Residual and Unmodeled-Signal Tests

A residual test subtracts a fitted general-relativistic waveform from calibrated multi-detector strain and asks whether coherent signal power remains. Excess power is not automatically new gravity: it can be detector noise, a glitch, calibration error, an incomplete source waveform, or bias from choosing the analysis after looking at the residual. The evidential requirement is therefore repeatable coherence across detectors, subtraction draws, off-source trials, waveform families, and independent residual statistics.

Helpful background. Weak-Field Gravity supplies the perturbative null signal; Probability and Statistical Inference supplies calibrated $p$-values and posterior prediction; Gravitational-Wave Data Analysis and Inference supplies strain likelihoods, calibration, and noise models; Tests of Gravity supplies the bounded conclusion; and Black-Hole Perturbation Theory identifies late-time source physics that a template might omit.

Residuals after uncertain subtraction

For detector $I$, define a residual for one source-parameter and calibration draw $s$ by

$$r_I^{(s)}(t)=d_I(t)-h_I(t;\vartheta^{(s)},C^{(s)},\mathcal M).$$

Subtracting a maximum-likelihood waveform gives a useful diagnostic, but its residual must be calibrated after the fitting step; it is not a draw from untouched detector noise. One robustness check propagates posterior draws of intrinsic and extrinsic parameters, calibration $C$, noise spectrum, and waveform model $\mathcal M$. It preserves the covariance created because the same data both fit the signal and define the residual. Alternatively, a frequentist injection study can repeat the complete fit-and-subtract procedure. Either approach must account for fitting; posterior draws are not the only valid construction.

The simplest example makes the issue visible. Suppose $N$ independent measurements obey $d_j=A+n_j$, with $n_j\sim\mathcal N(0,\sigma^2)$ and one unknown constant amplitude. Fitting $\hat A=\bar d$ gives $r_j=d_j-\bar d$, so

$$\sum_jr_j=0,\qquad \operatorname{Cov}(r_j,r_k) =\sigma^2\left(\delta_{jk}-\frac1N\right).$$

The residuals are correlated and have only $N-1$ independent directions. In this exactly linear Gaussian example, $\sum_jr_j^2/\sigma^2$ follows $\chi^2_{N-1}$, not $\chi^2_N$. A nonlinear waveform fit has no universal replacement obtained by simply subtracting the number of parameters; injections determine its relevant reference distribution.

A coherent wavelet reconstruction supplies a flexible statistic, such as the posterior upper quantile $\rho_{90}$ of coherent residual network signal-to-noise ratio. Off-source or noise-replica analyses define

$$p_\mathrm{res}=P\!\left(\rho_{90}^{\mathrm{noise}} \geq \rho_{90}^{\mathrm{event}}\mid\mathcal N,C,Q\right),$$

where $\mathcal N$ is the noise model and $Q$ the quality-selection rule. With a continuous statistic and a correctly matched frequentist reference distribution, predeclared event $p$-values are uniform under the null; finite simulated backgrounds instead give discrete ranks. A posterior-predictive tail area reuses the data through the posterior and is not generally uniform. Its null distribution needs its own calibration before a uniformity test is applied. A small value in one event is only a flag; trials, finite-background uncertainty, and the rule used to select the loudest morphology must be included.

The peer-reviewed GWTC-3 method reports residual signal-to-noise ratios, fitting factors, and background-derived $p$-values rather than identifying every leftover feature with a theory LIGO–Virgo–KAGRA 2023, §IV.A, Eqs. (1)–(2), Figs. 1–2, and Table III. The submitted GWTC-5 update follows the same logical separation with a larger selected set and a current waveform/noise treatment LIGO–Virgo–KAGRA 2026, submitted P2500781-v17, §4 and Table 4. Both released catalog implementations subtract a maximum-likelihood GR waveform; the draw-marginalized construction above is an additional robustness requirement, not a description of their released statistic.

Coherence that would survive mundane explanations

A candidate residual becomes physically interesting only if it passes several tests at once:

1. It is phase-coherent at the inter-site delays implied by the source sky position and follows the antenna response.
2. Its amplitude and phase persist across calibration draws, power-spectral-density estimates, gating choices, and reasonable analysis windows.
3. It survives subtraction with independently developed waveform families including precession, eccentricity, and higher modes where relevant.
4. A second method—such as a null stream or morphology-independent coherent reconstruction—finds compatible time–frequency support without being tuned on the first result.
5. GR injections and off-source data reproduce the advertised false-alarm coverage, while an injected coherent residual is recovered with the expected scaling.

Detector-local power, a feature confined to one calibration choice, or a residual absorbed by a better source model is evidence against a gravity interpretation. Conversely, a coherent excess would still not identify a theory: it would first establish that the tested waveform-plus-noise model is incomplete.

Reproducible benchmark specification

At the evidence cutoff 2026-08-19, strain, data-quality flags, segment boundaries, and parameter-estimation samples should come from GWTC-5/GWOSC stable release 9 GWOSC 2026, official parameter-estimation and strain documentation, while P2600410-v1 supplies the residual result products LIGO–Virgo–KAGRA 2026, official tests data release. File checksums and the relation between the two releases must be recorded. Selection includes only events and detectors satisfying the predeclared residual-pipeline cuts; this is not a complete sample of all 390 catalog detections. Frequencies below 10 Hz are excluded as uncalibrated, and 16 kHz source-matched products are not silently mixed with 4 kHz decimated products.

An observational benchmark comprises three cases: a GR injection, the same injection plus a controlled coherent sine-Gaussian, and off-source detector data. It records the BayesWave or equivalent code commit, container digest, compiler and MPI versions, LALSuite and sampler versions, waveform implementation, random seed, priors, calibration envelopes, and numerical precision. BayesWave distinguishes network-coherent wavelet signals from detector-local glitches Cornish and Littenberg 2015, §§II–III. A reproduction uses the executable instructions shipped with the selected release; the full LVK analysis has not been rerun for this exposition.

The following smaller calculation is executable with Python, NumPy, and SciPy. It verifies the constant-fit example above and then adds a unit-normalized shape orthogonal to the fitted constant:

import numpy as np
from scipy.stats import chi2

rng = np.random.default_rng(1729)
N, trials = 64, 20_000
data = 2.0 + rng.normal(size=(trials, N))
residual = data - data.mean(axis=1, keepdims=True)
statistic = np.sum(residual**2, axis=1)
print(statistic.mean(), np.mean(chi2.sf(statistic, N - 1) < 0.05))

shape = np.linspace(-1.0, 1.0, N)
shape -= shape.mean()
shape /= np.linalg.norm(shape)
injected = np.sum((residual + 5.0 * shape)**2, axis=1)
print(injected.mean(), np.mean(chi2.sf(injected, N - 1) < 0.05))

The expected null mean is $N-1=63$ and the rejection fraction is $0.05$. The injected statistic has noncentrality $5^2=25$, so its expected mean is $88$. A run with the stated seed gives approximately $(62.998,0.0492)$ and $(88.112,0.6257)$ respectively. Sampling uncertainty on the null rejection fraction is about $0.0015$. This checks fitting, noise calibration, and recovery of one added shape; it does not model detector coherence or certify an observational residual. Adding a constant instead would be absorbed entirely by the fitted amplitude, demonstrating a real blind direction of this test.

Refinement level	Numerical change	Required benchmark behavior
Coarse	200 off-source trials and baseline waveform draws	Detect gross failures and establish the statistic scale
Medium	800 trials, doubled subtraction draws, alternate spectrum estimate	Central residual quantiles and coverage stabilize within declared Monte Carlo error
Fine	3200 trials, second waveform family, alternate window and calibration draws	Null coverage and injected-residual recovery agree with the medium level

Convergence fails if the event rank, tail probability, or recovered injection changes beyond its binomial or Monte Carlo uncertainty. Other failure modes include glitch leakage, nonstationary spectral estimates, sampler nonconvergence, overly flexible wavelets fitting noise, overly rigid bases missing signal, inpainting artifacts, waveform truncation, detector correlations, and post-hoc changes to the time–frequency window. The normalized likelihood, priors, covariance, selection, and completeness statement must accompany any quoted statistic. Official data products and the exact software repository are necessary for reproduction; a figure alone is insufficient.

Interpretation and limits

The current submitted analysis reports residuals compatible with its noise null, but that is a statement about selected events, data calibration, templates, and residual algorithm—not proof that GR is uniquely correct. Missing source physics and new gravitational dynamics are competing explanations for a genuine coherent residual. An independent check must inject each separately and test whether the two residual morphologies can be distinguished without training on the candidate event. The conclusion remains conditional on the benchmark passing and should be reviewed when calibration, posterior products, waveform accuracy, or residual software changes.

Exercises

An off-source study finds $n=3$ trials at least as extreme as the event among $N=99$ trials. With a uniform prior for the exceedance probability, what is its posterior mean, and why is $3/99$ an incomplete report?

Solution

The binomial likelihood with a uniform prior gives a $\operatorname{Beta}(n+1,N-n+1)=\operatorname{Beta}(4,97)$ posterior. Its mean is $4/(4+97)=4/101\simeq0.0396$. This is an estimate of an unknown exceedance probability, not the frequentist Monte Carlo rank $p=(n+1)/(N+1)=4/100$, valid when the event and background statistics are exchangeable under the null. The raw fraction $3/99$ omits finite-background uncertainty. None of these quantities accounts automatically for trials across events or morphologies, subtraction covariance, or a mismatch between off-source and event noise. Therefore the reference ensemble and construction must accompany the number.

References

• Gravitational Wave Open Science Center (2026), “GWTC-5.0 Data Release Documentation,” stable release 9 parameter-estimation, strain, and data-quality products, official release, DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “Data Release for GWTC-5.0: Tests of General Relativity,” version P2600410-v1 (2026), official release.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “Tests of General Relativity with GWTC-3,” Physical Review D 107, 022005 (2023), DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “Tests of General Relativity with GWTC-5.0,” submitted manuscript P2500781-v17 (2026), official DCC record.
• Neil J. Cornish and Tyson B. Littenberg, “BayesWave: Bayesian Inference for Gravitational Wave Bursts and Instrument Glitches,” Classical and Quantum Gravity 32, 135012 (2015), DOI.