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Selection effects and injections

The catalogs are biased samples

Detections are not a fair sample of the binaries in the Universe. A heavy binary is louder and is detected out to much larger distances than a light one, so heavy binaries are strongly over-represented, and at a given mass nearby sources are favored. A population inferred from the raw catalog, a rate obtained by counting events, or a redshift inferred from the population, would all be wrong.

The correction needs the probability \(P_\text{det}(\theta)\) that the whole detection chain (detectors, noise, search pipelines, thresholds) detects a source of parameters \(\theta\). For real searches on real, non-stationary data with glitches, it cannot be computed analytically: the LVK measures it by simulation.

Search-sensitivity injections

The LVK draws a large number of simulated signals, the injections, with masses, spins, redshift, sky position and orientation drawn from a known density \(p_\text{draw}(\theta)\). For O3 and O4 the signals are added to the real detector data, and the same search pipelines that found the real events analyse them; each injection is recorded with the false-alarm rate (FAR) assigned by every pipeline. For O1 and O2 the LVK uses a semi-analytic estimate: the SNR each signal would have had in the measured noise spectrum of the time. The campaigns and their file format are described in Essick et al. 2025 [38]; the files are on Zenodo (Data sources).

Each injection file holds, per injection:

  • the source parameters;
  • the draw density \(\ln p_\text{draw}\) (in source-frame masses, redshift and Cartesian spin components);
  • a mixture weight: each observing period has its own draw, and the weights combine them into one set;
  • the FAR of each search, or the semi-analytic SNR;

and, globally, the number of signals drawn \(N_\text{gen}\) (including the many too weak to be recorded) and the total analysis time \(T\).

An injection is found if it passes the same criteria as the real events: lowest FAR below a threshold for the real injections, network SNR above a threshold for the semi-analytic ones.

Reweighting: one set of injections for every population

For a population model \(p_\text{pop}(\theta \mid \Lambda)\), the detectable fraction is

\[ \xi(\Lambda) = \int P_\text{det}(\theta)\, p_\text{pop}(\theta \mid \Lambda)\, d\theta \;\approx\; \frac{1}{N_\text{gen}} \sum_{j \,\in\, \text{found}} \frac{p_\text{pop}(\theta_j \mid \Lambda)}{p_\text{draw}(\theta_j)} . \]

The injections were drawn from \(p_\text{draw}\), not from the population being tested; weighting each found injection by \(p_\text{pop}/p_\text{draw}\) corrects for this (importance sampling). One set of injections thus serves for every population and every cosmology that an analysis tries, without running the searches again. With the mixture weights \(w_j\), the draw density of injection \(j\) is \(p_\text{draw}(\theta_j)/w_j\).

The sensitive volume-time of a population with a rate density \(R\) is

\[ \langle VT \rangle = T \cdot \frac{1}{N_\text{gen}} \sum_{j \,\in\, \text{found}} \frac{w_j\, p_\text{pop}(\theta_j)}{p_\text{draw}(\theta_j)}, \]

where \(p_\text{pop}\) includes the redshift distribution \(\frac{dV_c}{dz}\frac{1}{1+z}\) (see What a GW signal measures), so that the expected number of detections is \(N_\text{exp} = R \langle VT \rangle\).

The effective number of injections

The sum is a Monte Carlo estimate. If a few injections dominate it (the population puts its weight where few injections were found), the estimate is noisy. Its reliability is measured by the effective sample size

\[ n_\text{eff} = \frac{\left(\sum_j x_j\right)^2}{\sum_j x_j^2}, \qquad x_j = \frac{w_j\, p_\text{pop}(\theta_j)}{p_\text{draw}(\theta_j)} . \]

In a population analysis of \(N\) events, the selection term enters as \(\xi^N\), so its relative error is multiplied by \(N\). Farr (2019) [37] showed that \(n_\text{eff} > 4N\) keeps the resulting bias small; icarogw rejects the points of parameter space where this fails. The same statistic, computed over the PE samples of each event, measures the reliability of the per-event sums of the hierarchical likelihood.

A consequence: rare detections, common sources

Selection effects can invert the picture. Over O3–O4b there are 248 BBH detections for one BNS, yet the two merger rates per unit volume are comparable: a BNS is detectable only within a few hundred Mpc, a heavy BBH out to several Gpc, and the sensitive volume grows as the cube of the reach. In the rates mode, with a constant rate per comoving volume for both, ⟨VT⟩ is 0.045 Gpc³ yr for BNS and 6.0 Gpc³ yr for BBH, a factor ~130 that offsets most of the factor 248 in the counts (Merger rates). The LVK population analyses even find the BNS rate likely higher than the BBH one (GWTC-3: 10–1700 against 17.9–44 Gpc⁻³ yr⁻¹, Abbott et al. 2023 [12]).