Primordial Black Holes: Early Universe Probes

• Primordial black holes (PBHs) are black holes formed in the early universe via density perturbations and topological defects, serving as probes for cosmology and high-energy physics.
• Their mass spectrum, defined by horizon mass at formation, can be sharply peaked or extended, with constraints imposed by Hawking radiation, gravitational lensing, and cosmic background observations.
• PBHs influence gravitational wave signals and structure formation while offering a unique window into quantum gravity and early-universe phase transitions.

Primordial black holes (PBHs) are black holes formed in the early Universe through mechanisms independent of stellar evolution. As non-baryonic compact objects with a range of possible masses, PBHs provide a unique probe of early-universe cosmology, inflationary dynamics, high-energy particle physics, and are leading non-particle candidates for cold dark matter. They are subject to a wide array of cosmological, astrophysical, and particle-physics constraints owing to their gravitational, evaporative, and lensing signatures across cosmic time.

1. Formation Mechanisms

1.1. Collapse of Large Density Perturbations

During the radiation-dominated epoch, a comoving region of size $R$ can form a PBH upon horizon entry (at $t \sim R/c$) if its smoothed density contrast

$\delta = \frac{\delta\rho}{\rho}$

exceeds a critical value, $\delta_c \sim 0.3$–$0.5$. The mass of such a PBH is typically set by the horizon mass at that time: $$M_H(t) \simeq \frac{c^3 t}{G} \sim 10^{15}\Bigl(\frac{t}{10^{-23}\,\mathrm{s}}\Bigr)\,\mathrm{g}$$ (Green, 2014). The Press–Schechter formalism with Gaussian statistics gives the initial PBH formation fraction as

$$\beta(M) = \frac{1}{2} \mathrm{erfc}\left(\frac{\delta_c}{\sqrt{2}\,\sigma(M)}\right)$$

with $\sigma^2(M)$ the variance of $\delta$ on mass scale $M$. Critical-collapse numerics show the PBH mass–overdensity relation is

$$M_\mathrm{PBH} = \kappa\,M_H\,[\,\delta-\delta_c\,]^\gamma$$

where for radiation, $\gamma \sim 0.36$ and $\kappa \sim \mathcal{O}(1)$.

1.2. Topological Defect Collapse: Cosmic Strings and Bubble Collisions

• Cosmic-String Loops: Oscillating loops with tension $\mu$ can collapse if they contract within their Schwarzschild radius, forming PBHs of mass $M \sim \mu\ell$ with a broad spectrum $dn/dM \propto M^{-5/2}$. The collapse probability is $P \sim (G\mu)^p$, and $G\mu/c^2 \lesssim 10^{-6}$ is required by evaporation constraints (Green, 2014).
• Cusp Collapse on Strings: Cusps are generically present on cosmic string loops. A segment near a cusp collapses under the hoop conjecture if $2GM/r \gtrsim 1$, with rest-mass $m \sim (G\mu)^2 M_\mathrm{loop}$ and universal dimensionless spin $\chi = 2/3$. Constraints from PBH evaporation give nearly model-independent upper limits, $G\mu \lesssim 10^{-6}$, and the PBH DM fraction is suppressed below $10^{-10}$ (Jenkins et al., 2020).
• Bubble Collisions in First-Order Phase Transitions: Collisions of true-vacuum bubbles can trap regions of the false vacuum; rare multibubble collisions may lead to collapse for sufficiently slow phase transitions, producing PBHs with mass at the horizon scale at percolation (Green, 2014).

1.3. Scalar Field and Topological Soliton Mechanisms

A scalar condensate (e.g., along SUSY flat directions) can fragment into Q-balls; Poisson fluctuations in the Q-ball distribution can seed overdense regions that collapse to PBHs. For SUSY, $M_\mathrm{PBH} \lesssim 10^{23}$ g; a general charged scalar can yield black holes up to $\sim 100~M_\odot$—relevant for LIGO (Cotner et al., 2017).

Topological defect networks (defects with $\rho_\mathrm{defect} \propto a^{-n}$, $n<3$) can also generate large fluctuations and PBHs through Poisson statistics.

2. Mass Spectrum and Abundance

The PBH mass function, $\psi(M)$, for nearly instantaneous formation is sharply peaked,

$$\psi(M) \sim \beta(M) \frac{\rho_\mathrm{rad}(t_f)}{M} \delta(M - M_H(t_f))$$

(Green, 2014), or for critical collapse,

$$\psi(M) \propto \int d\delta\,P(\delta)\,\delta\bigl(M-\kappa\,M_H(\delta-\delta_c)^\gamma\bigr)$$

where $P(\delta)$ is the PDF of the density field. Non-instantaneous or broad mechanisms (e.g., scalar fragmentation, clustered formation) can yield extended or log-normal mass functions appropriate for interpreting observational constraints (Dolgov, 2017).

PBH abundance constraints are parametrized by $$f_\mathrm{PBH}(M) = \Omega_\mathrm{PBH}(M)/\Omega_\mathrm{DM}$$, with

$$f_\mathrm{PBH}(M) \simeq \left(\frac{\beta(M)}{10^{-8}}\right) \left(\frac{M}{M_\odot}\right)^{-1/2}$$

(Green, 2014).

3. Evolution and Evaporation Physics

PBHs lose mass via Hawking radiation at the rate

$$\dot{M}_\mathrm{evap} = -\frac{\hbar\,c^4}{15360\pi\,G^2\,M^2}$$

with evaporation time

$$\tau_\mathrm{evap}(M) = \frac{5120\,\pi\,G^2 M^3}{\hbar\,c^4} \simeq 10^{10}\left(\frac{M}{10^{15}\,\mathrm{g}}\right)^3~\mathrm{yr}$$

(Green, 2014, MacGibbon et al., 2015). PBHs with $M \lesssim 5 \times 10^{14}$ g have fully evaporated by the current epoch.

Accretion in the expanding universe is subdominant for cosmological mean densities: only PBHs with $M \gtrsim 10^{37}$–$10^{38}$ g grow significantly, reaching the Eddington limit and potentially growing to SMBH scales only in special, overdense environments (Rice et al., 2017). Hence, PBHs in the relevant cosmological mass range are generally relics with unchanged mass after formation, aside from Hawking evaporation for $M \lesssim 10^{15}$ g.

4. Observational Constraints

4.1. Hawking Radiation, $\gamma$-Rays, and BBN

• Diffuse $\gamma$-Ray Background: For $10^{13}\lesssim M/\,{\rm g}\lesssim 10^{15}$, non-detection of extragalactic $\gamma$-rays constrains $$\beta(M) \lesssim 10^{-27}\left(M/10^{15}\,{\rm g}\right)^{-5/2}$$ (Green, 2014).
• BBN Constraints: Evaporation products during $10^9\lesssim M/\,{\rm g}\lesssim 10^{13}$ affect light-element abundances; $\beta(M) \lesssim 10^{-23}$ (Green, 2014).
• CMB Spectral Distortions and Cosmic Ray Antiprotons: Place comparable or slightly weaker bounds.

4.2. Gravitational Lensing and Dynamical Effects

• Femtolensing: $10^{17}\lesssim M/\,{\rm g}\lesssim 10^{20}$: $f_\mathrm{PBH} \lesssim 1$.
• Microlensing: $10^{24}\lesssim M/\,{\rm g}\lesssim 10^{34}$: $f_\mathrm{PBH} \lesssim 0.1$ (EROS, MACHO, Kepler).
• Wide Binaries, Halo Friction, Disk-Heating:
  • Disruption of wide binaries: $10^{36}\lesssim M/\,{\rm g}\lesssim 10^{41}$, $f_\mathrm{PBH} \lesssim 0.4$.
  • MW halo friction/disk heating provide further exclusion at larger masses.

Clustering can relax microlensing bounds if PBHs form in clusters of $N_\mathrm{cl}\gtrsim10^2$; the microlensing event rate scales as $N_\mathrm{cl}^{-1/2}$, and the CMB bounds from accretion are also weakened due to enhanced PBH velocities in dense clusters. This opens a viable window $M\sim 1$–$10~M_\odot$ for clustered PBH DM (García-Bellido et al., 2017).

4.3. Dark Matter Window

Synthesizing all constraints, only the mass window

$10^{20}\lesssim M/\,{\rm g} \lesssim 10^{25}$

($10^{-13}M_\odot \lesssim M \lesssim 10^{-8}M_\odot$) remains where PBHs could still make up all of the dark matter (Green, 2014, Liang et al., 17 Jan 2025). For some broad mass functions and special formation scenarios, $1$–$10~M_\odot$ windows can survive, particularly if clustering is significant (García-Bellido et al., 2017, Dolgov, 2017).

5. PBHs as Probes of Cosmology and Early-Universe Physics

Formation of PBHs probes the primordial power spectrum at scales far smaller than the CMB, $10^{-2}\lesssim k/{\rm Mpc}^{-1}\lesssim 10^{23}$. For PBH formation to be significant,

$${\cal P}_{{\cal R}}(k) \lesssim 10^{-2}\text{--}10^{-1}$$

at these scales, several orders of magnitude above the $10^{-9}$ measured by Planck at CMB scales (Green, 2014). Stringent upper limits on $\beta(M)$ translate directly into bounds on small-scale power. Inflation models that predict "spikes" or "blue" spectra often violate these bounds. Non-Gaussian statistics can enhance PBH formation, further tightening joint constraints with non-Gaussianity parameters such as $f_\mathrm{NL}$.

PBHs from first-order phase transitions, magneto-hydrodynamic turbulence, Affleck–Dine condensate fragmentation, or string-defect collapse offer direct mapping from particle physics parameters (e.g., transition scale, string tension) onto PBH observables, establishing synergy between early-universe microphysics and late-time cosmological signatures (Dent et al., 6 Oct 2025, Liang et al., 17 Jan 2025, Cotner et al., 2017, Jenkins et al., 2020).

6. Role in Gravitational Wave and Structure Formation

PBHs in the stellar-mass range have garnered significant attention as progenitors of the binary BH merger events detected by LIGO/Virgo, capable of yielding high merger rates for $f_\mathrm{PBH} \sim 10^{-3}$–$1$, depending on their mass spectrum and spatial distribution (Sasaki et al., 2018). PBH binaries formed in the early universe, through spatial clustering or gravitational decoupling, yield characteristic eccentricity and spin distributions distinct from those of stellar-origin BHs.

High-mass PBHs ($M_\mathrm{PBH}\gtrsim 10^2~M_\odot$) with substantial abundance can accelerate the collapse of minihalos and advance Population III star formation, but are constrained by observations of early galaxies and 21-cm signals (Koulen et al., 6 Jun 2025). Lower-mass PBHs can delay or suppress star formation through tidal disruption of gas clouds.

Stochastic gravitational wave backgrounds are induced by both the PBH merger history and second-order effects from large scalar perturbations at PBH-forming scales, providing a target for future GW detectors (Inomata et al., 2021).

7. Quantum Gravity, Particle Physics, and Future Constraints

PBHs with $M \lesssim 10^{15}$ g evaporate via Hawking radiation, emitting a spectrum of photons, leptons, and hadrons. Stringent observational limits from the extragalactic $\gamma$-ray background, cosmic-ray antiprotons, and BBN restrict the PBH abundance at low masses (Carr, 2014, MacGibbon et al., 2015). Detection of final-stage PBH bursts would furnish unique evidence for quantum black hole thermodynamics.

The possibility of Planck-mass relics or exotic evaporation signatures constrains scenarios with new light degrees of freedom or quantum gravity effects. Fisher-matrix analyses combining GW, CMB spectral distortions, lensing statistics, and 21-cm cosmology are expected to further delimit PBH parameter space across multiple orders of magnitude in mass.

Across all scenarios, PBHs remain a multi-probe for the small-scale cosmological perturbations, high-energy particle physics, and the microphysical properties of the early universe. Ongoing and forthcoming observational programs targeting gravitational waves, CMB distortions, microlensing, and transient $\gamma$-ray bursts will empirically test the PBH hypothesis in the coming decade.