An interpreter who jumps at every rustle in the hall turns a one-hour meeting into two. That is roughly how gravitational-wave detectors behaved when analyzing long signals: each turn of the Earth added noise to the calculations, and processing a day-long neutron star merger threatened to stretch into weeks. New work shows how to bring order and hear the whole conversation.
When two neutron stars fall into each other, they emit gravitational waves, ripples in spacetime. Current detectors catch such signals in minutes, but future third-generation facilities, such as Cosmic Explorer and Einstein Telescope, will lower the sensitivity floor 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 that time, the detector rotates noticeably with the Earth, so its sensitivity to the direction of the incoming wave is no longer constant.
The author uses the exact five-harmonic decomposition by Jaranowski, Krolak, and Schutz (JKS) for the antenna response of a ground-based interferometer. This decomposition shows that the detector's response to the slow rotation of the Earth contains only the zeroth, first, and second harmonics of sidereal time. As a result, the dependence on source 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 set by the 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 waveform is approximated by a linear function, which reduces the likelihood computation to summing over a sparse grid of bins.
The result is impressive: for a GW170817-like signal 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 compared to direct computation on a grid that resolves the duration. The median absolute error in the log-likelihood is 0.245 with a bin tolerance of 0.05. 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.
The interpreter who used to jump at every rustle now listens only to the speech. The method can be directly combined with other fast approaches, such as likelihood surrogates based on radial basis functions. An open question remains about 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, and a separate generalization will be needed for that case. But already today, the result will speed up data processing for future third-generation detectors and allow faster localization of gravitational-wave sources for electromagnetic follow-up.
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.