Geometric approach for the modified second generation time delay interferometry

Time delay interferometry (TDI) is an algorithm proposed to suppress the laser frequency noise in spaceborne gravitational-wave detectors. As a post-processing technique, it is implemented by constructing a virtual equal-arm interferometer through an appropriate combination of the time-shifted data streams. Such an approach is tailored to the intrinsic feature of space-based gravitational-wave detection, namely, the distances between spacecraft are governed by orbital dynamics and thus cannot be held constant. Among different implementations, geometric TDI was introduced as a method of exhaustion to evaluate the second-generation TDI combinations. The applications of the algebraic approach based on computational algebraic geometry, on the other hand, are mostly restricted to first- and modified first-generation TDI. Besides, geometric TDI furnishes an intuitive physical interpretation of the synthesis of the virtual optical paths. In this paper, geometric TDI is utilized to investigate the modified second-generation TDI combinations in conjunction with a ternary search algorithm. The distinction between second-generation and modified second-generation TDI solutions is elaborated regarding the rate of change of the arm lengths with respect to the opposite cyclic directions. For the 16-link combinations, 40 second-generation TDI solutions are recovered, among which nine are identified as the modified second-generation ones. Furthermore, we explore the properties of the modified second-generation TDI solutions, which turn out to be potentially preferable in practice. Regarding the Taylor expansion of arm lengths in time, the expressions for the leading-order optical path residuals for the relevant geometric TDI combinations are derived, which are further specified using the Keplerian orbits of the spacecraft for the LISA detector constellation. The response function, noise power spectral density, and signal-to-noise ratio of the TDI solutions are given analytically and discussed. We obtain three distinct sensitivity curves among nine 16-link modified secondgeneration TDI combinations, while eight sensitivity curves are encountered out of 31 second-generation ones. It is argued that the modified second-generation TDI solutions present a quantitative advantage over their second-generation counterparts. Even though the noise suppressions of both scenarios are found to be at the same level, owing to the cancellations in the response function caused by the temporal symmetry of the arm lengths, the magnitude of the gravitational-wave signals is less pronounced for the secondgeneration TDI solutions. Moreover, analytic analysis confirms that the alternative modified secondgeneration TDI solutions are desirable as they possess fewer zeros in the average response function and the noise power spectral density, in accordance with previous findings.