Detection Thresholds and False-Alarm Accounting

A catalog threshold turns a continuous ranking statistic into a reported set. Its false-alarm interpretation depends on the search background, analyzed time, source class, pipeline, and number of opportunities for noise to cross the threshold. A false-alarm rate is not the probability that a particular candidate is noise, and a threshold chosen for one search cannot be transplanted to another without recalibration.

Helpful background. False-Alarm Rates and Trials derives search-level background calibration. From Search Triggers to Catalog Events explains how several pipelines and candidate records enter one release.

Background survival and false-alarm rate

Let $x$ be a ranking statistic for which larger values are more signal-like. If a background procedure supplies effective noise time $T_b$ and $N_b(x)$ background triggers with rank at least $x$, an elementary estimate is

$$\widehat{\mathrm{FAR}}(x) =\frac{N_b(x)+\kappa}{T_b},$$

where $\kappa$ records the finite-background convention. The exact estimator and uncertainty must be stated; adding one event, fitting a tail, or pooling background changes the meaning.

If false alarms above threshold form a Poisson process with rate $r$, then in analyzed foreground time $T$ the expected count is

$$\mu=rT,$$

and the probability of at least one is

$$P(N\ge1)=1-e^{-\mu}.$$

These formulas assume a stationary Poisson description over the declared analysis. Correlated glitches, changing detector states, overlapping trials, and tail extrapolation require empirical calibration rather than blind use of the ideal expression.

The Gaussian ideal, likelihood-ratio statistics, and their null distributions are reviewed by Jaranowski and Królak 2012, §§ 3–6; catalog false-alarm accounting must additionally model the real search background.

Trials enter through the search definition

Searching more templates, source classes, time periods, or pipelines creates more chances for a noise fluctuation. If $K$ independent tests each have small tail probability $p$, then

$$P(\text{at least one exceedance}) =1-(1-p)^K \simeq Kp.$$

Real trials are not independent. Template banks overlap, pipelines share data, and observing epochs have different backgrounds. A valid global false-alarm statement therefore comes from a background construction that mirrors the complete search or from a declared combination rule with correlations bounded. Multiplying a single-template $p$-value by a visually estimated trials factor is not equivalent.

Thresholds express a scientific loss function

No threshold is optimal without a purpose. A high-purity catalog may tolerate fewer marginal signals; a population analysis may admit lower-ranked candidates if it includes a validated signal–noise mixture; a rapid-alert stream may trade purity against latency and follow-up cost. The threshold should therefore state:

• search and ranking-statistic version;
• source class and detector configuration;
• analyzed foreground and effective background time;
• data-quality and coincidence selections;
• false-alarm estimator and tail treatment;
• whether the rule is per pipeline or combined; and
• which downstream likelihood conditions on the threshold.

Changing the threshold changes both contamination and selection efficiency. A catalog cannot lower its threshold while reusing a selection function measured for the old rule.

FAR, false-alarm probability, and astrophysical probability

These quantities should remain separate:

Quantity	Conditional meaning	Needs a signal-population model?
Ranking statistic $x$	Pipeline ordering of triggers	Usually implicit in training or weighting
$\mathrm{FAR}(x)$	Noise-event rate at least as extreme	No explicit astrophysical rate, but needs a background model
False-alarm probability	Chance of one or more background exceedances in declared trials/time	No explicit astrophysical rate
$p_{\mathrm{astro}}$	Mixture probability that this trigger belongs to a modeled signal class	Yes, plus noise and rate models

A small FAR does not equal $1-p_{\mathrm{astro}}$. The latter changes if the assumed astrophysical rate or signal distribution changes even when the trigger and noise background are fixed.

Current catalog boundary and checks

At the evidence cutoff 2026-08-18, the current catalog threshold and background definitions must be taken from GWTC-5.0 Methods, search and sensitivity sections, with inclusion scope checked against GWTC-5.0 Observations, §§ 2–4 and release tables. This page deliberately gives no timeless candidate count or universal numeric cutoff.

An independent threshold check uses a versioned toy or public background table. It should reproduce the chosen estimator, vary background duration and tail treatment, propagate counting uncertainty, and inject a signal population to estimate efficiency under the same rule. The calculation fails if a reported global FAR was calculated by treating correlated pipelines as independent, if foreground time includes unanalyzed data, or if the selection injections use a different threshold.

Common pitfalls

Interpreting inverse FAR as a waiting time for this source. It characterizes the search’s background tail under stated conditions, not the probability or astrophysical recurrence time of the candidate.

Counting catalog rows as independent trials. Trial factors arise from the search process that generated candidates. The number of published rows is an outcome, not generally the number of opportunities.

Comparing FARs across pipelines as if identically calibrated. Distinct ranking statistics and backgrounds can yield comparable units but different uncertainties and selection functions.

Exercises

1. Expected contamination. A threshold has a calibrated FAR of $0.02\,\mathrm{yr}^{-1}$ in an analyzed time of $1.5\,\mathrm{yr}$. Under the Poisson approximation, find the expected number of false alarms and the probability of at least one.

Solution

The expected count is

$$\mu=(0.02\,\mathrm{yr}^{-1})(1.5\,\mathrm{yr})=0.03.$$

Therefore

$$P(N\ge1)=1-e^{-0.03}\simeq0.0296.$$

This is a search-level probability under the stationary Poisson model, not the noise probability of any particular candidate.

2. Correlated trials. Two pipelines analyze the same data and are perfectly correlated. Why is doubling the single-pipeline tail probability wrong?

Solution

Perfectly correlated pipelines cross threshold together, so they create one effective opportunity rather than two independent ones. The union probability equals the single-pipeline probability. A factor of two would overstate the trials penalty. Real pipelines are partially correlated, which is why the combined background or correlation-aware rule must be calibrated empirically.

References

• Jaranowski, P., and Królak, A. (2012). “Gravitational-Wave Data Analysis. Formalism and Sample Applications: The Gaussian Case.” Living Reviews in Relativity 15, 4. DOI.
• 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.