All-sky searches for continuous gravitational waves (CWs) — long-lived nearly monochromatic signals expected from spinning non-axisymmetric neutron stars or boson clouds around black holes — report upper limits on the strain amplitude band by band. The loosely coherent Falcon search of Dergachev and Papa on LIGO O1 data (Phys. Rev. Lett. 123, 101101 (2019)) acknowledged two limitations: (i) the sub-100 Hz band is degraded by narrow-band instrumental artifacts, notably a 0.25 Hz comb of lines, which significantly limit the sensitivity of any search there; (ii) only a heuristic proxy was given for population-average upper limits, because true per-band limits require computationally expensive injection-and-recovery campaigns. The paper's outlook anticipated continued deployment of the method on new data, and the follow-up literature 2019–2026 has substantially delivered this — Falcon has been deployed on O2 and O3 data up to 1700 Hz and, in 2025, on binaries with substellar companions. Limitation (ii) remains open: as of the 2026 low-frequency atlas, per-band population-average upper limits are still proxy-based estimates, and the artifact-contaminated low-frequency bands are precisely where rigorous confidence levels are hardest to establish. The open problem is to obtain, for a loosely coherent all-sky search on Advanced LIGO observing-run data, true population-average upper limits band by band rather than heuristic proxies or empirical scaling factors, with the narrow-band artifact contamination below 100 Hz as the principal acknowledged difficulty for constructing such limits at low frequency.

Background

A continuous gravitational wave (CW) is a long-lived, nearly monochromatic signal with roughly constant frequency and slow spin-down, expected from a rapidly rotating neutron star with a non-axisymmetric mass distribution (conventionally parametrized by the equatorial ellipticity ϵ\epsilon), or from ultralight boson clouds around black holes. The signal amplitude is the strain h0h_0, which for a fixed ellipticity scales with the square of the rotation frequency, making low-frequency sources intrinsically weak. Because no such signal has yet been detected, all-sky searches — which scan unknown sky position, frequency, and spin-down — set upper limits on h0h_0 as a function of frequency band, and translate these into limits on ellipticity or boson-cloud parameters.

All-sky CW searches are semi-coherent: the data are divided into segments of some coherence length, each segment is matched coherently against a signal template, and segment results are combined incoherently. A longer coherence length improves sensitivity but explodes the computational cost, because the number of templates grows with the parameter-space resolution it induces. The Falcon pipeline of Dergachev and Papa is a loosely coherent method: instead of demanding exact phase agreement with a template, its bilinear detection kernel allows a small controlled phase drift between data and template. This widens the family of detectable signals per template and permits efficient vectorized implementations, so a much longer initial coherence length becomes affordable; the sensitivity of the first, broadest stage largely determines the sensitivity of the whole hierarchical search.

Upper limits come in two flavors. Worst-case (and best-case, circular-polarization) limits are computed with the universal statistic of Dergachev and hold for any signal waveform, at the cost of being conservative. Population-average upper limits instead specify an amplitude at which a stated fraction (the confidence level) of a simulated population of signals — averaged over sky position and orientation — would have been detected. Most other pipelines, such as the Einstein@Home searches, report population-average limits, so comparability motivated their construction in the source paper.

The source work (Dergachev and Papa, Phys. Rev. Lett. 123, 101101 (2019); arXiv:1902.05530) ran the Falcon search on data from the first Advanced LIGO observing run (O1), improving on the previous PowerFlux O1 all-sky limits (Abbott et al., Phys. Rev. D 96, 062002 (2017)) by about 30% and setting the tightest limits in the 100–200 Hz band. It also acknowledged two limitations. First, the O1 data below 100 Hz contain many instrumental artifacts, including a 0.25 Hz comb of lines (a family of narrow spectral lines spaced 0.25 Hz apart, associated with data-acquisition electronics), which significantly limit sensitivity there and produce large band-to-band scatter; the contemporaneous identification-and-mitigation study of Covas et al. (Phys. Rev. D 97, 082002 (2018)) is the standing approach to such artifacts, and the search itself accommodated the contamination by reporting a reduced recovery rate (80% below 50 Hz versus 95% above). Second, a rigorous population-average upper limit in every frequency band would require extensive fake-signal injection-and-recovery simulations, which the authors judged computationally impractical; they instead introduced a heuristic weighted average of per-polarization limits as a proxy, spot-verified with 1440 injected signals over 20–200 Hz. The paper's conclusion was forward-looking — the successful deployment "sets the stage for a new era of semi-coherent robust continuous wave searches on the next sets of gravitational wave data" — and related subsequent work characterized the effect of binary orbital parameters on search sensitivity (Singh, Papa, and Dergachev, Phys. Rev. D 100, 024058 (2019)).

Subsequent work has substantially delivered that anticipated deployment: the extended O1 survey completing 20–600 Hz (Dergachev and Papa, Phys. Rev. D 101, 022001 (2020)); Falcon searches on O2 data at 20–500 Hz and 500–1700 Hz (Phys. Rev. D 104, 043003 (2021); Phys. Rev. Lett. 125, 171101 (2020); Phys. Rev. D 103, 063019 (2021)); the O3 frequency-resolved atlas series (Phys. Rev. X 13, 021020 (2023); Phys. Rev. D 109, 022007 (2024); the expanded 20–1700 Hz atlas, Phys. Rev. D 112, 042005 (2025); the low-frequency 20–200 Hz atlas, Phys. Rev. D 113, 042001 (2026)); and the first loosely coherent search including binary orbital parameters, for sources with substellar companions (Phys. Rev. D 112, 062005 (2025)). The population-average-limit gap, however, persists: the 2024 full-O3 PowerFlux search still only estimates population-averaged limits by scaling circular-polarization limits with an empirical factor (Tripathee and Riles, Phys. Rev. D 109, 043049 (2024)), and the 2026 low-frequency atlas still substitutes lock-level-proxy quantiles for actual per-band population-average limits, stating that computationally expensive test-signal recovery simulations would be required for the real confidence levels, while searches continue to exclude or suffer degraded sensitivity in strongly line-contaminated bands.

Problem Statement

For the loosely coherent (Falcon-type) all-sky search for continuous gravitational waves on the publicly available Advanced LIGO O3 strain data (the expanded atlas covering 20–1700 Hz, Dergachev and Papa, Phys. Rev. D 112, 042005 (2025)), establish band-by-band upper limits on the strain amplitude h0h_0: for the search's bands spanning 100–200 Hz together with at least one representative band below 100 Hz that is contaminated by the narrow-band artifacts (following that search's own band-reporting and detection conventions), provide a true population-average upper limit at 90% confidence — the amplitude at which 90% of the population of simulated signals defined by that search's own injection-and-recovery convention, averaged over sky position and polarization, would be recovered by the search — rather than a heuristic proxy (such as the weighted average of per-polarization limits) or an empirical scaling factor. Extending the same per-band limits to every remaining band of the full 20–1700 Hz search is the natural follow-up. The low-frequency band below roughly 100 Hz, where the data are contaminated by narrow-band instrumental artifacts such as combs of lines and disturbed spectral lines, constitutes the principal acknowledged difficulty: the contamination depresses signal-recovery rates, degrades the limits, and produces large band-to-band scatter, so establishing rigorous per-band confidence levels in the disturbed bands is the hardest part of the same objective. The related task of validating and deploying the loosely coherent method itself on larger datasets and extended parameter spaces, including binary orbital parameters, has been substantially accomplished in the 2019–2026 follow-up literature and is therefore not the objective of this task, though an answer may naturally build on those validated deployments.

References

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