Merger Rates and Sensitive Spacetime Volume

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An observed count becomes an astrophysical merger rate only after dividing by the spacetime region in which the search could have found the population. That region is not a geometric sphere fixed by a single “range”: it depends on masses, spins, orientation, sky location, redshift, detector state, pipeline, threshold, waveform family, and the assumed source population. The correct quantity is a selection-weighted spacetime volume, estimated with the same search that created the catalog.

Required background. Probability and Statistical Inference supplies Poisson likelihoods and marginalization. Helpful background. Hierarchical Population Likelihoods develops the point-process framework, and Selection Functions and Injection Campaigns explains catalog-specific efficiency.

From a source-frame rate to detected counts

Let $\lambda$ collect intrinsic source parameters and let $\mathcal R(z,\lambda\mid\Lambda)$ be a merger-rate density per source-frame time, comoving volume, and $d\lambda$ . During detector-frame observing time $T_{\rm obs}$ , the expected detected count is

\mu(\Lambda)=T_{\rm obs} \int dz\,d\lambda\, \frac{dV_c}{dz} \frac{\mathcal R(z,\lambda\mid\Lambda)}{1+z} p_{\rm det}(z,\lambda).

The factor $(1+z)^{-1}$ converts source-frame time to detector-frame time. The comoving-volume element and the map between luminosity distance and redshift require a declared cosmology. Detector-frame masses obey $m^{\rm det}=(1+z)m^{\rm src}$ , so sensitivity and population modeling cannot generally be separated by changing a label after the integral.

For a local, non-evolving rate $R_0$ and a normalized parameter distribution $\pi(\lambda)$ , define

\langle VT\rangle_\Lambda =T_{\rm obs}\int dz\,d\lambda\, \frac{dV_c}{dz}\frac{1}{1+z} \pi(\lambda\mid\Lambda)p_{\rm det}(z,\lambda),

so that $\mu=R_0\langle VT\rangle_\Lambda$ . The dimensions are volume times source-frame time. A quoted detector “range” is an orientation-averaged distance diagnostic under a specified signal model; $4\pi D_{\rm range}^3T/3$ is not automatically the search’s $\langle VT\rangle$ .

If $N$ signals are known with certainty, the count likelihood is

p(N\mid R_0,\Lambda) =\frac{(R_0\langle VT\rangle_\Lambda)^N}{N!} e^{-R_0\langle VT\rangle_\Lambda}.

The intuitive estimator $N/\langle VT\rangle$ is the maximum-likelihood rate for fixed $\Lambda$ , but a reported interval also depends on the rate prior, uncertainty in $\Lambda$ , uncertain candidate membership, and Monte Carlo error in $\langle VT\rangle$ . Rate and population shape must therefore be inferred together whenever sensitivity varies across the population.

Measuring sensitive spacetime volume with injections

Suppose injections are drawn from a normalized density $p_{\rm inj}(z,\lambda)$ , placed uniformly over the detector live time and its observing states, and recovered with indicator $I_k$ . Importance sampling gives

\langle VT\rangle_\Lambda \approx \frac{1}{N_{\rm inj}} \sum_{k=1}^{N_{\rm inj}} I_k \frac{ T_{\rm obs}(dV_c/dz)_k(1+z_k)^{-1} \pi(\lambda_k\mid\Lambda) }{p_{\rm inj}(z_k,\lambda_k)},

This is the local, non-evolving rate model defined above; a redshift-evolution factor belongs in the numerator when that model is generalized. Equivalent estimators may instead include injection time explicitly or absorb the spacetime measure into the injection distribution. The numerator and denominator must then change together so that live time is counted exactly once.

An injection is “found” only if it satisfies the same data-quality selection, coincidence logic, ranking statistic, and catalog threshold as a real candidate. Software injections test the analysis response but not every instrumental coupling; hardware injections provide valuable end-to-end checks but sparsely sample source parameters. Both differ from adding an analytic waveform to ideal Gaussian noise.

The estimator also has an effective sample size. If recovered weights are $w_k$ , then

N_{\rm eff}=\frac{(\sum I_kw_k)^2}{\sum I_kw_k^2}.

Large total injection counts can hide poor support at a mass edge, high redshift, or an extreme spin. A rate posterior extending into a region with small $N_{\rm eff}$ is partly an extrapolation of the injection proposal.

Live time is part of sensitivity

Detector duty cycle is not captured by multiplying a stationary horizon distance by calendar duration. Sensitivity changes with detector combinations, noise spectra, calibration epochs, vetoed intervals, and pipeline configurations. A defensible analysis partitions time into sufficiently homogeneous segments or injects across the real observing state. Network sensitivity also depends on the scientific selection: a coincident compact-binary search, a coherent burst search, and a low-latency alert stream can assign different outcomes to the same simulated source.

For a union of epochs $a$ ,

\langle VT\rangle=\sum_a \langle VT\rangle_a,

provided the epochs are disjoint and each uses its own search configuration. Averaging PSDs first is generally not equivalent because detection probability is nonlinear in signal-to-noise ratio and background.

Uncertain candidates and rate mixtures

Near threshold, the count is not simply an integer number of astrophysical events. Signal and terrestrial triggers form competing point processes. Their rate likelihood has factors of the form

\mu_{\rm s}\ell_{{\rm s},i}+\mu_{\rm n}\ell_{{\rm n},i},

with Poisson normalization for both intensities. Summing catalog $p_{\rm astro}$ values is a useful descriptive effective count, but it is not a universal replacement for this likelihood because $p_{\rm astro}$ depends on the assumed source population and pipeline background.

The GWTC-5.0 search, candidate, and injection products are the current example through the evidence cutoff 2026-08-18. A numerical rate based on them is conditional on the exact catalog version, observing segments, inclusion criterion, search pipelines, waveform injections, calibration state, cosmology, and population family LVK 2026, GWTC-5.0 population analysis, §§2–3 and selection appendices. This page therefore teaches the conversion rather than freezing a mutable rate interval into an evergreen statement.

Failure tests

Distance-only sensitivity. Replace the injection selection by a sharp luminosity-distance cutoff. A substantial change identifies orientation, mass, noise, or threshold structure that the sphere approximation erased.
Population dependence. Reweight the same recovered injections to at least two plausible mass–spin–redshift models. If $\langle VT\rangle$ changes, quote a jointly inferred rate rather than a population-free value.
Epoch closure. Sum independently estimated epoch sensitivities and compare with an injection set spanning the full run.
Mock-count coverage. Draw Poisson counts from a known rate, include background triggers, and verify interval coverage or posterior calibration under the declared procedure.
Waveform and calibration stress. Repeat recovery with plausible signal-model and calibration perturbations when they change ranking or parameter support.

Exercises

1. A Euclidean limit. A search detects every isotropically distributed source within distance $D$ and none outside, operates for time $T$ , and has a constant local source-frame rate. Neglect cosmological corrections. Find $\langle VT\rangle$ .

Solution

The detection probability is the indicator of the sphere, so

\langle VT\rangle =T\int_0^D4\pi r^2dr =\frac{4\pi}{3}D^3T.

This familiar expression is valid only because detectability was assumed independent of orientation and source parameters and the spacetime was taken as Euclidean.

2. Why the rate changes with the mass model. Population A has larger detector-frame chirp masses than population B and is visible to greater distance. Both models are fitted to the same effective signal count. Which infers the smaller rate?