Self-Similar Mass Spectra of Hierarchical Black Hole Mergers
Abstract
Black holes that merge repeatedly carry a record of that history in their mass distribution. We treat hierarchical merging as a coagulation problem, in which a single kernel encodes how the merger rate depends on the masses involved and on cosmic time, and we derive the late-time spectra that such populations approach. The framework yields closed scaling laws, an exactly solvable benchmark, and a classification of merger environments by a single exponent measuring how strongly mergers favor or suppress massive participants. Applying it to primordial black holes requires care in translating published merger rates into kernel form; done correctly, three of the four standard binary-formation channels map onto superlinear kernels and are therefore candidates for runaway growth, in which the heaviest objects would dominate and no steady mass-conserving spectrum exists; establishing physical gelation, however, requires the full population-dependent kernel and its finite coagulation history, neither of which we settle here. Only the early three-body channel maps into the nongelling regime and admits slow self-similar growth. Numerical solutions across thirteen suppressive, nongelling kernels show that the shape of the spectrum is not fixed by the scaling exponent alone, and that accounting for the energy radiated at each merger measurably changes the high-mass cutoff and drains the population's total black-hole mass.
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BibTeX
@misc{blum2026selfsimilar,
title = {{Self-Similar Mass Spectra of Hierarchical Black Hole Mergers}},
author = {Liam Blum and Stefano Profumo and Ryan Schantz and Quoc Ha Tran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21033},
eprint = {2609.21033},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21033},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Self-Similar Mass Spectra of Hierarchical Black Hole Mergers
AU - Liam Blum
AU - Stefano Profumo
AU - Ryan Schantz
AU - Quoc Ha Tran
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21033
UR - https://arxiv.org/abs/2609.21033
AB - Black holes that merge repeatedly carry a record of that history in their mass distribution. We treat hierarchical merging as a coagulation problem, in which a single kernel encodes how the merger rate depends on the masses involved and on cosmic time, and we derive the late-time spectra that such populations approach. The framework yields closed scaling laws, an exactly solvable benchmark, and a classification of merger environments by a single exponent measuring how strongly mergers favor or suppress massive participants. Applying it to primordial black holes requires care in translating published merger rates into kernel form; done correctly, three of the four standard binary-formation channels map onto superlinear kernels and are therefore candidates for runaway growth, in which the heaviest objects would dominate and no steady mass-conserving spectrum exists; establishing physical gelation, however, requires the full population-dependent kernel and its finite coagulation history, neither of which we settle here. Only the early three-body channel maps into the nongelling regime and admits slow self-similar growth. Numerical solutions across thirteen suppressive, nongelling kernels show that the shape of the spectrum is not fixed by the scaling exponent alone, and that accounting for the energy radiated at each merger measurably changes the high-mass cutoff and drains the population's total black-hole mass.
ER -