Third-generation gravitational-wave detectors such as Cosmic Explorer and Einstein Telescope will lower the low-frequency sensitivity limit from tens of hertz to a few hertz. For binary neutron stars, this stretches the observed signal from minutes to nearly a day: a source like GW170817 would enter the band at 2 Hz about 21 hours before merger. Over such a time, the detector rotates noticeably with Earth, and its antenna response is no longer constant. Related work has already taken steps to speed up inference for long signals: Smith et al. applied reduced-order quadrature with Earth's rotation, Guttman extended these methods to longer durations, and Baral included Earth's rotation in inference for a single Cosmic Explorer. However, all these approaches are either limited to signals shorter than a day or use an injected waveform as a reference, requiring prior knowledge of the source's sky position.
The author uses the exact five-harmonic Jaranowski–Krolak–Schutz (JKS) decomposition for the antenna response of a ground-based interferometer. This decomposition shows that the detector's response to slow Earth rotation contains only the zeroth, first, and second harmonics of sidereal time. As a result, the dependence on sky position, polarization, orbital inclination, and distance is captured in five coefficients, while all time variation is described by five known functions. To speed up calculations, the waveform is computed on a non-uniform frequency grid whose step is determined by phase curvature rather than the total signal duration. Then relative binning is applied: within each frequency interval, the ratio of the trial waveform to the reference is approximated by a linear function, reducing the likelihood computation to a sum over a sparse grid of bins.
For a signal similar to GW170817, with component masses of 1.46 and 1.27 solar masses, in the band from 2 to 1800 Hz and with ET-D sensitivity, the method achieves a speedup of 15,000 times compared to direct computation on a grid that resolves the duration. The median absolute error of the log-likelihood is 0.245 with a bin tolerance of 0.05. A full Bayesian inference over eight parameters on 64 cores takes about 312 seconds, achieving an accuracy of 0.36 parts per million in chirp mass, 0.45% in mass ratio, and 8.7% in distance amplitude. The 90% credible sky area for a single detector with LIGO Hanford geometry is 404.6 square degrees.
This work shows that the long-signal barrier for Bayesian inference can be overcome by exploiting the finite harmonic structure of the rotating detector response. Unlike neighboring approaches, where Earth's rotation is either approximated or requires knowledge of the source position, here the separation is achieved without prior sky information. This makes day-long binary neutron star signals accessible to full parameter estimation on a single computing node.
The method can be directly combined with other fast approaches, such as radial basis function surrogates for the likelihood. An open question remains the finite arm length of the Cosmic Explorer interferometer: at 40 km, the frequency dependence of the response and its coupling to the source direction break the exact five-term form, requiring a separate generalization.
The result will speed up data processing for future third-generation detectors and enable faster localization of gravitational-wave sources for electromagnetic follow-up.
The next step will be to adapt the method to detectors with finite arm lengths and test it on realistic noisy data.
This work directly addresses the unresolved problem of fast Bayesian inference for long gravitational-wave signals in third-generation detectors, where standard methods become computationally prohibitive.
A binary neutron star signal at 2 Hz lasts almost a sidereal day, 21.3 hours, during which the detector completes nearly a full rotation with Earth.