All-Sky Continuous-Wave Searches

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An all-sky continuous-wave search gives up an external ephemeris and searches over sky position, frequency evolution, and sometimes binary motion. Fully coherent coverage over an observing run is usually computationally prohibitive. The central design problem is therefore quantitative: choose a semicoherent hierarchy that maximizes detection probability at fixed computing cost while controlling template mismatch, instrumental lines, and the enormous trials factor.

Helpful background. Targeted Continuous-Wave Searches supplies the phase model and coherent statistics. Continuous-Wave Phase Models develops the signal model independently of search implementation.

The parameter-space metric

Let $\lambda^i$ denote frequency, spindowns, sky coordinates, and any orbital parameters. For a small offset $\Delta\lambda^i$ , the fractional loss of squared coherent signal-to-noise ratio is approximately

\mu \simeq g_{ij}\Delta\lambda^i\Delta\lambda^j, \qquad g_{ij} = \left\langle \partial_i\Phi\,\partial_j\Phi \right\rangle -\left\langle\partial_i\Phi\right\rangle \left\langle\partial_j\Phi\right\rangle ,

where the average is over the coherent span. A template bank covers parameter space with cells whose metric radius is set by a maximum mismatch. Longer coherent time narrows frequency and spindown cells and resolves finer Doppler structure on the sky. Sensitivity improves, but the number of templates—and the cost per template—can rise steeply.

For example, if $\Phi=2\pi ft$ on $-T/2\leq t\leq T/2$ with uniform weights, then $g_{ff}=4\pi^2\langle t^2\rangle=\pi^2T^2/3$ . Thus $\mu\simeq(\pi T\Delta f)^2/3$ , the small-offset expansion of the sinc-squared loss on the preceding page. The uniform phase metric approximates the full antenna- and noise-weighted statistic; gaps and varying sensitivity change the weights.

The quadratic metric is local. Near coordinate singularities, binary boundaries, or long spans where higher phase derivatives matter, bank coverage must be checked with direct signal injections rather than assumed from the approximation.

Semicoherent sensitivity at finite cost

Divide total live time $T_{\rm obs}$ into $N$ segments of duration $T_{\rm coh}$ and sum coherent powers:

\mathcal S(\lambda) = \sum_{a=1}^{N}2\mathcal F_a(\lambda_a), \qquad N\simeq\frac{T_{\rm obs}}{T_{\rm coh}}.

The parameter track $\lambda_a$ accounts for the same sky and spin evolution across segments, while phase between segments is not retained. In an idealized equal-noise search, a fixed false-alarm and false-dismissal probability gives the rough scaling

h_0^{\rm sens} \propto \sqrt{S_n}\, \left(T_{\rm coh}T_{\rm obs}\right)^{-1/4}.

The scaling follows from a simple counting argument. Independent noise-only segments give $\mathcal S\sim\chi^2_{4N}$ , with mean $4N$ and standard deviation $\sqrt{8N}$ . A weak signal adds total noncentrality $\rho^2\propto h_0^2T_{\rm obs}/S_n$ . Requiring that excess to grow like $\sqrt N$ yields $h_0\propto\sqrt{S_n}(T_{\rm coh}T_{\rm obs})^{-1/4}$ . This assumes large $N$ and a fixed single-track false-alarm probability; changing the searched template count also changes the threshold.

The scaling is weaker than coherent $T_{\rm obs}^{-1/2}$ sensitivity, but permits far broader coverage. Real optimization includes unequal detector noise, gaps, template count, resampling cost, memory traffic, and candidate follow-up.

A hierarchy typically proceeds from a wide, shorter-coherence stage to progressively longer coherent follow-ups. A true signal should sharpen around one phase-evolution track and grow with the expected noncentrality. An instrumental line may remain loud without following the sky-dependent Doppler pattern; a transient line may disappear or migrate across stages.

Trials, lines, and upper limits

The loudest template is expected to be unusual when billions of correlated templates are searched. Significance must use the distribution of the maximum statistic for the actual bank and veto sequence, estimated analytically where justified and with noise realizations, frequency slides, or off-source bands where possible.

Line-robust statistics compare a coherent astrophysical hypothesis with single-detector or persistent-line alternatives. Vetoes must be frozen before evaluating a surviving candidate, and their effect on real signals measured by injections. Coincidence between detectors is valuable only after accounting for shared timing, possible environmental correlations, and the coincidence trials.

A population-averaged sensitivity $h_0^{p}(f)$ is an amplitude at which a fraction $p$ of simulated signals drawn from stated sky, orientation, spindown, and binary distributions meets a declared recovery criterion. This is often used in setting search upper limits, but sensitivity, a confidence bound derived from the observed statistic, and a Bayesian credible bound are distinct constructions. State which one is reported and whether injections test candidate selection alone or the complete follow-up. A local sensitivity depth cannot replace these recovery checks.

Reproducible optimization and convergence

For each stage, record the parameter bounds, lattice or stochastic-bank construction, maximum mismatch, segment timestamps, statistic implementation, detector weighting, vetoes, candidate quotas, and computing hardware. Report wall time or normalized operations so the sensitivity–cost choice can be reproduced.

Convergence and failure checks should include:

inject a synthetic coherent tone at cell centers and near worst-case cell boundaries;
halve frequency, spindown, and sky spacings and verify recovery changes consistently with the metric;
vary $T_{\rm coh}$ and segment selection while holding an explicit cost model fixed;
repeat with lines, combs, gaps, and spectral leakage added to otherwise controlled noise;
follow pure-noise outliers through the entire hierarchy to calibrate the trials tail;
verify recovered phase parameters converge and signal-to-noise grows at longer-coherence stages.

The useful benchmark is not merely whether code runs, but whether measured mismatch, false dismissal, and cost agree with the declared design within tolerance.

Observational status through 2026-08-18

The O4a all-sky search for isolated neutron stars selected at this cutoff reports no credible continuous-wave detection and publishes configuration-dependent upper limits over its searched frequency and spindown domain LVK 2026, O4a isolated all-sky search. A complementary O4a all-sky binary search finds no credible astrophysical signal and reports population-averaged sensitivity estimates. Its 95% criterion measures simulated signals that pass the specified candidate-selection thresholds within its searched orbital parameter space and non-vetoed bands LVK 2025, O4a binary search, §VI and Appendix B.

These are null results with quantified sensitivity, not weak detections. They do not cover every spin evolution, wandering source, binary orbit, or duty cycle. The authoritative releases give the exact bands, vetoed regions, confidence construction, and data products. A new official observing run or calibration could change the sensitivity or interpretation of these searches.

Exercises

Holding $T_{\rm obs}$ and $S_n$ fixed, an idealized semicoherent search increases $T_{\rm coh}$ by a factor of $16$ . By what factor does the amplitude sensitivity improve?

Solution

Since $h_0^{\rm sens}\propto T_{\rm coh}^{-1/4}$ ,

\frac{h_{0,\rm new}}{h_{0,\rm old}} = 16^{-1/4} = \frac12.

The detectable amplitude is halved. This ignores the potentially steep increase in template count, which is why finite-cost optimization is essential.

A one-dimensional bank uses spacing $\Delta\lambda$ with constant metric $g$ . If the worst point lies halfway between templates, find the maximum mismatch and the new spacing required to reduce it by a factor of four.

Solution

The maximum offset is $\Delta\lambda/2$ , so

\mu_{\max}=g\left(\frac{\Delta\lambda}{2}\right)^2.

Because mismatch is quadratic, halving the spacing gives

\mu_{\max}'=g\left(\frac{\Delta\lambda}{4}\right)^2 = \frac{\mu_{\max}}{4}.

The number of templates doubles in this one-dimensional example; in $D$ comparable dimensions it grows roughly as $2^D$ .

References

LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025). “All-sky search for continuous gravitational-wave signals from unknown neutron stars in binary systems in the first part of the fourth LIGO–Virgo–KAGRA observing run.” arXiv:2511.16863.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). “All-sky Searches for Continuous Gravitational Waves from Isolated Neutron Stars in the Data from the First Part of the Fourth LIGO–Virgo–KAGRA Observing Run.” arXiv:2603.14168.