Accretion, Clustering, and Merger Histories

The abundance produced in the early universe is not yet a prediction for a late-time merger catalog. Accretion can move objects through mass and spin space; clustering changes encounter rates; and early binaries can be disrupted by the very environments that enhance late captures. The central result is therefore a conditional population integral, not a universal rate. A bounded deterministic benchmark below verifies normalization and quadrature sensitivity while deliberately omitting the astrophysics needed for a forecast.

Helpful background. Einstein Equations supplies gravitational dynamics; Relativistic Matter and Compact Objects supplies accretion and environmental physics; Black-Hole Dynamics and Collapse supplies binary and merger evolution; Relativistic Cosmology supplies expansion and halo growth; and Probability and Statistical Inference supplies population distributions, selections, and uncertainty propagation.

Population histories are conditional integrals

Write $u=\ln(M/m_c)$ and normalize a lognormal distribution per logarithmic mass,

$$\psi(u)=\frac{1}{\sqrt{2\pi}\sigma} \exp\!\left(-\frac{u^2}{2\sigma^2}\right), \qquad \int_{-\infty}^{\infty}\psi(u)\,du=1.$$

A schematic observable assembled from two PBHs and a redshift-dependent channel is

$$\mathcal I_X= \int du_1\,du_2\,dz\; \psi(u_1)\psi(u_2)w(z) K_X(u_1,u_2,z\mid\boldsymbol\theta),$$

where $w$ is normalized over the declared redshift interval and $\boldsymbol\theta$ contains environmental assumptions. Turning $\mathcal I_X$ into a physical merger rate requires a dimensionful normalization, PBH fraction, spatial correlation function, binary-formation probability, disruption history, cosmology, detector-frame selection, and avoidance of double-counted channels.

Early binaries may decouple from expansion through nearest-neighbor gravity and external tidal torques. Later halos permit two-body or few-body captures. Primordial or Poisson clustering can raise encounter densities, yet dense environments can ionize early binaries. Accretion changes masses, spins, drag, and detectability. Green and Kavanagh 2021, § II.A.5 and §§ III.D, III.F review these opposing effects and their model dependence.

A bounded deterministic benchmark

The checked repository benchmark uses Node.js 24.18.0, IEEE-754 binary64 arithmetic, and the standard library only—no external packages. Because the declared toy kernels factor into one-dimensional Gaussian moments and redshift moments, it applies composite Simpson integration separately on $u\in[-8\sigma,8\sigma]$ and $z\in[0,10]$ rather than constructing a three-dimensional grid, with $m_c=30M_\odot$ and $\sigma=0.6$. The mass scale is a label only because every kernel is dimensionless. Results should agree across conforming runtimes to ordinary floating-point rounding; platform agreement would still test implementation portability, not add astrophysical evidence.

The redshift weight is $w(z)=0.35e^{-0.35z}/(1-e^{-3.5})$ on $0\le z\le10$. Its finite-interval integral is one, which the code also verifies numerically. A toy accretion switch replaces $x=M/m_c=e^u$ by

$$x_{\rm eff}=\exp\!\left[u+g\frac{z}{1+z}\right], \qquad g=0.2.$$

The isolated early-binary and environmental-capture verification kernels are

$$K_E=(x_{1,{\rm eff}}x_{2,{\rm eff}})^{0.25},$$ $$K_C=(x_{1,{\rm eff}}x_{2,{\rm eff}})^{0.6} \left[1+c\left( \frac{(u_1-u_2)^2}{2\sigma^2}+\frac{z}{1+z} \right)\right], \qquad c=0.75.$$

These exponents and switches are diagnostic choices, not calibrated population physics. With $g=c=0$, independence of the two Gaussian $u$ variables gives exact reference values $\mathcal I_E=e^{0.25^2\sigma^2}$ and $\mathcal I_C=e^{0.6^2\sigma^2}$.

Download the MIT-licensed standard-library benchmark, then use the standalone download-and-run recipe:

curl -fL https://blackhole.org/code/cosmology/cosmology-benchmarks.mjs -o cosmology-benchmarks.mjs
node cosmology-benchmarks.mjs --case pbh-population

Convergence and exact null-limit checks

The checked execution on this environment produced:

Simpson panels $N$	Mass normalization result	Redshift normalization result	Combined capture result	Successive change
16	0.995205416447	1.000012648852	2.9082879264	—
32	0.999999998216	1.000000793929	2.9019470085	$2.180292\times10^{-3}$
64	1.000000000000	1.000000049674	2.9020684004	$4.183119\times10^{-5}$
128	1.000000000000	1.000000003105	2.9020767112	$2.863748\times10^{-6}$
256	1.000000000000	1.000000000194	2.9020772439	$1.835583\times10^{-7}$
512	1.000000000000	1.000000000012	2.9020772774	$1.154689\times10^{-8}$

At $N=512$, the early null kernel is $1.022755034177$ versus the analytic $1.022755034164$; the capture null kernel is $1.138372943079$ versus $1.138372943065$. Relative to their own null cases, accretion changes the early kernel by $6.1656\%$; accretion alone changes capture by $15.5454\%$; clustering alone by $119.6680\%$; and both switches by $154.9320\%$. Thus numerical refinement is controlled far below the imposed model sensitivity.

Results and limitations through 2026-08-19

As of 2026-08-19, the existence of binary formation and environmental evolution is established, but the PBH contribution to observed merger populations remains unresolved and model-dependent.

Formulation or evidence	Result	Scope and limitation
Analytic early-binary models	Map nearest-neighbor separations and tidal torques into a merger-time distribution	Commonly assume initial spatial statistics, negligible later disruption, and simplified mass functions
Halo-capture and clustering models	Dense environments can enhance late encounters and hierarchical mergers	Halo profiles, substructure, relaxation, ejection, and non-Poisson clustering are uncertain
Accretion calculations	Can move masses and spins and modify radiation histories	Geometry, feedback, relative velocities, duty cycle, and Eddington treatment strongly affect results
Gravitational-wave catalogs and null searches	Constrain selected population mixtures and unusual mass regimes	Selection, astrophysical backgrounds, mass/spin priors, and channel degeneracies prevent a universal PBH fraction
Deterministic benchmark on this page	Verifies normalization, analytic moments, switch isolation, and quadrature convergence	Dimensionless toy kernels are not rates, likelihoods, population synthesis, or catalog reproduction

The competing positions include initially Poisson versus primordially clustered populations, durable versus disrupted early binaries, late halo capture, and suppressed versus enhanced accretion. Escrivà, Kühnel, and Tada 2025, § II.M and §§ VII.A–VII.D survey how these assumptions propagate into gravitational-wave signatures. Agreement with one observed rate under one parameter regime does not identify a primordial channel.

The interpretation could change after validated population simulations converge across disruption and clustering prescriptions, or after a versioned gravitational-wave catalog and public likelihood isolates a population feature that materially narrows the dominant environmental and selection uncertainties.

Failure modes and interpretation

A production calculation must vary mass and spatial distributions, time step, environmental cutoff, merger-tree resolution, random seed, and selection function. Accretion transports the same objects through mass bins, whereas a merger replaces two objects by one remnant and radiates part of their energy. The evolution must account for those number and energy changes, and prevent the same binary from entering both an early and capture channel. The deterministic benchmark catches quadrature, normalization, and switch-coupling errors only. It cannot validate a physical kernel, quantify theoretical uncertainty, or supply detector-frame rates.

Read a converged toy result correctly

At the finest grid, the combined capture integral changes by $1.15\times10^{-8}$ on refinement but lies $154.9320\%$ above its no-effect kernel. Does this establish a precise physical enhancement?

Solution

No. It establishes that Simpson quadrature accurately integrates the declared toy kernel. The much larger switch response shows that the code transmits the chosen accretion and clustering factors, not that those factors describe nature. A physical claim would additionally require calibrated dimensionful rates, initial clustering, binary disruption, accretion feedback, catalog selection, and uncertainty propagation. Numerical convergence can eliminate one failure mode while leaving model error dominant.

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.