Horizon Mass, Critical Collapse, and Mass Functions

A primordial-black-hole mass is set jointly by the cosmological scale entering the horizon and by how far its perturbation lies above the collapse threshold. Consequently, even a sharply peaked primordial spectrum need not produce a monochromatic black-hole population. This page derives the mapping and separates the formation probability, formation energy fraction, and present dark-matter mass function—quantities that are often denoted by the same symbol but are not interchangeable.

Helpful background. Einstein Equations supplies the Friedmann constraint; Relativistic Matter and Compact Objects supplies the fluid equation of state; Black-Hole Dynamics and Collapse supplies critical scaling; Relativistic Cosmology supplies horizon entry and redshifting; and Probability and Statistical Inference supplies changes of random variables and normalized distributions.

Horizon mass fixes the cosmological scale

In a spatially flat FLRW background, let $t_H$ denote Hubble-radius crossing for the comoving scale of interest and define

$$M_H(t_H)=\frac{4\pi}{3}\rho(t_H)H(t_H)^{-3}.$$

Using $H^2=8\pi G\rho/3$ gives the useful dimensional identity

$$M_H=\frac{1}{2GH}.$$

For radiation domination, $H=1/(2t)$, so $M_H=t_H/G$ in units $c=\hbar=k_{\mathrm B}=1$. Associate a comoving mode with this reference mass by the declared crossing condition $k=a_HH_H$. Since $H\propto a^{-2}$ during ideal radiation domination, $M_H\propto k^{-2}$; entropy release adds a weak dependence on the relativistic degree count. The trapped surface generally appears later, at an apparent-horizon time $t_{\rm AH}\gt t_H$. “Horizon mass” is therefore the conventional FLRW Hubble-volume label at crossing, not the mass inside an already formed black-hole horizon.

The numerical coefficient in a quoted $M_H(k)$ relation depends on the crossing definition, $a_0$ normalization, thermal history, and whether $M_H$ or the eventual black-hole mass is reported. Those inputs must accompany any conversion between megaparsecs and grams or solar masses.

Critical collapse broadens a single formation scale

Near the threshold, relativistic collapse gives

$$M=K(q,w)M_H(t_H)\left(\delta_m-\delta_c\right)^{\gamma(w)}, \qquad \delta_m\gt\delta_c.$$

Here $\delta_m$ is the peak compaction amplitude, $q$ characterizes its shape, $K$ is dimensionless and profile-dependent, and $\gamma$ depends mainly on the equation of state. Radiation-fluid calculations give $\gamma\simeq0.36$. The law is asymptotic: the cited calibration identifies $\delta_m-\delta_c\lesssim10^{-2}$ as the critical regime, with deviations of order $10$–$15\%$ at larger amplitudes. Escrivà, Kühnel, and Tada 2025, § II.E, printed pp. 24–29, Eq. (II.30), and § III.A, printed pp. 93–95 document the numerical scaling and its use in abundance integrals, while the general critical-phenomena derivation belongs to Mass Scaling and Critical Exponents.

For one Hubble-crossing mass and a declared probability density $p(\delta_m)$, normalized over its amplitude domain, the probability in $d\delta_m$ is not yet a mass fraction: each selected patch contributes $M/M_H$. Normalize to the background Hubble patch at $t_H$, and call this reference fraction $\beta_H$. With the conventional Press–Schechter factor two,

$$d\beta_H=2\frac{M}{M_H}p(\delta_m)\,d\delta_m.$$

Invert the critical law and change variables:

$$\delta_m(M)=\delta_c+ \left(\frac{M}{K M_H}\right)^{1/\gamma},$$ $$\frac{d\beta_H}{d\ln M} =\frac{2}{\gamma}\frac{M}{M_H} \left(\frac{M}{K M_H}\right)^{1/\gamma} p\!\left[\delta_m(M)\right].$$

Dimensions now check term by term: $M/M_H$, $K$, $\delta_m$, and $\beta_H$ are dimensionless. The normalization can be checked without choosing a special $p$. Changing variables back from $M$ to $\delta_m$ gives

$$\int d\ln M\;\frac{d\beta_H}{d\ln M} =2\int_{\delta_c}^{\delta_{\max}}d\delta_m\; \frac{M(\delta_m)}{M_H}\,p(\delta_m) =\beta_H.$$

Here $\delta_{\max}$ is the upper end of the amplitude domain over which the profile calibration and probability density are being used; it is not silently infinity. At this single crossing scale, $\widehat\psi_H(M)=\beta_H^{-1}d\beta_H/d\ln M$ is therefore normalized. The hat distinguishes this unit-normalized shape from an unnormalized formation-abundance density used in a constraint. A mass-weighted diagnostic obeys $\langle X\rangle_H=\int d\ln M\,\widehat\psi_H(M)X(M)$. This check fails if the Jacobian or $M/M_H$ factor is omitted.

If $p$ is smooth and nonzero at threshold, the low-mass asymptote is $d\beta_H/d\ln M\propto M^{1+1/\gamma}$. The extra power of $M$ is the energy weight. A number distribution instead scales as $M^{1/\gamma}$ per logarithmic mass. Thus even one formation scale produces an extended tail, while its small masses need not dominate the energy density.

From formation fraction to a normalized mass function

PBHs redshift as nonrelativistic matter after formation, while radiation redshifts faster. The abundance normalization and the starting epoch must be translated together. For the fixed final-mass assignment above, conserved comoving patch number gives, at an epoch $a$ after collapse has completed,

$$\frac{d\beta(a)}{d\ln M} =\frac{\rho_b(a_H)}{\rho_b(a)} \left(\frac{a_H}{a}\right)^3 \frac{d\beta_H}{d\ln M}.$$

In ideal radiation domination this factor is $a/a_H$. If instead a simulation reports the actual fraction $\beta_f$ at apparent-horizon formation $a_f=a(t_{\rm AH})$, evolve that fraction by $a/a_f$. For this same fixed-mass patch assignment, $\beta_f=\beta_H a_f/a_H$, so the two descriptions agree. Using $\beta_H$ but starting its redshifting at $a_f$ would miss that factor. An amplitude-dependent formation time must be handled inside the population integral. Changes in particle degrees of freedom, subsequent mass evolution, or mergers require the full density and number transport rather than a single $a$ ratio.

For surviving populations transported to the present, denote the fraction density per logarithmic present mass by $\phi(M)$. Then

$$f_{\rm PBH}=\int d\ln M\;\phi(M), \qquad \psi(M)=\frac{\phi(M)}{f_{\rm PBH}}, \qquad \int d\ln M\;\psi(M)=1.$$

This page uses $\psi$ per logarithmic mass. A density defined per unit mass differs by $1/M$. That Jacobian matters when a likelihood or an “extended-mass-function” prescription expects $\int dM$ instead of $\int d\ln M$.

A sharp consistency check is the monochromatic limit. If critical scaling is deliberately replaced by $M=\alpha M_H$ and the primordial spectrum selects one horizon mass, then

$$\psi(M)\longrightarrow \delta_{\rm D}\!\left[\ln M-\ln(\alpha M_H)\right].$$

Integration over $d\ln M$ gives unity, while integration over $dM$ requires the corresponding $1/M$ Jacobian. A proposed mass-function formula that fails either normalization is not a different PBH model; it is inconsistent bookkeeping.

For a narrow curvature spectrum, one can integrate the critical-collapse distribution at essentially one $M_H(t_H)$. For a broad spectrum, contributions from many smoothing scales and correlated peaks overlap. Simply binning $\mathcal P_{\mathcal R}(k)$, computing independent mass functions, and summing can double-count the same fluctuation. A consistent peak/window prescription and thermal-history mapping are required. Green and Kavanagh 2021, §§ II.A.2–II.A.4, printed pp. 3–5 explain the Hubble-crossing mass, critical-collapse, and present-abundance conversions.

Results and limitations through 2026-08-19

As of 2026-08-19, extended PBH mass functions are expected generically, but their exact shapes remain unresolved outside specified model classes.

Formulation or evidence	Result	Scope and limitation
Friedmann horizon-mass mapping	Gives $M_H=(2GH)^{-1}$ and $M_H\propto k^{-2}$ in radiation domination	Depends on horizon-entry convention and the thermal history through $g_*$ and entropy transfer
Near-critical fluid simulations	Support $M\propto M_H(\delta_m-\delta_c)^\gamma$	$K$, $\delta_c$, and the valid amplitude interval depend on profile and equation of state
Narrow-spectrum analytic mapping	Produces an extended low-mass tail even at one $M_H$	Assumes a declared perturbation PDF and neglects overlap among formation scales
Broad-spectrum peak or binned treatments	Can generate broad and multimodal present distributions	Window choice, peak assignment, non-Gaussianity, and later accretion or mergers remain model-dependent

Monochromatic proxies are useful for publishing one-dimensional constraints; critical-collapse functions are physically better near threshold; and peak-theory mappings are needed when profile statistics matter. These are competing approximations over different parameter regimes, not mutually exclusive claims. The strongest open question is how accurately a broad primordial spectrum can be mapped to a mass function without double-counting correlated peaks or hiding thermal-history and nonlinear-collapse uncertainties.

The interpretation could change after a new collapse calibration changes $K$, $\gamma$, or $\delta_c$ across relevant equations of state, or after a validated broad-spectrum calculation demonstrates a materially different normalized mass function under the same primordial statistics and thermal history.

Check the critical-scaling interpretation

At fixed $M_H$, $K$, and radiation-era $\gamma=0.36$, compare perturbations with $\delta_m-\delta_c=0.0025$ and $0.0100$. Does the second PBH become four times as massive?

Solution

No. The ratio is

$$\frac{M_2}{M_1} =\left(\frac{0.0100}{0.0025}\right)^{0.36} =4^{0.36}\simeq1.65.$$

The amplitude excess grows by four, but critical scaling is sublinear because $\gamma\lt1$. Both chosen excesses lie within the stated near-threshold domain. The comparison says nothing about their relative abundance, which also requires the probability density evaluated at the two amplitudes.

Scope boundary

The result maps formation scale and collapse amplitude into a mass distribution. It does not prove the universal critical exponent, select an inflationary spectrum, evolve the population through accretion and mergers, or combine observational bounds. Those tasks require the canonical collapse treatment and the later pages in this chapter.

References

• Escrivà, Albert, Florian Kühnel, and Yuichiro Tada. 2025. “Primordial Black Holes.” arXiv:2211.05767v5. Open PDF.
• Green, Anne M., and Bradley J. Kavanagh. 2021. “Primordial Black Holes as a Dark Matter Candidate.” Journal of Physics G 48: 043001. DOI.