Gravitational-wave responses and noise correlations in geometric time-delay interferometry

Abstract

Space-based gravitational-wave observations will probe the formation and growth of massive black holes and the evolution of compact binaries in the millihertz band. We develop a geometric-link representation of time-delay interferometry using six round-trip and six non-round-trip two-link blocks. Starting from the one-way Doppler response, we derive the elementary block responses and a dictionary that expresses 45 known second-generation combinations in terms of these blocks and residual one-way measurements. Representative Michelson and Relay observables admit compact forms, from which analytical sky-averaged tensor responses are obtained for a fixed, equal-arm constellation. We also derive the noise covariance of the blocks, retaining cross-spectra between blocks that share measurement streams. The resulting noise spectra agree with direct one-way assembly. We apply this formulation to a narrowband compact-binary signal in an equal-arm LISA instrument model. For the Relay example, neglecting the shared-link cross-spectrum changes the low-frequency noise scaling and overestimates the conditional Fisher uncertainties in amplitude or distance by factors of 10.71 at 1~mHz and 3.37 at 3~mHz.