Time delay interferometry combination with zero-response

In space-based gravitational wave detection, gravitational wave signals are detected by measuring the Doppler frequency shift of laser exchange between three satellites. Although the laser is pre-stabilized, the laser phase noise is significantly higher than the gravitational wave signal, approximately 7–8 orders of magnitude higher. To reduce the otherwise-overwhelming laser phase noise, the time delay interferometry (TDI) technique is employed by properly time-shifting and linearly combining raw data. Different TDI combinations have various abilities in suppressing laser phase noise, and the response function of gravitational wave signal as well as the power spectral density of noise floor (including test mass acceleration noise and optical path noise) are also different. In this work, we present a unique class of TDI combinations, denoted as [ZX]116,[ZX]216,[ZX]316,[ZX]416, and [ZX]516, which exhibit a distinct geometric space–time diagrams composed of Michelson interferometers as the fundamental constituents. Specifically, when the detector arm lengths are assumed to be equal, these combinations yield response functions and noise power spectral densities of zero. To evaluate the detection performance of these combinations, we investigate their response functions and noise power spectral densities when the detector arm lengths are unequal, and furthermore, derive the analytical expressions for their sensitivity functions. Through our analysis, we have discovered some intriguing results. When the detector arm lengths are unequal, both the response functions and noise spectral densities of these combinations are correlated with the difference in arm lengths L3−L2. However, the sensitivity functions of these TDI combinations remain unaffected by the specific values of the arm length difference, and are consistent with the sensitivity functions of widely-used TDI combinations. For instance, the sensitivity function of [ZX]116 combination is the same as that of the Michelson combination, while the sensitivity functions of [ZX]216 and [ZX]416 combinations match with that of the Sagnac combination. Furthermore, the [ZX]316 and [ZX]516 combinations share the same sensitivity function as the fully symmetric Sagnac combination, enabling the identification of instrument noise and stochastic gravitational wave backgrounds. The discovery of this class of combinations provides support for the diversity of data processing.