Completeness and Search Heterogeneity

A gravitational-wave catalog is never simply “complete.” Completeness is conditional on a source population, detector time, search family, ranking threshold, waveform support, and reporting rule. Different observing epochs and pipelines cover different regions of parameter space, so a cumulative catalog is a union of heterogeneous selection functions rather than a uniform census.

Helpful background. Selection Functions from Injections defines detection efficiency, while From Search Triggers to Catalog Events separates pipeline records, de-duplication, and the parameter-estimation subset.

Completeness is a conditional probability

For source parameters $\theta$, data state $x$, search collection $\mathcal P$, and catalog rule $\mathcal S$, define

$$C(\theta,x;\mathcal P,\mathcal S) =P(\text{included}\mid\theta,x,\mathcal P,\mathcal S).$$

This is a selection function with an interpretation tailored to the catalog claim. A statement such as “complete above a mass” is incomplete unless it also fixes distance or redshift, spins, orientation, waveform morphology, detector sensitivity, analyzed time, and threshold. Even a source that would be loud when optimally oriented may be missed in an unfavorable orientation or detector state.

A useful bounded claim has the form:

For population family $\pi(\theta\mid\Lambda)$, analyzed epochs $E$, search versions $\mathcal P$, and inclusion rule $\mathcal S$, the injection-estimated efficiency exceeds a stated value over a declared parameter region, with a stated Monte Carlo uncertainty.

It is an empirical statement about the complete procedure, not a theorem inferred from nominal horizon distance.

Heterogeneous observing epochs

Let epoch $e$ have analyzed duration $T_e$ and efficiency $p_{\mathrm{det},e}(\theta)$. For a source rate stationary in detector time, the exposure-weighted efficiency is

$$\bar p_{\mathrm{det}}(\theta) = \frac{\sum_e T_e p_{\mathrm{det},e}(\theta)}{\sum_e T_e}.$$

This average remains valid at fixed $\theta$ when the astrophysical rate evolves with redshift: cosmic evolution is not a change between months of detector observing time. The cosmological rate, volume, and time-dilation factors enter when integrating over $\theta$. A genuinely changing rate or population across detector epochs instead requires rate-weighted exposure. Calibration, detector network, duty cycle, PSD, data-quality cuts, bank coverage, and threshold can all change between epochs. Preserving $p_{\mathrm{det},e}$ separately is usually safer than compressing everything into one representative sensitivity.

Multiple searches are not independent

For searches $P_1,\ldots,P_K$, catalog inclusion is a union:

$$p_{\mathrm{det}}^{\mathrm{union}}(\theta) =P\!\left(\bigcup_{k=1}^K D_k\middle|\theta\right).$$

Only under conditional independence would this reduce to

$$p_{\mathrm{det}}^{\mathrm{union}} =1-\prod_{k=1}^K\bigl(1-p_{\mathrm{det},k}\bigr).$$

Searches using the same strain are correlated through signal strength, detector state, glitches, and overlapping waveform banks. The union efficiency should therefore be measured by passing common injections through the actual searches and catalog association rule. Adding per-pipeline efficiencies can exceed unity and double-count sources.

Different searches can still be complementary. A modeled compact-binary search may be highly efficient for quasicircular signals inside its bank, while a burst search may recover unusual morphologies. “Complete for compact binaries” then depends on which waveform departures remain inside the combined injection support.

Four kinds of incompleteness

Source	Mechanism	Required diagnostic
Detector-time gaps	No valid network data or insufficient duty cycle	Analyzed-time and network-state accounting
Search support	Bank or morphology does not represent the signal	Injection recovery across the claimed source domain
Threshold and background	Candidate falls below the catalog rule	Rank/FAR-dependent efficiency using the same rule
Product heterogeneity	Event is included but lacks uniform parameter analysis	Explicit event-product mask and version provenance

The last row matters because a catalog may be complete under an inclusion rule yet incomplete for a downstream question requiring homogeneous posterior samples. A population analysis must either model that second-stage selection or restrict itself to a documented subset.

The role of selection normalization in event and hierarchical likelihoods is derived by Thrane and Talbot 2019, § 5 and Appendices E–F, corrected PDF v9.

Comparing cumulative catalogs

Catalog versions often combine several observing runs and multiple pipelines. A scientifically meaningful comparison keeps separate:

• newly analyzed detector time;
• changed search or background methods;
• changed inclusion thresholds or source classes;
• revised association and de-duplication;
• updated calibration or data quality;
• added or revised parameter-estimation products; and
• corrected or withdrawn records.

A larger catalog need not be uniformly more complete in every source region. New data can add exposure while a changed pipeline or threshold alters classification at the boundary. Completeness is assessed through the version-matched injection ensemble, not row count.

What current results establish and how to check them

The empirical cutoff is 2026-08-18. Current catalog scope and version-specific inclusions come from GWTC-5.0 Observations, §§ 2–4 and release tables; the contributing searches, backgrounds, and sensitivity studies come from GWTC-5.0 Methods, §§ 2–3 and the official release products. No current count or completeness boundary is promoted to an evergreen claim.

An independent mock-catalog check draws a known population, assigns injections to the actual epoch distribution, runs or emulates each declared search and threshold, applies the same de-duplication, and compares recovered with injected distributions. It should vary epoch weights, pipeline combinations, thresholds, and injection proposals. Failure is indicated by biased recovery, low effective injection support, or an apparently stable aggregate hiding an epoch- or source-class-specific deficit.

Exercise

A catalog combines two disjoint epochs. Epoch 1 has analyzed time $T_1=0.4\,\mathrm{yr}$ and detection efficiency $0.30$ for a specified source population. Epoch 2 has $T_2=0.6\,\mathrm{yr}$ and efficiency $0.70$.

1. Find the exposure-weighted efficiency for a stationary rate.
2. Explain why averaging the efficiencies without the durations is wrong.

Solution

The weighted efficiency is

$$\bar p_{\mathrm{det}} =\frac{(0.4)(0.30)+(0.6)(0.70)}{0.4+0.6} =0.54.$$

The unweighted mean would be $0.50$ and would pretend that the two detector states contributed equal exposure. Duration weighting is valid here because the source rate is assumed stationary and the epochs are disjoint; a time-dependent rate or overlapping searches requires the corresponding generative measure.

References

• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). “GWTC-5.0: Methods for Identifying and Characterizing Gravitational-Wave Transients.” arXiv:2605.27224. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). “GWTC-5.0: Observations from the Second Part of the Fourth LIGO-Virgo-KAGRA Observing Run and Updates to the Gravitational-Wave Transient Catalog.” arXiv:2605.27225. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). GWTC-5.0 Data Release. DOI 10.7935/bk00-6a89. Official release.
• Thrane, E., and Talbot, C. (2019). “An Introduction to Bayesian Inference in Gravitational-Wave Astronomy: Parameter Estimation, Model Selection, and Hierarchical Models.” Publications of the Astronomical Society of Australia 36, e010. DOI. Open PDF v9, including the selection erratum.