Coincident and Coherent Network Searches

A detector network can be searched by requiring independent single-detector triggers to coincide, or by fitting one sky-delayed gravitational wave coherently to all calibrated channels. Coincidence is modular and robust; coherence uses phase, amplitude, and antenna information more completely and creates null combinations that expose inconsistency. Neither approach is intrinsically superior under every glitch and calibration regime. This page derives their common likelihood geometry and gives a reproducible network projection test, with software and observing configurations bounded by 2026-08-18.

Required background. Probability and Statistical Inference supplies multivariate Gaussian likelihoods, projections, and sampling distributions.

Helpful background. Gravitational Radiation and Waveforms supplies the two tensor polarizations, while Computational Relativity supplies waveform products and independent numerical checks.

One signal in several calibrated channels

For detectors $a=1,\ldots,D$, write the frequency-domain data as

$$\widetilde d_a(f) =\sum_{A\in\{+,\times\}} F_a^A(\boldsymbol n,\psi) e^{+2\pi i f\tau_a(\boldsymbol n)} \widetilde h_A(f) +\widetilde n_a(f).$$

Here $\boldsymbol n$ is sky direction, $\psi$ is the polarization-basis angle, and $\tau_a$ is the actual arrival delay at detector $a$ relative to the geocenter: a delayed signal is $h(t-\tau_a)$. The plus sign in the phasor follows the transform $\widetilde x(f)=\int x(t)e^{+2\pi i f t}\,dt$. The quantity $F_a^A$ is the antenna response for the documented detector configuration. Assuming independent stationary Gaussian detector noises gives the network inner product

$$(\boldsymbol x\mid\boldsymbol y)_{\rm net} =\sum_a4\operatorname{Re}\int_{f_{\min}}^{f_{\max}} \frac{\widetilde x_a(f)\widetilde y_a(f)^*}{S_{n,a}(f)}\,df.$$

If environmental correlations are consequential, the diagonal PSDs must be replaced by the full cross-spectral covariance. The coherent Gaussian likelihood and multi-detector signal subspace follow from this network geometry Jaranowski and Królak 2012, §§4–6.

Coherent and null projections

Whiten the data and collect the two polarization responses into a matrix $F$. At fixed sky position and waveform time dependence,

$$\boldsymbol d=F\boldsymbol a+\boldsymbol n,$$

where $\boldsymbol a$ contains the fitted polarization amplitudes. If $F^\dagger F$ is invertible, the maximum-likelihood signal projection is

$$P=F(F^\dagger F)^{-1}F^\dagger, \qquad E_{\rm coh}=\boldsymbol d^\dagger P\boldsymbol d.$$

The complementary null energy is

$$E_{\rm null}=\boldsymbol d^\dagger(I-P)\boldsymbol d.$$

For ideal circular complex whitened Gaussian noise normalized by $\mathbb E[n_an_b^*]=\delta_{ab}$, a rank-$r$ complex projector contains $2r$ real quadratures:

$$2E_{\rm coh}\sim\chi^2_{2r}, \qquad 2E_{\rm null}\sim\chi^2_{2(D-r)}.$$

Equivalently, one may formulate the projection on an explicit real quadrature vector and count its real rank. A true signal represented by $F$ raises coherent energy but not the expected null energy. A glitch confined to one detector generally excites both when a nontrivial null space exists.

For two detectors with two freely fitted complex polarizations and rank $r=2$, $P=I$ and there is no null stream at that frequency. A third independent channel, a response degeneracy, or an additional waveform/polarization constraint is needed to leave a null direction. The equal-response benchmark below imposes one signal direction and therefore has a null stream even with two detectors; it should not be mistaken for a generic two-polarization network.

The inverse becomes ill-conditioned when the network has weak sensitivity to one polarization. Singular-value decomposition exposes this directly. Truncating a small singular value changes the fitted signal subspace and null degrees of freedom, so the threshold and background must be recalibrated rather than borrowed from the full-rank formula.

Coincidence and coherence compared

Question	Coincident search	Coherent search
Basic object	Single-detector triggers joined by time and parameter windows	One network likelihood evaluated with sky delays and responses
Phase and amplitude consistency	Usually partial or applied after triggering	Built into the fitted network signal subspace
Glitch robustness	Can require independent support and detector-specific vetoes	Gains null and residual tests but can fit coherent-looking artifacts
Missing or insensitive detector	Naturally handled by participation rules	Requires the appropriate response and covariance for the active network
Background	Time shifts or off-source trigger coincidences	Time shifts, sky shifts, or off-source coherent statistic

Coincidence windows must cover timing uncertainty, inter-site light travel, and template uncertainty without becoming so wide that accidental pairs dominate. Coherence requires accurate relative timing, calibration phase, antenna response, and sky delay. Hybrid searches often create coincident candidates and then apply a coherent follow-up; the categories and thresholds belong to the search definition.

A two-detector projection benchmark

Consider two whitened detector samples with a single equal-response signal direction

$$\boldsymbol u=\frac{1}{\sqrt2}(1,1), \qquad \boldsymbol v=\frac{1}{\sqrt2}(1,-1).$$

The coherent and null coordinates are $c=\boldsymbol u^T\boldsymbol d$ and $q=\boldsymbol v^T\boldsymbol d$. Noise-only theory predicts $\mathbb E[c^2]=\mathbb E[q^2]=1$. Adding coherent amplitude three predicts $\mathbb E[(c+3)^2]=10$. A fixed-seed calculation with Python 3.9.6 and NumPy==2.0.2 gives:

Monte Carlo samples	Coherent null result	Null-stream result	Injected coherent result
10000	1.016706	1.016279	9.935950
100000	1.000121	1.000250	9.982011
1000000	1.000455	0.998965	10.002309

All three means approach their independent analytic values. The control verifies the projection normalization; it does not model non-Gaussian glitches, sky grids, or frequency-dependent detector tensors.

Executable rerun

import numpy as np

for samples in (10_000, 100_000, 1_000_000):
    rng = np.random.default_rng(314159)
    noise = rng.normal(size=(samples, 2))
    coherent = (noise[:, 0] + noise[:, 1]) / np.sqrt(2.0)
    null = (noise[:, 0] - noise[:, 1]) / np.sqrt(2.0)
    injected = coherent + 3.0
    print(samples,
          f"{np.mean(coherent**2):.6f}",
          f"{np.mean(null**2):.6f}",
          f"{np.mean(injected**2):.6f}")

Save this as network_projection_check.py, then run:

python3 -m venv .venv
. .venv/bin/activate
python3 -m pip install numpy==2.0.2
python3 network_projection_check.py

For a physical rerun, the inputs expand to versioned calibrated strain, detector geometry, calibration-response samples, quality segments, per-detector PSDs, a sky grid, waveform version, timing convention, ranking definition, and off-source background. The public file hashes and all active detector channels must be recorded.

Search evidence, selections, and failure modes

The detector-noise guide explains how calibration, data quality, glitches, and signal consistency condition a network analysis LIGO Scientific Collaboration and Virgo Collaboration 2020, §§6–8, PDF. At the cutoff, GWTC-5.0 documents versioned O4b coincident and coherent search components, sensitivities, and candidate checks LVK Collaboration 2026, §§2–4, PDF. Those methods define their own active networks, thresholds, background livetime, injection populations, and pipeline versions.

Important failures are a wrong delay sign, inconsistent detector timestamps, calibration phase error, an obsolete detector tensor, omitted correlated noise, a nearly singular response treated as full rank, time slides that preserve a terrestrial correlation, and tuning a veto on the foreground candidate. A network statistic also conditions the sample: requiring two detectors can discard signals confidently seen in one sensitive instrument while another is unavailable or poorly oriented.

The result of a coherent fit is evidence for consistency with the supplied network response and waveform family. It is not by itself proof of astrophysical origin; an empirically calibrated false-alarm analysis and data-quality assessment remain necessary.

Check your understanding

For whitened data $\boldsymbol d=(4,2)$ and the equal-response basis above, compute the coherent energy, null energy, and total energy. What does the decomposition diagnose?

Solution

The projected coordinates are

$$c=\frac{4+2}{\sqrt2}=3\sqrt2, \qquad q=\frac{4-2}{\sqrt2}=\sqrt2.$$

Hence

$$E_{\rm coh}=c^2=18, \qquad E_{\rm null}=q^2=2, \qquad \lVert\boldsymbol d\rVert^2=20.$$

Most energy lies in the allowed common signal direction, while a smaller inconsistency remains in the null direction. Its significance still depends on the calibrated noise distribution and on whether the assumed equal response is appropriate for the sky position and detector configuration.

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.