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 ${\mathcal{P}}_{i_1{i}_2{i}_3\dots i_{n-1}{i}_n}$, 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 ($\alpha ,\beta ,\gamma$), Monitor, Beacon, Relay, and fully symmetric Sagnac ($\zeta$) 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 $(\alpha ,\beta ,\gamma )$ and $\zeta$ combinations and the $T$ channel, all sensitivity curves exhibit a flat section; (v) the averaged response, noise PSD, and sensitivity of $\zeta$ 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.

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