Primordial Black Hole Formation Fraction Calculator

What this calculator estimates

Primordial black holes (PBHs) are hypothetical black holes that could have formed in the early universe from unusually large density fluctuations. In the simplest horizon-reentry picture, a region collapses into a PBH when its smoothed overdensity δ exceeds a critical threshold δ_c at the time the perturbation reenters the horizon. If δ is below that threshold, pressure gradients and cosmic expansion prevent collapse.

This page provides a quick, educational estimate of PBH abundance using a Gaussian statistics + Press–Schechter tail integral. You enter the RMS fluctuation amplitude σ, the collapse threshold δ_c, and a PBH mass scale M (in solar masses). The tool returns two quantities: β, the fraction of the total energy density that collapses into PBHs at formation, and f_PBH, an approximate present-day fraction of dark matter in PBHs at that mass.

The key idea is that PBH formation is a rare-event problem: β is controlled by the extreme tail of the overdensity distribution. That makes the result extraordinarily sensitive to σ. A small change in σ can change β by many orders of magnitude, which is why PBH scenarios are often described as “fine-tuned” in simple Gaussian models.

How to use the calculator

1. Enter the density fluctuation RMS σ (dimensionless). In many inflationary models consistent with CMB observations, σ on PBH-forming scales is tiny; PBH production typically requires an enhanced small-scale power spectrum.
2. Enter the collapse threshold δ_c. Numerical simulations in radiation domination often suggest values around 0.4–0.5; the default here is 0.45.
3. Enter the PBH mass M in solar masses (M_☉). In this simplified tool, mass is used only to convert β into an approximate present-day dark-matter fraction.
4. Click Compute. Use Copy Result to copy the summary line (β and f_PBH) to your clipboard.

Practical tip: if you are scanning parameter space, vary σ in small increments (for example 0.10 → 0.105 → 0.11). Because the calculation probes a Gaussian tail, coarse steps can skip over the transition from “negligible” to “dominant” abundance.

Formula, definitions, and assumptions

The calculation assumes the smoothed overdensity δ is Gaussian distributed with variance σ²:

Formula: P(δ) = 1 / (sqrt(2 π σ^2)) e^−δ^2/(2σ^2)

$P\left(\delta \right)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{\delta }^2}{2{\sigma }^2}}$

The Press–Schechter estimate identifies the PBH formation fraction with the probability that δ exceeds the threshold δ_c. Integrating the Gaussian tail yields:

Formula: β = 1 / 2 erfc(δ_c / (sqrt(2) σ))

$\beta =\frac{1}{2}\mathrm{erfc}\left(\frac{{\delta }_c}{\sqrt{2}\sigma }\right)$

To translate β into a rough present-day dark-matter fraction at mass M (in solar masses), the calculator uses a common scaling for radiation domination and a standard thermal history:

Formula: f_PBH ≈ 1.06 × 10^8 β (M/M_☉)^−1/2

$f_{\text{PBH}}\approx 1.06\times {10}^8\beta {\left(\frac{M}{M_{☉}}\right)}^{-\frac{1}{2}}$

In code, this is implemented as f = (1.06e8 * beta) / sqrt(mass) where mass is in solar masses. The output also includes a qualitative classification (negligible/minor/significant/overcloses) based on the computed value of f_PBH.

Worked example (step-by-step)

Use the default threshold and mass: δ_c = 0.45 and M = 30 M_☉. Now try σ = 0.10. The calculator evaluates the complementary error function tail and returns a β of order 10⁻⁸. Plugging that into the scaling for f_PBH gives a present-day fraction of order 10⁻², meaning a percent-level contribution to dark matter.

Next, keep δ_c and M fixed but increase σ slightly, for example to 0.12. Because β depends on δ_c divided by σ, the tail probability increases rapidly. It is common to see β jump by many orders of magnitude for such a change. This is the main reason PBH abundance is a sensitive probe of the small-scale primordial power spectrum.

Reference table (intuition builder)

The table below lists β and f_PBH for a few representative values of σ at fixed mass 30 M_☉ and δ_c = 0.45. Values of f_PBH much larger than 1 would imply PBHs exceed the observed dark-matter density (often described as “overclosing” in this simplified context).

These values highlight the steep dependence on σ: moving from 0.05 to 0.10 increases β by roughly thirty orders of magnitude. In practice, researchers compare such estimates with observational constraints (microlensing, CMB anisotropies and spectral distortions, dynamical heating, wide binaries, accretion limits, and gravitational-wave merger rates). This calculator does not apply those constraints; it is intended to help you understand the mapping from fluctuation amplitude to abundance.

How to interpret β and f_PBH

The quantity β is defined at the time of PBH formation (roughly horizon reentry of the relevant scale). It is not the same as the present-day fraction of dark matter in PBHs. PBHs behave like nonrelativistic matter, so their energy density redshifts more slowly than radiation. As a result, even a tiny β at early times can correspond to a much larger fraction later. The approximate conversion used here captures that growth in a compact way, but it is still a simplified scaling.

The quantity f_PBH in this tool is best read as “the order-of-magnitude fraction of today’s dark matter that would be in PBHs of mass M if the formation fraction were β.” If f_PBH is far below 1, PBHs are a negligible component of dark matter at that mass. If f_PBH is near 1, PBHs could in principle make up most of dark matter, but only if the scenario is consistent with observational bounds. If f_PBH is much larger than 1, the parameter combination would overproduce PBHs in this simplified picture.

Limitations (what this model leaves out)

• Gaussian assumption: the β formula used here assumes a Gaussian distribution for δ. Even small non-Gaussianity can strongly enhance or suppress the tail probability.
• Single-threshold collapse: real PBH formation depends on perturbation shape, critical collapse effects, and the equation of state. The effective δ_c can vary with scale and model.
• Mass mapping: the conversion from β to f_PBH is an approximate scaling and does not include extended mass functions, accretion, evaporation, or changes in relativistic degrees of freedom.
• Constraints not applied: the calculator does not enforce observational bounds. A computed f_PBH near 1 does not imply viability without checking constraints at that mass.
• Numerical behavior: for extremely small σ, β may underflow to 0 in floating-point arithmetic; for very large σ, β approaches 0.5.

FAQ (common questions)

Introduction: Why does the result change so much when I change σ?

The complementary error function erfc computes the probability in the far tail of a Gaussian. When δ_c is fixed, the argument is proportional to 1/σ. Decreasing that argument slightly (by increasing σ) can move you from an exponentially suppressed regime to a much less suppressed one. That is why β can jump by many orders of magnitude.

What does “overcloses the universe” mean here?

In this simplified context, it means the computed f_PBH is greater than 1, i.e., the model would predict more PBH dark matter than the total observed dark-matter density. Real cosmological analyses are more nuanced, but f_PBH > 1 is a clear sign that the chosen parameters are inconsistent with a standard cosmological history.

Is δ_c always 0.45?

No. The threshold depends on the equation of state and on the shape of the perturbation. Values around 0.4–0.5 are often quoted for radiation domination, but different definitions of δ and different collapse criteria can shift the effective threshold. Treat δ_c as a model parameter.

Does this calculator include an extended PBH mass function?

No. It treats the input mass as a single representative scale and uses it only in the approximate conversion from β to f_PBH. Many realistic scenarios produce a distribution of PBH masses; in that case, constraints and abundances must be integrated over the mass function.

If you want to connect PBH abundance estimates to other black-hole observables, you can also explore the black hole shadow angular size calculator, estimate jet power with the Blandford–Znajek jet power calculator, or check detectability with the gravitational wave strain calculator.

Glossary (quick definitions)

PBH

Primordial black hole, a black hole that could form in the early universe from large density perturbations rather than from stellar collapse.

Overdensity δ

A dimensionless measure of how much denser a region is compared with the cosmic mean, after smoothing on a chosen scale.

σ (RMS)

The root-mean-square amplitude of δ on the smoothing scale. In this calculator it is treated as an input parameter.

δ_c (threshold)

The critical overdensity above which collapse to a PBH occurs in the simplified model.

β

The fraction of the total energy density that collapses into PBHs at the time of formation.

f_PBH

An approximate fraction of today’s dark matter in PBHs at the specified mass, computed from β using a standard scaling.