Press–Schechter Estimate for Primordial Black Hole Birth
Primordial black holes (PBHs) are hypothetical black holes that formed in the very early universe, long before stars and galaxies came into existence. Unlike stellar-mass black holes that arise from the collapse of massive stars, PBHs could emerge wherever extremely overdense regions exceeded a critical threshold at the time those regions reentered the cosmological horizon. In the simple spherical-collapse picture, an overdensity with amplitude collapses to a black hole if it is greater than some critical value . If the overdensity is lower than this threshold, pressure forces disperse the region without black hole formation. Because the early universe’s density fluctuations are believed to be nearly Gaussian, the probability of a region exceeding the threshold can be treated statistically. The dimensionless variance of those fluctuations, typically denoted σ, then directly controls the expected mass fraction of the universe collapsing into black holes. Our calculator implements this idea using the classic Press–Schechter formalism. Users supply the root-mean-square amplitude σ, the collapse threshold δc, and the mass scale of interest. The tool then returns two quantities: β, the fraction of total energy density that goes into PBHs at the time of formation, and , the fraction of the present-day dark-matter density composed of PBHs of the specified mass.
The Press–Schechter approach, although originally designed to describe galaxy halo formation, can be adapted to PBHs by treating the problem as an instance of threshold statistics for random Gaussian fields. The central assumption is that the primordial curvature perturbations generated during inflation translate into matter overdensities at horizon reentry. In this context the probability distribution for the overdensity δ at a given smoothing scale is . The mass fraction collapsing into PBHs is then identified with the tail of this distribution beyond δc, because only those rare peaks with δ ≥ δc become black holes. We integrate the Gaussian tail and obtain
This expression is extremely sensitive to σ, because it probes the exponentially suppressed tail of the Gaussian. Consequently, even modest changes in the primordial power spectrum can lead to orders-of-magnitude differences in PBH abundance. The characteristic mass of the black holes produced is related to the horizon mass at the time the relevant perturbation scale reenters the horizon. In a radiation-dominated universe that horizon mass is approximately , so a PBH with 30 solar masses traces back to fluctuations entering the horizon about one second after the Big Bang. The parameter today depends on how the energy density in PBHs redshifts relative to the rest of the cosmic energy budget. A standard approximate scaling gives . Our JavaScript implementation follows this approximation to offer a quick estimate of the potential contribution of PBHs to dark matter.
Choosing appropriate values for σ and δc is a nontrivial task. The threshold δc is not a universal constant but depends on the shape of the perturbation, the equation of state of the universe at the time of collapse, and the exact collapse dynamics. Numerical simulations of PBH formation in radiation domination typically find δc between 0.4 and 0.5. We adopt 0.45 as a default but allow users to adjust it. The fluctuation amplitude σ is determined by the primordial curvature power spectrum. In the simplest single-field inflation models tuned to match cosmic microwave background measurements, σ on small scales is tiny, leading to negligible PBH production. However, a variety of inflationary scenarios, including features in the potential or non-standard reheating histories, can amplify power on select scales, raising σ and enhancing PBH abundance. The debate over whether PBHs could make up all or part of the dark matter hinges on how large σ can plausibly become without conflicting with other cosmological and astrophysical constraints.
The utility of a calculator like this extends beyond speculative cosmology. Press–Schechter estimates appear whenever Gaussian random fields cross critical thresholds, so the same mathematics applies to rare events in large-scale structure, gravitational wave burst rates, and even condensed matter problems involving phase transitions. Yet PBH formation is particularly sensitive because the observable consequences of overproduction, such as excessive microlensing, CMB distortions, or gravitational wave backgrounds, strongly constrain σ. Conversely, the absence of PBH signatures places limits on inflationary models, illustrating a deep connection between small-scale primordial physics and present-day astrophysics. The calculation of β also enters in forecasts for second-generation gravitational wave detectors, which could probe PBH mergers if they constitute a significant fraction of black hole binaries. Accurately estimating β and fPBH therefore informs observational strategies across multiple domains.
To provide intuition, the table below lists β and fPBH for a few representative values of σ at a fixed mass of 30 M☉ and δc = 0.45. Even slight increases in σ dramatically raise the expected PBH abundance. Values of fPBH much larger than unity would overclose the universe, meaning that such parameter combinations are ruled out. Conversely, very small fPBH implies negligible contribution to dark matter, though even a tiny fraction could lead to interesting astrophysical signatures.
| σ | β | fPBH |
|---|---|---|
| 0.05 | ≈ 1.6×10−38 | ≈ 3×10−31 |
| 0.10 | ≈ 3.7×10−8 | ≈ 0.02 |
| 0.15 | ≈ 1.4×10−3 | ≈ 1.7×105 |
These numbers illustrate the steep dependence on σ; the jump from 0.05 to 0.10 increases β by thirty orders of magnitude. Such sensitivity underscores why any prediction of PBH abundance must carefully account for uncertainties in the primordial power spectrum. The β values also hint at the so-called fine-tuning problem in PBH dark matter scenarios: achieving fPBH ≈ 1 requires σ to be tuned to within a few percent in many models. Beyond astrophysical considerations, PBHs intersect with fundamental physics through potential links to baryogenesis, dark matter, and even the information paradox. Some theories suggest that evaporating PBHs could leave behind stable Planck-mass relics that themselves constitute dark matter. Others explore how PBHs could seed structure formation or explain puzzling observations such as the origin of supermassive black holes. Whatever the case, estimating β is a first step toward quantifying these ideas.
While our calculator embraces a simplified analytic model, researchers often go beyond Press–Schechter by employing peaks theory, non-Gaussian statistics, or detailed numerical simulations that track the nonlinear evolution of overdensities. These approaches can shift the predicted β by orders of magnitude. Nevertheless, the Press–Schechter method remains a valuable pedagogical tool and a quick way to scan parameter space. By coupling the analytic estimate with the transformation to present-day dark matter fraction, the calculator offers a compact yet informative window into a vast literature. Users can experiment with exotic scenarios, explore parameter constraints, or simply build intuition about how PBH formation responds to small changes in the primordial fluctuation spectrum.