False-Alarm Rates and Trials

A false-alarm rate measures how often a fully specified search produces noise events at least as highly ranked as a candidate. It is a repeated-search property, not the probability that this particular candidate is noise. This page connects single-trial tails to searched backgrounds, derives the conversion between rate and false-alarm probability, and explains how time slides, category trials, finite background, and analysis selection enter the result.

Required background. Probability and Statistical Inference supplies survival functions, order statistics, Poisson processes, hypothesis tests, and Monte Carlo uncertainty.

Helpful background. Gravitational Radiation and Waveforms supplies the signal families whose search space creates trials, while Computational Relativity supplies simulated waveforms used in search validation.

From a null tail to a search rate

Let $R(d)$ be the ranking statistic returned by a fixed search applied to data $d$, with larger values more signal-like. For a statistic value $R_*$, the local tail probability

$$p_\mathrm{local}=P_0\!\left(R\ge R_*\right)$$

refers to one prespecified trial under the noise hypothesis $H_0$. A search usually maximizes over event time, templates, sky points, durations, detectors, and categories. If $R_{\max}$ denotes that complete maximum, its null tail is instead

$$p_\mathrm{search}=P_0\!\left(R_{\max}\ge R_*\right).$$

For $N$ independent, identically distributed trials,

$$p_\mathrm{search} =1-\left(1-p_\mathrm{local}\right)^N \simeq Np_\mathrm{local}$$

when $Np_\mathrm{local}\ll1$. Real trials overlap and are correlated, so a nominal template count is not generally the effective $N$. The practical solution is to run the complete pipeline on a background that preserves its correlations and selections. The distinction between a statistic’s ideal null distribution and a search-level false alarm is developed in Jaranowski and Królak 2012, §§4.1–4.3.

False-alarm rate and observation probability

Suppose background events above $R_*$ form a stationary Poisson process. A count-based estimate of its rate is

$$\widehat{\operatorname{FAR}}(R_*) =\frac{N_\mathrm{bg}(R\ge R_*)}{T_\mathrm{bg}},$$

where $T_\mathrm{bg}$ is the analyzed background livetime after all category and quality selections. For a specified true rate $\operatorname{FAR}$ and on-source observing time $T_\mathrm{obs}$, the probability of at least one such noise event is

$$\operatorname{FAP}(R_*;T_\mathrm{obs}) =1-\exp\!\left[-\operatorname{FAR}(R_*)T_\mathrm{obs}\right].$$

This equation follows because a Poisson process has zero-count probability $\exp(-\lambda)$ with $\lambda=\operatorname{FAR}T_\mathrm{obs}$. For $\lambda\ll1$, $\operatorname{FAP}\simeq\lambda$. Substituting an estimated rate gives an estimated FAP whose uncertainty still comes from the background. For a time-dependent Poisson rate, replace the product by the integrated rate over the selected foreground. The conversion must use the foreground exposure and category definition to which the candidate belongs. Quoting “one per century” without the searched observation time does not specify a false-alarm probability.

Neither FAR nor FAP equals $P(H_0\mid d)$. Reversing the conditioning requires prior odds and a signal population model. A posterior astrophysical probability therefore answers a different question and may depend on the inferred foreground and background rate densities as functions of the ranking statistic.

Time-shifted network backgrounds

For transient network searches, time shifts chosen beyond the intersite light-travel time and coincidence tolerance can destroy genuine astrophysical coincidence while retaining each detector’s single-site noise behavior. Long-lived signals need correspondingly careful shifts or different background methods. Repeating many shifts produces an effective coincident background livetime much longer than the zero-lag observation. The search statistic, templates, gates, vetoes, detector combinations, category boundaries, and clustering must be identical in foreground and background.

Time slides do not manufacture independent single-detector data. Reusing the same triggers creates correlated background trials. A short correlated terrestrial transient can disappear under a slide just as a true signal does, making the shifted background underestimate the zero-lag noise tail; slowly varying correlations may instead survive some shifts. Statistical uncertainty is therefore not determined solely by the nominal number of slides. Search teams test shift families, environmental correlations, and stability under alternative backgrounds. The treatment of data quality and time-shifted search backgrounds is summarized in LIGO Scientific Collaboration and Virgo Collaboration 2020, §§6.2 and 8.4, PDF.

If zero background events exceed $R_*$ in livetime $T_\mathrm{bg}$, the empirical point estimate is zero but the true rate has not been proved zero. Under a Poisson model, the one-sided 90% upper bound solves

$$\exp(-\operatorname{FAR}_{90}T_\mathrm{bg})=0.1, \qquad \operatorname{FAR}_{90}=\frac{2.302585\ldots}{T_\mathrm{bg}}.$$

Extrapolating beyond the loudest measured background requires an explicit tail model and validation; simply reporting an infinite significance discards finite-sample uncertainty.

Categories, clustering, and the look-elsewhere effect

A pipeline may report separate backgrounds for detector networks, template-duration bins, observing epochs, or data-quality categories because their noise distributions differ. If a candidate could have been selected from any of $K$ categories, a global false alarm must account for that selection. For disjoint Poisson backgrounds with rates $\lambda_k(R_*)$,

$$\operatorname{FAR}_\mathrm{global}(R_*) =\sum_{k=1}^{K}\lambda_k(R_*).$$

This sum is valid only when each category’s threshold means the same declared selection event and duplicate triggers have been clustered consistently. Multiplying the most favorable category tail by an arbitrary trials factor can be either too conservative or anti-conservative.

The same principle applies to tuning. If the on-source candidate is used to choose a veto, binning, waveform family, or tail fit and the background is not reprocessed through that choice, the nominal tail omits an analysis-selection trial. A blinded or frozen selection protects the interpretation; a post-hoc choice requires recalibration of the full procedure.

Evidence boundary and reproducible record

The stable equations above are analytic. The observational example is the versioned O4b search methodology described in the GWTC-5.0 methods paper, with an evidence cutoff of 2026-08-18 LVK Collaboration 2026, §§2–4, PDF. This page does not extract catalog trigger counts or reproduce a collaboration FAR.

An empirical FAR result is reproducible only after recording the catalog or strain release, detector network, GPS segments, calibration version, data-quality masks, sample rate, search and software version, waveform and template-bank versions, ranking statistic, clustering window, category map, foreground exposure, time-shift scheme, background livetime, surviving counts, tail estimator, uncertainty interval, and any extrapolation. Systematic checks should vary calibration response, PSD estimation, vetoes, shift families, category boundaries, and tail model. A public-data rerun must additionally retain GWOSC file hashes and the exact commands that construct both foreground and background.

No public strain dataset is analyzed on this page. Its independently reproducible control is the Poisson conversion itself: a threshold with $\operatorname{FAR}=0.01\,\mathrm{yr}^{-1}$ observed for one year has

$$\operatorname{FAP}=1-e^{-0.01}=9.950166\times10^{-3},$$

whereas observing for ten years gives $1-e^{-0.1}=9.516258\times10^{-2}$. The same FAR produces different experiment-level probabilities because the exposures differ.

Common pitfalls

A small tail is not a signal posterior. A p-value or FAP is conditioned on the noise model. It does not include prior odds or a foreground population and therefore cannot be read as the probability that the candidate is terrestrial.

Background livetime is not foreground livetime. Time slides may create centuries of effective background, but the FAP conversion uses the actual searched foreground exposure. The background duration controls rate resolution and uncertainty.

The loudest bin cannot be chosen for free. Selecting the most favorable category after seeing the candidate changes the experiment. Include all eligible categories in the background procedure or state that the result is local.

Check your understanding

A frozen search finds one candidate with an estimated FAR of one per 100 years. The analyzed foreground duration is one year. Compute the Poisson FAP. Then explain why neither “99% astrophysical” nor “$5\sigma$” follows from the FAR statement.

Solution

The expected number of equally or more highly ranked noise events is

$$\lambda=\frac{1\ \mathrm{yr}}{100\ \mathrm{yr}}=0.01.$$

Thus

$$\operatorname{FAP}=1-e^{-0.01}=0.009950166\ldots,$$

or about $0.995\%$. The complement $1-\operatorname{FAP}$ is the probability that this noise experiment produces no event above threshold; it is not $P(\text{signal}\mid d)$. An astrophysical posterior needs foreground and background rate models and prior odds. A $5\sigma$ label would require a specified Gaussian-tail convention and a search-level tail near $10^{-7}$, far smaller than $9.95\times10^{-3}$.

References

• Jaranowski, Piotr, and Andrzej Królak. “Gravitational-Wave Data Analysis: Formalism and Sample Applications: The Gaussian Case.” Living Reviews in Relativity 15 (2012): 4. DOI.
• LIGO Scientific Collaboration and Virgo Collaboration. “A Guide to LIGO–Virgo Detector Noise and Extraction of Transient Gravitational-Wave Signals.” Classical and Quantum Gravity 37 (2020): 055002. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Methods for Identifying and Characterizing Gravitational-Wave Transients.” arXiv:2605.27224 [gr-qc] (2026). Open PDF.