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 $\mathcal{P}_{i_{1}i_{2}i_{3}\ldots i_{n-1}i_{n}}$, 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.