Analytical sensitivity curves of the second-generation time-delay interferometry

Abstract Forthcoming space-based gravitational-wave (GW) detectors will employ second-generation time-delay interferometry (TDI) to suppress laser frequency noise and achieve the sensitivity required for GW detection. We introduce an inverse light-path operator 𝒫 i 1 i 2 i 3… i n-1 i n , which enables simple representation of second-generation TDI combinations and a concise description of light propagation. Under the instantaneous equal-arm, equilateral triangular-constellation approximation, analytical expressions and high-accuracy approximate formulas are derived for the sky- and polarization-averaged response functions, noise power spectral densities (PSDs), and sensitivity curves of TDI Michelson, Sagnac-type ( α,β,γ ), Monitor, Beacon, Relay, and fully symmetric Sagnac ( ζ ) combinations, as well as their orthogonal A, E, T channels. The validity of the analytical results is confirmed via Monte Carlo integration with 10 4 randomly sampled sky locations and polarization angles. Our results show that: (i) second-generation TDIs have the same sensitivities as their first-generation counterparts; (ii) the A, E, T sensitivities and the optimal sensitivity are independent of the TDI generation and specific combination; (iii) the A and E channels have equal averaged responses, noise PSDs, and sensitivities, while the T channel has much weaker response and sensitivity at low frequencies; (iv) except for the ( α,β,γ ) and ζ combinations and the T channel, all sensitivity curves exhibit a flat section; (v) the averaged response, noise PSD, and sensitivity of ζ scale with those of the T channel. These analytical and approximate formulations provide useful equal-arm benchmarks for instrument optimization and data-analysis preparatory studies for future space-based GW detectors.