asteroid-mass primordial black holes (around 10⁻¹⁵ M⊙) are about the size of an atomic nucleus but weigh as much as a small mountain, and their Hawking evaporation makes them practically invisible to current instruments.
🎬 The notion that a new universe might lurk inside a black hole resonates with the imagery of H.P. Lovecraft's 'The Shadow Out of Time' and the science fiction of Stephen Baxter, where black holes serve as gateways to other realms.
Key numbers
- Peak gravitational wave frequency for PBH dark matter: 10⁻⁵–10⁻² Гц
- Stochastic background amplitude Ωp GW h²: ~10⁻⁸
- Asteroid-range PBH mass: 10⁻¹⁵ M⊙
- Phase transition temperature for full dark matter: 4×10²–10⁴ ГэВ
- Domain wall tension energy scale σ^(1/3): 10⁶–10⁸ ТэВ