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

TL;DR

This paper develops analytic and high-accuracy approximate expressions for the sky- and polarization-averaged response and sensitivity of second-generation time-delay interferometry (TDI) for space-based GW detectors. By introducing an inverse light-path operator and decomposing TDI combinations into orthogonal $A,E,T$ channels, it demonstrates that second-generation TDIs retain the sensory performance of first-generation designs, with A and E channels sharing equal response and noise, while the T channel is comparatively weaker at low frequencies. The authors provide closed-form and asymptotic formulas for Michelson, $\alpha,\beta,\gamma$, Monitor, Beacon, Relay, Sagnac, and their corresponding $A,E,T$ channels, and show that the sensitivity curves exhibit a characteristic flat region in a broad frequency range, along with a proportionality of the Sagnac response to the T channel. These results furnish practical benchmarks for instrument optimization and data-analysis pipelines for future space-based GW detectors such as LISA, Taiji, and TianQin.

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. 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, ($α,β,γ$), 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 ($2πfL/c\lesssim3$); (iv) except for the $(α,β,γ)$ and $ζ$ combinations and the $T$ channel, all sensitivity curves exhibit a flat section in the range $f_{n}<f\lesssim 1.5/(2πL/c)$, where the noise-balance frequency $f_{n}$ separates the proof-mass- and optical-path-dominated regimes, while the response-transition frequency $\sim 1.5/(2πL/c)$ separates the response function's low- and high-frequency behaviors; (v) the averaged response, noise PSD, and sensitivity of $ζ$ 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.

Figures (9)

• Figure 1: Spacecraft configuration and definition of the one-way frequency measurements $y_{ij}$.
• Figure 2: Five basis sky- and polarization-averaged response integrals $\left|I^{+|\times}_{i}\right|$ as functions of the dimensionless frequency $u = 2\pi fL$. The solid curves represent the exact numerical evaluations, while the dashed lines show the low‑ and high‑frequency asymptotic approximations listed in Table \ref{['tab:asymptotics']}.
• Figure 3: Light paths of the second-generation TDI combinations $X_{2}$ and $\alpha_{2}$.
• Figure 4: Analytical (solid) and approximate (dashed) sky‑ and polarization‑averaged response functions of the second‑generation TDI combinations, along with their orthogonal channels. Each solid line is accompanied by a vertical dotted line of the same color, marking the response-transition frequency—the boundary between the response function’s low- and high-frequency behaviors.
• Figure 5: Noise PSDs of the second‑generation TDI combinations for the LISA mission, together with the PSDs of the corresponding orthogonal combinations. Each solid line is accompanied by a vertical dotted line of the same color, marking the noise-balance frequency $f_{n}$—the boundary between the proof-mass–dominated (low-frequency) and optical-path–dominated (high-frequency) noise regimes.