Detection Statistics and Signal Searches

A gravitational-wave search must do three things at once: collect the signal power predicted by a source model, reject detector artifacts that violate that model, and calibrate how often the complete selection would succeed on noise. The ideal Gaussian likelihood supplies a common mathematical spine, but real sensitivity and significance depend on the searched waveform family, detector network, data conditioning, trials, and background. This chapter builds that argument from a single matched filter through empirical false-alarm rates and injection-measured sensitivity.

Helpful background. Probability and Statistical Inference supplies likelihood ratios, null distributions, p-values, and Monte Carlo error. Gravitational Radiation and Waveforms supplies signal families and detector response, while Computational Relativity supplies numerical waveforms and validation principles.

The detection problem

For data $d$, compare a noise model $H_0$ with a signal-plus-noise model $H_1$. In stationary Gaussian noise with known covariance, the log likelihood ratio for a fixed waveform $h(\vartheta)$ is

$$\ln\Lambda(\vartheta) =(d\mid h(\vartheta)) -\frac12(h(\vartheta)\mid h(\vartheta)).$$

The inner product weights frequency bins by the inverse one-sided noise PSD. Maximizing or marginalizing this expression over phase, time, intrinsic parameters, sky location, and polarization produces different statistics with different null distributions. The likelihood-ratio construction and its Gaussian geometry are developed in Jaranowski and Królak 2012, §§3–6.

Three statements must remain separate:

• a ranking statistic orders triggers according to the search model;
• a false-alarm probability is a noise-model tail for the complete searched procedure; and
• a signal probability additionally needs foreground and background populations and prior odds.

A high matched-filter signal-to-noise ratio is therefore not, by itself, a calibrated discovery claim.

Route through the chapter

Read the pages in order; each enlarges one part of the statistical experiment.

1. Likelihood Ratios and Matched Filtering derives the optimum linear filter for a known waveform in stationary Gaussian noise and checks its normalization and null distribution.
2. Template Banks and Signal Manifolds turns intrinsic-parameter maximization into a geometric covering problem and verifies a controlled mismatch metric.
3. Coincident and Coherent Network Searches compares per-detector coincidence with a network likelihood and derives coherent and null projections.
4. Burst and Excess-Power Searches replaces a detailed phase model by energy in selected time–frequency subspaces and exposes the associated clustering trials.
5. False-Alarm Rates and Trials connects local null tails, time-shifted backgrounds, searched exposure, category trials, and finite-background uncertainty.
6. Injection Campaigns and Search Sensitivity measures recovery under the frozen pipeline and derives importance-weighted sensitive spacetime volume.

The sequence separates two independent questions. Background trials estimate how often noise crosses a threshold. Injections estimate how often a specified signal population crosses the same threshold. Neither calculation can replace the other.

A common Gaussian thread

Whitening transforms ideal detector noise to unit covariance. A modeled signal then occupies a subspace of whitened data:

• one normalized template gives a standard-normal matched-filter output;
• two phase quadratures give a $\chi^2_2$ energy;
• a network response spans at most two tensor-polarization directions for a fixed sky position;
• a weakly modeled burst occupies a larger time–frequency subspace; and
• template banks assemble many nearby subspaces into a searched manifold.

This picture supplies useful analytic controls. It also identifies failure modes: PSD estimation changes the whitening map, glitches create heavy tails, calibration errors rotate network responses, waveform errors move a signal outside the modeled subspace, and maximization changes the relevant null distribution. Operational pipelines therefore add signal-consistency tests, data-quality selections, clustering, and empirical background. FINDCHIRP gives a detailed compact-binary implementation of filtering and bank iteration Allen et al. 2012, §§II–VIII, PDF, while the detector-noise guide connects ideal statistics to calibration, conditioning, glitches, and searches LIGO Scientific Collaboration and Virgo Collaboration 2020, §§4–8, PDF.

What a reproducible search result contains

A scientific result begins with a defined dataset: release and minor version, detector GPS segments, sample rate, calibration product, quality masks, and file hashes. It then fixes the conditioning and PSD estimation, waveform or burst family, template or sky grid, software commit and environment, ranking statistic, vetoes, clustering, categories, threshold, foreground exposure, background construction, and injection proposal. The result reports background counts and livetime, false-alarm uncertainty, efficiency and sensitive-volume uncertainty, and systematic variations of calibration, waveforms, data selection, and tail treatment.

The empirical discussion in this chapter has an evidence cutoff of 2026-08-18. The current observational anchor is the versioned O4b methodology described in LVK Collaboration 2026, §§2–4, PDF. No page here claims to reproduce a collaboration search or promote its release-specific thresholds into universal constants. The executable examples instead use declared synthetic distributions with analytic answers so their normalizations and convergence can be checked independently.

Review the chapter

1. Ranking, significance, and sensitivity. A candidate has a larger ranking statistic than 99.9% of a fixed search’s noise trials. The same pipeline recovers 40% of injections drawn from a declared source distribution. What do the two percentages mean, and what can neither establish?

Solution

The first percentage estimates a search-level noise tail of $10^{-3}$ only if the background trials faithfully repeat the complete selection and have sufficient effective size. The second estimates detection efficiency averaged over the stated injection population, detector times, and threshold. The first does not give $P(\text{noise}\mid d)$, and the second does not apply to a different population without reweighting or new injections. Neither establishes a signal posterior, because foreground rates and prior odds are absent.

2. Exposure and finite background. A search obtains no background event above a threshold in 5000 years of effective livetime. Find the one-sided 90% Poisson upper limit on FAR. What is the corresponding upper bound on FAP for one year of foreground?

Solution

For zero events,

$$\operatorname{FAR}_{90}=\frac{-\ln0.1}{5000\ \mathrm{yr}} =4.60517\times10^{-4}\ \mathrm{yr}^{-1}.$$

For one foreground year,

$$\operatorname{FAP}_{90} =1-\exp(-4.60517\times10^{-4}) =4.60411\times10^{-4}.$$

The inequality is an uncertainty statement under the Poisson background model, not proof that the actual rate equals the bound. Correlated time slides or an invalid tail model can require a larger uncertainty.

References

• Allen, Bruce, et al. “FINDCHIRP: An Algorithm for Detection of Gravitational Waves from Inspiraling Compact Binaries.” Physical Review D 85 (2012): 122006. DOI. Open PDF.
• 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.