primordial_black_holes

Upper limits on f_PBH, the fraction of the dark matter in primordial black holes of a single mass, from the merger rates: a PBH population merges at a rate that grows with f_PBH, and that rate may not exceed what the data allow. The physics is explained in Primordial black holes and sub-solar masses.

gwtc_analysis primordial_black_holes                       # 0.35-150 Msun, the default choices
gwtc_analysis primordial_black_holes --suppression lvk     # the suppression fit of the LVK sub-solar papers
gwtc_analysis primordial_black_holes --attribute none      # the observed events taken as astrophysical

Method

Merger rates of a PBH population

For PBHs of mass m, a fraction f of the dark matter, the mode uses two formation channels.

Early binaries, formed in the radiation era and merging today (Raidal et al. 2019 [123]; Hütsi et al. 2021 [124], eq. 2.1), with M = 2m and η = 1/4:

\[ R_\text{early} = 1.6\times10^{6}\ \text{Gpc}^{-3}\,\text{yr}^{-1}\; f^{53/37} \left(\frac{t}{t_0}\right)^{-34/37} \left(\frac{M}{M_\odot}\right)^{-32/37} \eta^{-34/37}\, S(f). \]

S < 1 is the suppression: binaries disrupted by other PBHs (S₁, through the number N̄ of PBHs around the pair) or by later encounters (S₂). Two choices, --suppression:

--suppression	S	Where it holds
hutsi (default)	S₁ S₂ of [124]: eqs. 2.2, 2.3, A.5 for S₁ (monochromatic: ⟨m²⟩/⟨m⟩² = 1, N̄ = 2f / (f + σ_M), σ_M = 0.004) and the z = 0 fit 2.7 for S₂	every f
lvk	2.3 × 10⁻³ f^−0.65, the fit of the LVK sub-solar papers [101], [122]	equal to S₁ S₂ at f = 1; 1.3 times smaller at f = 0.1, 2.9 times at f = 0.01; above 1, which is unphysical, below f ≈ 10⁻⁴

The two give the same limits near f = 1 and differ by up to ~6 at f ~ 0.01: the LVK numbers for the sub-solar searches are reproduced with lvk, while the catalog limits, at f ~ 10⁻⁵–10⁻³, need hutsi.

Late binaries, formed by capture in PBH clusters (Phukon et al. 2021 [125], eq. 2; Clesse & García-Bellido [126]): R_late = f × 420 Gpc⁻³ yr⁻¹ (2m)^(10/7) / (m²)^(5/7), linear in f and taken constant in time.

The limit on f is where the PBH rate equals the rate limit R₉₀ (solved in closed form for lvk and for the late binaries, numerically for hutsi). Values above 1 mean no constraint.

Rate limits from the data

Masses	Data	Rate limit
0.35–1 M☉	the sensitive volume-time of the LVK sub-solar-mass searches, O4a + O4b combined, per pipeline (GstLAL, MBTA, PyCBC) and chirp-mass bin (results-PBH.zip of Zenodo 23346454, [122]); m = 2^(1/5) × the bin's chirp mass	no significant candidate: R₉₀ = −ln(1 − 0.9) / VT = 2.30 / VT, as in the LVK papers
above 1 M☉	the search-sensitivity injections of rates (--sensitivity-release), reweighted to binaries with both masses log-uniform in [m e^−w, m e^w] (w = --mass-window); the GWOSC events with FAR ≤ --far-threshold whose median chirp mass lies in 2^(−1/5) [m e^−w, m e^w]	the 90% Poisson upper limit for n events, Γ⁻¹(0.9; n + 1) / VT (2.30 / VT for none)

• Only the sub-solar bins up to 1 M☉ are used, as in the LVK papers: the searches require a secondary below 1 M☉, so their higher chirp-mass bins hold unequal binaries, not the equal-mass binaries of a single PBH mass.
• Events are matched by chirp mass, the best-measured mass. The mass ratio is poorly measured and its median is pulled away from 1 by the priors, so a window on both component masses would miss equal-mass sources and give limits too tight by up to ~10. An unequal binary with the right chirp mass also counts, which is conservative for an upper limit.
• --attribute window (default) lets every such event be primordial; --attribute none takes n = 0, i.e. all the observed events are astrophysical: a tighter limit (early binaries down to 8.3 × 10⁻⁵ at 50 M☉ with the default injections, late binaries ~3 × 10⁻⁵) and a much stronger assumption.
• The early-binary volume includes the rate evolution (t/t₀)^(−34/37) (Planck15), so that VT × R(z = 0) is the expected number of detections; spins as in rates (|a| < 0.4 below 2.5 M☉, else 0.99).

Results

With the defaults (gwtc5 injections, 259 events with FAR ≤ 1/yr):

PBH mass	Early binaries, hutsi	Early binaries, lvk	Published
0.35 M☉ (sub-solar)	0.061–0.142	0.096–0.178	0.10–0.18 [122] (LVK fit)
1 M☉ (sub-solar)	0.0028–0.0059	0.018–0.032	0.018–0.032 [122] (LVK fit)
1.2–2.5 M☉ (catalog)	0.0013–0.0017	0.004–0.007	~2 × 10⁻³ at 1.5 M☉ [102] (astrophysics only)
12–38 M☉ (catalog)	0.0006–0.0008	0.0007–0.0015	~1.5–2 × 10⁻⁴ [102] (astrophysics only)
~100 M☉ (catalog)	1.1 × 10⁻⁴	(S > 1: not valid)	~1 × 10⁻⁴ at 50–90 M☉ [102] (astrophysics only)

Ranges over the three pipelines for the sub-solar searches. Two checks:

• The LVK sub-solar limits are reproduced exactly with their suppression fit (--suppression lvk): 0.0183–0.0324 at 1 M☉, 0.0958–0.178 at 0.35 M☉; late binaries cross f = 1 at 0.62–0.74 M☉ (linear interpolation between the bins, as in the LVK script), the paper's 0.62–0.74 M☉.
• The two data sets agree where they meet: at 1–2.6 M☉ the sub-solar VT (unequal binaries, not used above 1 M☉) and the GWTC injections give the same early-binary limit, 0.0013–0.0028.

Late binaries give weaker limits below ~50 M☉ and tighter ones above: for equal masses their rate, 420 × 2^(10/7) f ≈ 1130 f Gpc⁻³ yr⁻¹, does not depend on m, while the early-binary rate falls as M^(−32/37).

Relation to Andrés-Carcasona et al.

The catalog limits of [102] (O1–O4a, 161 events with FAR < 1/yr) use the same early-binary rate and the same suppression: their eq. A.1 is eq. 2.1 of Hütsi et al. [124], with the suppression "from the analytical prescriptions" of that paper, i.e. --suppression hutsi. The rest differs:

	[102]	primordial_black_holes
PBH channels	early two-body + early three-body (γ = 1, K = 4)	early two-body; late binaries in clusters
Mass function	monochromatic and log-normal	monochromatic
Detection probability	semi-analytic, single-detector SNR > 8 with representative noise spectra	the LVK search-sensitivity injections, FAR ≤ 1/yr (as in rates)
Statistics	hierarchical Bayesian likelihood (dynesty, bilby)	Poisson upper limit in a chirp-mass window

They treat the observed events in three ways:

1. Astrophysics only: all events are astrophysical; the 95 % bound is where the expected number of PBH detections N < 3. These are the solid curves of their Fig. 1 and the [102] values in the tables (~10⁻² at 0.5 M☉, ~2 × 10⁻³ at 1 M☉, ~2 × 10⁻⁴ at 10 M☉, ~1 × 10⁻⁴ at 50–90 M☉; their text's "between 10⁻² and 10⁻⁴" spans 0.5–90 M☉). Counterpart here: --attribute none (2.30 / VT, 90 %).
2. Agnostic: any subset of events may be primordial (10⁵ random subsets, or a subset-marginalised likelihood); slightly weaker at 2–20 M☉. Rough counterpart: --attribute window (every event in the window may be primordial).
3. Astrophysically informed: a joint fit of a power law + 2 peaks (Multi Peak) astrophysical population and a log-normal PBH population. No counterpart.

Why the mode does not reproduce their analysis. It extends the method of the LVK sub-solar searches, a rate limit from the published search sensitivity, which it reproduces exactly below 1 M☉, to the catalog masses with the same LVK injections as rates: one framework from 0.35 to 150 M☉, from public data, in minutes. Their case 1 is a comparable computation (and the three-body channel or a log-normal mass function could be added at little cost); cases 2 and 3 need the posterior samples of all events and nested sampling over ~16 population parameters, or 10⁵ subsets, hours to days of computing.

Comparison with their case 1. --attribute none with --sensitivity-release gwtc4 (O3 + O4a, their data period) against the monochromatic constraint of their Fig. 1 (read from the figure, ±20–30 %):

PBH mass	this mode, none, gwtc4 (90 %)	[102], astrophysics only (95 %)
1.5 M☉	1.8 × 10⁻³	~1.8 × 10⁻³
3 M☉	9.1 × 10⁻⁴	~8 × 10⁻⁴
10 M☉	3.2 × 10⁻⁴	~2.3 × 10⁻⁴
20 M☉	1.8 × 10⁻⁴	~1.5 × 10⁻⁴
30 M☉	1.3 × 10⁻⁴	~1.2 × 10⁻⁴
50 M☉	1.1 × 10⁻⁴	~1.0 × 10⁻⁴
90 M☉	1.4 × 10⁻⁴	~1.1 × 10⁻⁴
120 M☉	2.0 × 10⁻⁴	~1.5 × 10⁻⁴

Agreement within a factor 1.0–1.4, despite the different detection probability (injections against SNR > 8), the missing three-body channel and 90 % against 95 %. The ~10⁻⁵ level of their Fig. 1 is the dashed "sensitivity floor" (expected detections N > 0.05), not a constraint. The default --attribute window is 2–4 times weaker at 10–50 M☉, where events fall in the window, as their agnostic case is weaker than their case 1.

Limitations

• Not the analysis of Andrés-Carcasona et al. [102]; see the section below. Monochromatic mass function only, no three-body channel, and the catalog part is a Poisson count in a mass window, not a hierarchical fit.
• No search: the mode uses the published search sensitivities and the catalog; a sub-solar candidate below the catalog threshold, or an alert (HasSSM), does not enter.
• A sub-solar merger need not be primordial: collapsar disks may fragment into sub-solar neutron stars that merge (Metzger et al. 2024 [103]); unlike PBHs, that channel shines.
• The S₂ fit of [124] is for z = 0; its redshift dependence is not included.

Options

Option	Default	What it provides	When to change it
Masses
--masses	22 values from e^w to 150 M☉	the PBH masses of the catalog part; each must satisfy m e^−w ≥ 1 M☉, where the injections start	a finer grid, or a few masses of interest
--mass-window	0.2	w, the half-width of the window in ln m: binaries in [m e^−w, m e^w] for the VT, chirp masses in 2^(−1/5) [m e^−w, m e^w] for the events	smaller for a sharper mass function (fewer injections: the effective sample size is in the TSV)
--no-sub-solar	off	leave out the sub-solar searches (0.35–1 M☉)	to skip the download of the sub-solar release
Physics choices
--suppression	hutsi	the suppression of the early-binary rate: S₁ S₂ of Hütsi et al. 2021, or the LVK fit 2.3 × 10⁻³ f^−0.65	lvk to reproduce the LVK sub-solar papers
--attribute	window	which observed events may be primordial: those with their chirp mass in the window, or none	none for the limit if all the events are astrophysical
Data
--far-threshold	1	FAR threshold [1/yr] for the found injections and the events	as in rates
--sensitivity-release	gwtc5	the LVK search-sensitivity injections of the catalog part, as in rates	gwtc4 for O3 + O4a only
--sensitivity-file	—	a local injection file instead of the release	offline runs
--data-repo	zenodo	where the injections and the sub-solar release are downloaded from: zenodo, s3 (the MinIO mirror) or galaxy	s3 for faster downloads
Outputs
--out-limits	pbh_limits.tsv	one row per mass (and pipeline): VT, effective sample size, events counted, R₉₀, the early- and late-binary limits	—
--out-report	pbh_limits.html	the HTML report	—
--plots-dir	pbh_plots	the plot of the limits against the mass	—