Mass and Mass-Ratio Distributions

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Compact-binary catalogs do not display the source mass distribution directly. They display uncertain detector-frame parameters after a strongly mass-dependent selection. Population features become credible only when they survive event uncertainty, source-frame conversion, correlated pairing models, catalog selection, waveform systematics, and reasonable changes in functional form. As of 2026-08-18, the data support structured black-hole mass distributions, but neither every fitted peak nor a unique stellar-evolution explanation is model independent.

Required background. Probability and Statistical Inference supplies density transformations and hierarchical modeling. Helpful background. Hierarchical Population Likelihoods supplies the selection-normalized likelihood; Parameter Degeneracies explains what an individual waveform constrains.

The joint mass distribution

Order component masses as $m_1\ge m_2$ and define $q=m_2/m_1\le1$ . A population density must specify a measure, for example

p(m_1,q\mid\Lambda)\,dm_1\,dq.

It is not numerically identical to $p(m_1,m_2)$ because $dm_2=m_1\,dq$ at fixed $m_1$ . Likewise, a distribution uniform in component masses is not uniform in chirp mass

\mathcal M=\frac{(m_1m_2)^{3/5}}{(m_1+m_2)^{1/5}}

or symmetric mass ratio $\eta=q/(1+q)^2$ . The Jacobian and domain boundaries are part of the model, not presentation details.

Event likelihoods primarily measure redshifted masses,

m_a^{\rm det}=(1+z)m_a^{\rm src}.

Converting to a source-frame mass distribution therefore couples the assumed cosmology and distance posterior to the population redshift model. A high detector-frame mass can mean a high source-frame mass, a larger redshift, or both.

A common factorization is

p(m_1,q\mid\Lambda) =p(m_1\mid\Lambda_m)\, p(q\mid m_1,\Lambda_q),

but independence of mass ratio from primary mass is an assumption. It can turn a genuinely two-dimensional ridge into a one-dimensional peak or force an excess into a “subpopulation.” Joint models should therefore be checked whenever the scientific claim concerns pairing.

Parametric and flexible descriptions

A truncated power law tests broad slope and endpoints. A power law plus a smooth peak tests a localized excess. Broken power laws allow slope changes; tapers avoid discontinuities that the data cannot resolve. Splines, Gaussian processes, histograms, and autoregressive processes relax the shape but introduce smoothing scales, binning, or stochastic-process priors.

Flexibility trades bias for variance. A strongly parametric family can report a narrow feature because the alternative shapes were never allowed. A highly flexible family can follow Monte Carlo noise or create prior-driven boundary structure. The appropriate question is not “parametric or nonparametric?” but whether the conclusion persists across models that can represent the relevant counterexample. Callister and Farr demonstrate this role for an autoregressive population model Callister and Farr 2024, §§ II–III.

The selection correction is model dependent:

\alpha(\Lambda)=\int dm_1\,dq\,dz\,d\xi\, p_{\rm det}(m_1,q,z,\xi) p(m_1,q,z,\xi\mid\Lambda),

where $\xi$ includes spins, orientation, and other parameters. More massive binaries may be visible farther away, yet very high total mass can move inspiral power below the analysis band. Unequal masses, higher modes, spin, and redshift further alter recovery. Correcting a mass histogram by a one-dimensional “volume” can therefore imprint false structure.

What survives current cross-checks

The official GWTC-5.0 analysis uses strongly and weakly parameterized models, event posteriors, catalog-specific injections, and a declared significance selection. Its broad conclusion is that the binary-black-hole mass distribution is structured rather than a single featureless power law. It reports a robust low-mass-scale peak and a higher-mass change of slope; the sharper interpretation of the latter as a peak, turnover, or gap edge depends on the joint model LVK 2026, GWTC-5.0 population analysis, §§2–3 and 5–7.

Evidence line at the cutoff	What it supports	What it does not establish
Official parametric and flexible reconstructions agree on broad structure	The source population is not well described by one unbroken power law over the analyzed domain	Every local maximum is a distinct birth channel
Joint $m_1$ – $m_2$ models resolve mass-dependent pairing structure	Mass ratio should not always be modeled independently of primary mass	A unique binary-evolution mechanism
Alternative flexible models broaden or move some features	Model uncertainty is scientifically material	That all structure is an artifact
Gap-targeted and waveform-systematics studies admit different edge behavior	A sharp pair-instability boundary is not a direct observable	The absence of pair-instability physics

The current release also finds no clear, model-independent empty pair-instability gap in the primary-mass distribution. That statement is weaker than “pair instability is absent.” Stellar physics predicts remnant-mass structure conditional on metallicity, nuclear rates, mass loss, rotation, binarity, and whether merger remnants or other channels refill the interval. Those formation calculations belong in Black-Hole Astrophysics; the detector result here is a distribution of merging objects.

Mass ratio is a conditional measurement

At leading inspiral order the chirp mass is measured much better than $q$ . Mass ratio competes with aligned spin in the phasing and with inclination and higher modes in amplitude structure. Population inference can sharpen $p(q)$ by pooling events, but it can also borrow strength from a restrictive component-mass or spin model.

A robust pairing claim should compare at least:

a separable $p(m_1)p(q)$ model;
a mass-dependent $p(q\mid m_1)$ model;
a joint two-dimensional reconstruction;
waveform variants with and without the relevant precession and higher modes;
selection injections that cover the low- $q$ region.

An apparent preference for equal masses may partly reflect detection efficiency and the finite support $m_2\le m_1$ . Conversely, a low- $q$ tail can be exaggerated where few injections carry enormous importance weights. Both directions require injection effective-sample-size maps.

Adversarial checks and resolution criteria

Mock-feature recovery. Inject a smooth distribution and fit a peak model; the false-feature rate calibrates how often selection and finite catalogs manufacture peaks.
Feature deletion. Generate from a distribution containing a known gap or peak and fit progressively more flexible families. Failure to recover it identifies smoothing bias.
Waveform perturbation. Repeat event and population inference with credible waveform alternatives. A mass feature moving more than its statistical width is not yet robust. Controlled injection and mock-catalog tests demonstrate how waveform differences can produce such shifts Das and Mukherjee 2026, §§ III–IV, PDF v1; that study is a failure-mode demonstration, not a reanalysis establishing which features in GWTC-5.0 are artifacts.
Leave-one-out analysis. Remove influential events and recompute the feature. This distinguishes a population-supported structure from an outlier classification.
Posterior predictive checks. Draw detected mock catalogs from the hyperposterior, pass them through selection, and compare two-dimensional mass summaries to the observed catalog.

The evidence ranking should be refreshed after a superseding official catalog, corrected event likelihoods or injections, a credible waveform-systematics result, or a flexible reconstruction that changes which features persist. Resolution of an astrophysical origin additionally requires forward formation models that predict masses, spins, redshift evolution, and rates together.

Exercises

1. The missing Jacobian. If $q=m_2/m_1$ , derive the relation between densities $p(m_1,m_2)$ and $p(m_1,q)$ .

Solution

At fixed $m_1$ , $m_2=m_1q$ , so the absolute Jacobian is

\left\lvert\frac{\partial(m_1,m_2)}{\partial(m_1,q)}\right\rvert=m_1.

Probability invariance gives $p(m_1,q)=m_1p(m_1,m_2)$ on the transformed domain. Omitting this factor changes the population prior and can change an inferred mass-ratio slope.

2. Selection and a false high-mass excess. Explain why directly comparing detected counts in equal source-mass bins can overstate the abundance of higher-mass binaries.