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

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. 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, ($\alpha,\beta,\gamma$), Monitor, Beacon, Relay, and Sagnac combinations, as well as their orthogonal $A, E, T$ channels. 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 ($2\pi fL/c\lesssim3$); (iv) except for the $(\alpha,\beta,\gamma)$ and $\zeta$ combinations and the $T$ channel, all sensitivity curves exhibit a flat section in the range $f_{n}<f\lesssim 1.5/(2\pi L/c)$, where the noise-balance frequency $f_{n}$ separates the proof-mass- and optical-path-dominated regimes, while the response-transition frequency $\sim 1.5/(2\pi L/c)$ separates the response function's low- and high-frequency behaviors; (v) the averaged response, noise PSD, and sensitivity of $\zeta$ scales with those of the $T$ channel. These analytical and approximate formulations provide useful benchmarks for instrument optimization and data-analysis studies for future space-based GW detectors.