Primordial Black Hole Formation Fraction Calculator: β and f_PBH from σ

What the primordial black hole formation fraction estimate means

This calculator estimates how efficiently unusually large early-universe density fluctuations could seed primordial black holes (PBHs). In the horizon-reentry picture, a patch of the universe collapses only when the smoothed overdensity δ rises above the critical threshold δ_c at the moment that scale reenters the horizon. If δ stays below the threshold, expansion and pressure support keep the region from becoming a PBH.

The page uses a Gaussian-statistics Press–Schechter tail approximation to connect the RMS fluctuation amplitude σ to two outputs. You provide σ, the collapse threshold δ_c, and a PBH mass scale M in solar masses. The calculator then reports β, the early-time formation fraction, and f_PBH, an approximate present-day dark-matter fraction for PBHs at that mass.

The most important feature of this model is its extreme sensitivity to the tail of the overdensity distribution. That makes σ the dominant lever in the result: a small upward shift can push many more regions above δ_c, while a small downward shift can suppress collapse almost entirely. In simple Gaussian PBH models, that steep response is why abundance estimates are often described as finely balanced.

How to use the primordial black hole calculator

1. Start with the density-fluctuation RMS σ. For PBH-forming scales, viable models usually involve a much stronger small-scale fluctuation than the one inferred on CMB scales, so the input here is meant to represent that enhanced early-universe patch.
2. Set the collapse threshold δ_c. In radiation domination, simulations often place it near 0.4–0.5, and the default on this page is 0.45.
3. Choose the PBH mass M in solar masses (M_☉). In this simplified calculator, mass only enters the conversion from β into a rough present-day dark-matter fraction.
4. Click Compute, then use Copy Result if you want to transfer the β and f_PBH summary into notes, a draft, or a spreadsheet.

Practical tip: when you scan PBH scenarios, vary σ in small steps rather than large jumps. Because the calculation probes the far tail of a Gaussian, a coarse grid can miss the narrow transition between a negligible PBH abundance and a scenario that quickly becomes dominant.

PBH tail formula, definitions, and assumptions

This PBH estimate treats the smoothed overdensity δ as a Gaussian random variable with variance σ²:

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

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

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

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

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

In the code, the conversion 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: a PBH collapse estimate at the default threshold

Use the default collapse threshold δ_c = 0.45 and the default PBH mass M = 30 M_☉ as a baseline, then compare nearby values of σ. Even without pinning the result to a single number, you can see the pattern immediately: the tail probability is dominated by the ratio of δ_c to σ, so a modest increase in σ can noticeably raise β.

The useful lesson is not that one particular input set gives one particular abundance, but that the response is extremely steep. If you keep δ_c and M fixed and compare two nearby σ values, the darker matter classification can move from negligible to minor or significant much faster than most other parameters would suggest. That is exactly the behavior researchers look for when judging whether a PBH scenario is tuned too tightly to the Gaussian tail.

Reference table: qualitative σ-to-β behavior at 30 M_☉

The table below does not try to pin down exact β values. Instead, it shows the direction of change you should expect at fixed mass 30 M_☉ and δ_c = 0.45. The point is to compare how the tail opens up as σ moves from suppressed to less suppressed.

The important feature is the steepness of the response rather than any one row. In practice, researchers compare the calculator's output with observational constraints from 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 meant to show how σ maps into abundance in the simplified Press–Schechter picture.

How to interpret β and f_PBH in this PBH model

The quantity β is defined at the time PBHs would form, which is roughly the horizon-reentry moment for the relevant scale. It is not the same thing 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, a tiny early-time β can correspond to a much larger late-time share. The approximate conversion used here captures that growth in a compact way, but it is still only a simplified scaling.

In this calculator, f_PBH 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 only a minor component at that mass. If f_PBH is near 1, PBHs could in principle make up most of the dark matter, but only if the scenario survives the relevant observational bounds. If f_PBH is much larger than 1, the chosen parameter combination would overproduce PBHs in this simplified picture.

Limitations of the primordial black hole estimate

• Gaussian assumption: the β formula used here assumes a Gaussian distribution for δ. Even a small amount of 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.

PBH formation fraction FAQ

Why does the PBH formation fraction change so much when σ changes?

The complementary error function erfc measures the probability in the far tail of a Gaussian. When δ_c is fixed, the argument is proportional to 1/σ. Increasing σ therefore pushes more weight into the collapse tail, while decreasing it pulls the distribution away from the threshold. That is why β can change by many orders of magnitude from a very small adjustment in σ.

What does “overcloses the universe” mean for PBHs?

In this simplified context, it means the computed f_PBH is greater than 1, so 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 still a clear sign that the chosen parameters are inconsistent with a standard cosmological history.

Is δ_c always 0.45 in this calculator?

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 rather than a universal constant.

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, the abundance must be integrated over the mass function and compared against the relevant constraints across the full range.

If you are comparing primordial black hole abundance estimates with other compact-object calculators, 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.

PBH formation fraction glossary

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.