The geometry of lunar gravitational wave detection
The Lunar Gravitational Wave Antenna (LGWA) is a planned gravitational wave detector on the Moon, targeting the deci-Hertz band and expected to deliver breakthrough discoveries across several science cases, including the Moon's interior structure and astrophysics. In this work, we show that adopting a frame comoving with the Solar System barycenter (SSB), but with its origin at a location that minimizes timing uncertainty, reduces the sampling time by an order of magnitude. We present a systematic post-processing procedure to identify the optimal origin within the Solar System for any given signal. We explore alternative timing parametrizations beyond the merger time, and find that they have only a minor impact on parameter uncertainties. Using the stellar-mass black hole binary GW250114 as a case study, we illustrate how these geometrical considerations translate into improved parameter constraints. Two minutes before its merger, the LGWA would have measured its chirp mass to a precision of 0.0002 solar masses (90% symmetric) and constrained its sky position to within 65 square degrees (90% HPD area); these constraints are tighter than those obtained by the LIGO-Virgo-KAGRA (LVK) detectors, despite a lower signal-to-noise ratio. We connect our results to an analytical approximation proposed by Wen and Chen, which relates the area spanned by the orbital motion of a detector to its efficacy in constraining the sky position of a source. We verify its qualitative validity for compact binary sources with a series of injections, identifying the regimes in which its underlying assumptions break down. Our results demonstrate that inference for long-duration GW signals with the LGWA must be treated as a geometrical problem, in which detector motion, reference-frame choice, and signal evolution jointly determine both parameter constraints and computational efficiency.