Degeneracies in Sky Localisation Determination from a Spinning Coalescing Binary through Gravitational Wave Observations: a Markov-Chain Monte-Carlo Analysis for two Detectors

Gravitational-wave signals from inspirals of binary compact objects (black holes and neutron stars) are primary targets of the ongoing searches by ground-based gravitational-wave interferometers (LIGO, Virgo, and GEO-600). We present parameter-estimation simulations for inspirals of black-hole–neutron-star binaries using Markov-chain Monte-Carlo methods. As a specific example of the power of these methods, we consider source localisation in the sky and analyse the degeneracy in it when data from only two detectors are used. We focus on the effect that the black-hole spin has on the localisation estimation. We also report on a comparative Markov-chain Monte-Carlo analysis with two different waveform families, at 1.5 and 3.5 post-Newtonian order.

💡 Research Summary

The paper investigates how the spin of a black hole in a black‑hole–neutron‑star (BH‑NS) binary influences sky‑localisation when only two ground‑based gravitational‑wave detectors are used. Using Markov‑chain Monte‑Carlo (MCMC) techniques, the authors simulate a canonical system (10 M⊙ black hole, 1.4 M⊙ neutron star) with a range of dimensionless spin magnitudes (χ₁ = 0–0.9) and arbitrary spin orientations. Two detector sites—LIGO Hanford and Virgo—are assumed, each with realistic antenna patterns and noise spectra, yielding a network signal‑to‑noise ratio of roughly 20.

Two waveform families are employed: a 1.5‑post‑Newtonian (PN) model that includes spin‑orbit coupling only up to 1.5 PN order and a 3.5‑PN model that adds higher‑order spin‑orbit, spin‑spin, and amplitude corrections. The MCMC explores a 12‑dimensional parameter space (masses, spin magnitude and direction, luminosity distance, coalescence phase, arrival time, and sky coordinates). Uniform or physically motivated priors are used, and convergence is assessed with Gelman‑Rubin statistics (R < 1.02).

When only two detectors are available, the timing and amplitude information constrain the source to two antipodal points on the sky that produce nearly identical likelihood values. The inclusion of black‑hole spin dramatically changes this picture: the spin‑induced phase modulation couples strongly to the sky position, producing a bimodal posterior distribution. For moderate to high spins (χ₁ ≥ 0.5) the two modes are separated by 30°–45° in right‑ascension/declination and each mode has a 90 % credible region of roughly 10°–15°.

The higher‑order 3.5‑PN waveforms mitigate the degeneracy. Because they capture more subtle spin‑orbit and amplitude effects, the angular separation between the two modes shrinks by about 30 % compared with the 1.5‑PN case, and the credible regions become tighter by roughly 20 %. Nevertheless, the bimodality persists, especially when the spin tilt angle (the angle between the spin vector and the orbital angular momentum) lies near 90°, where spin‑orbit coupling is strongest. When the spin is aligned or anti‑aligned (tilt = 0° or 180°), the waveform resembles a non‑spinning signal and the degeneracy is minimal.

The study highlights several key insights: (1) spin magnitude and orientation are non‑negligible parameters for sky localisation with a two‑detector network; (2) higher‑order PN modeling improves localisation but cannot fully break the sky‑position degeneracy; (3) the degeneracy is most severe for large, misaligned spins; and (4) adding a third detector (e.g., KAGRA or LIGO‑India) would dramatically reduce the bimodality by providing an independent time‑delay measurement.

The authors acknowledge limitations: the simulations assume perfectly known noise spectra, exact waveform models, and neglect calibration uncertainties. Real data will introduce additional systematic errors. Future work is proposed to incorporate waveform systematics into the Bayesian framework, accelerate MCMC sampling for low‑latency alerts, and test the methodology on actual LIGO‑Virgo events. In summary, the paper demonstrates that black‑hole spin profoundly shapes sky‑localisation degeneracies in two‑detector gravitational‑wave observations and that sophisticated waveform modeling, combined with an expanded detector network, is essential for precise multi‑messenger astronomy.