Gravitational-wave response functions for space-borne detectors based on multiple geometric time-delay interferometry links

Abstract: The primary challenge for space-borne gravitational wave (GW) detectors lies in extracting the weak GW signal from instrumental noise that exceeds the signal level by many orders of magnitude. Time-delay interferometry (TDI) addresses this by suppressing the dominant laser phase noise through recombination of time-delayed measurement data. The detector's response to a GW signal is represented in the frequency domain by a response function. Currently, the GW signal response is first expressed in terms of the Doppler frequency shift in a single detection arm, and this formulation is then incorporated into specific TDI combinations to derive the corresponding response function. This paper introduces a generalized formulation for TDI combinations based on multiple geometric links. By extending the representation of the laser Doppler frequency shift to include various geometric configurations, such as round-trip and non-round-trip links, we reformulate 45 second-generation TDI combinations. For several of these, the new formulation significantly streamlines their mathematical expressions and enhances physical clarity. Our results demonstrate that the proposed link-mapping rules not only enable efficient construction of response functions for these TDI combinations but also reduce computational complexity. This approach provides a reliable theoretical and algorithmic foundation for data processing in future space-borne GW missions.

• The paper introduces a novel geometric multi-link approach for TDI that significantly enhances laser noise suppression in space-based gravitational wave detectors.
• It employs both round-trip and non-round-trip link formulations to derive concise analytical expressions for second-generation TDI observables.
• The methodology improves computational efficiency and facilitates robust noise characterization, directly impacting detector design and calibration.

Gravitational-Wave Response Functions for Space-Borne Detectors Using Multiple Geometric TDI Links

Introduction: Space-Borne Gravitational Wave Detection and Noise Suppression

The detection of gravitational waves in the low-frequency band (0.1 mHz–1 Hz) necessitates space-borne interferometric observatories such as LISA, Taiji, and TianQin due to the insurmountable seismic and Newtonian noise in ground-based detectors. Space-borne detectors form large laser-interferometric constellations with baselines spanning from $10^5$ to $10^7$ km. A defining technical challenge in these missions is the suppression of laser phase noise, which exceeds the astrophysical signal by approximately seven orders of magnitude in each arm. Time-delay interferometry (TDI) is the paradigm for noise suppression, constructing virtual equal-arm interferometric observables by time-shifting and linearly combining heterodyne phase measurements.

Geometric Formulation of TDI Observables

Traditional TDI formulation builds observables from single-link Doppler frequency measurements, incorporating appropriate delay operators to synthesize laser noise cancellation across unequal arm lengths. Previous studies have employed two main approaches for response modeling:

• The algebraic approach constructs TDI combinations by solving for polynomial coefficients in delay operators to ensure cancellation of laser phase noise terms [Algebraic-tdi-2002-Dhurandhar].
• The geometric approach identifies paths with equal optical lengths, providing an intuitive means to enumerate all feasible TDI combinations [geometric-tdi-2005-Michele-PRD].

This paper systematically extends the response function formalism by employing multiple geometric links—both round-trip and non-round-trip paths—as the elementary building blocks for constructing and analyzing TDI observables. This generalization enhances formal transparency and computational efficiency, especially for the complex, second-generation TDI combinations required to suppress laser noise under realistic, time-varying arm-lengths and constellation rotation.

Analytical Derivation of GW Response Functions for Multiple Geometric Links

Single-Link Response

For a laser link from spacecraft $a$ to $b$ (direction $\hat{n}_{ab}$, arm length $L$), the gravitational-wave-induced fractional Doppler frequency shift in the frequency domain is given by

$$F_{a\rightarrow b,A}(\Omega) = \frac{ e^{i\Omega(L+\hat k\cdot\vec r_a)/c} }{ 2(1-\mu_{ab}) } \left[ 1-e^{-iu(1-\mu_{ab})} \right]\xi_{ab,A},$$

where $u = \Omega L/c$, $\mu_{ab} = \hat{k} \cdot \hat{n}_{ab}$, $A$ denotes GW polarization, and $10^7$0 are polarization geometric factors.

Round-Trip and Non-Round-Trip Links

• Round-trip link $10^7$1 combines two single-link signals traversing $10^7$2 and $10^7$3, allowing for more compact observable expressions.
• Non-round-trip link $10^7$4 considers continuous propagation through three spacecraft, mapping directly to several advanced TDI combinations.

Analytical sky-averaged responses for these geometric links are derived, e.g., the long-wavelength limits are

$10^7$5

This reveals the relative sensitivities of different link combinations below the transfer frequency.

Recasting TDI Observables

By establishing mapping rules from individual Doppler measurements to geometric link expressions, the paper reformulates 45 second-generation TDI combinations. For several core observables, the mathematical structure becomes significantly more concise. For example, the second-generation Michelson observable $10^7$6 is compactly written as a difference of two round-trip observables (distinct optical paths):

$10^7$7

Here, $10^7$8 is a delay operator in the frequency domain. This construction elucidates the correspondence between TDI topology and GW response, and clarifies how second-generation TDI achieves superior laser phase noise suppression by exploiting additional delay degrees of freedom.

Numerical Results and Analytical Benchmarks

Explicit analytical expressions for the response functions of fundamental TDI observables, including $10^7$9 (Michelson), $a$0 (Relay), and their mapping to geometric links, are presented under the equal-arm approximation. The paper highlights:

• Direct algebraic relationships between second-generation and first-generation TDI responses, e.g., $a$1.
• Quantitative differences in frequency response curves, demonstrating how multi-link observables extend the high-frequency sensitivity of the TDI observables.
• The framework enables computation of sky-averaged or source-dependent response functions for arbitrary TDI combinations with decreased computational complexity.
• The formalism allows for direct inclusion of practical design features, such as laser phase locking, into the response analysis without affecting the GW signal propagation.

Implications for Space-Borne GW Data Analysis

The generalization of the GW response function formalism to multiple geometric links has several critical implications:

• Algorithmic Efficiency: Complex TDI combinations, required for higher-order laser noise suppression, can be constructed and their sensitivities evaluated directly from concise geometric-link representations, reducing storage and computation costs.
• Physical Insight: The explicit optical path mapping clarifies the relationship between geometrically meaningful paths and algebraically constructed TDI channels, facilitating the identification of optimal noise-suppressing and null-stream channels for detector characterization and calibration.
• Practical Design: The framework is naturally compatible with frequency planning, laser phase lock strategies, and real-world arm length variations expected in next-generation missions.
• Noise Characterization: Interchangeability between single-link, round-trip, and non-round-trip responses enables rigorous assessment of not only GW signal responses but also the transfer functions for secondary noise sources such as TTL and OMS noise.

Theoretical and Future Perspectives

This geometric multi-link formalism unifies and extends previous algebraic and geometric TDI theory, providing a robust mathematical foundation for the modeling and optimization of complex space-based GW observatories. In principle, the response function recipes given here are extendable to arbitrary array geometries, higher-generation TDI, and alternative GW polarizations, facilitating systematic exploration of detector upgrades and mission concepts.

Possible theoretical extensions include:

• Integration with full end-to-end mission simulation frameworks for LISA, Taiji, and TianQin, supporting accurate forward modeling of data products.
• Automated derivation of null-stream and optimal sensitivity channels, informed by the physical mapping of geometric-link paths and their response characteristics.
• Systematic analysis of secondary noise projections, leveraging the link-based formalism for efficient subtraction or mitigation modeling.

Conclusion

This paper presents a comprehensive geometric formalism for the construction and analysis of gravitational wave response functions in space-borne detectors, leveraging multiple geometric time-delay interferometry links. The systematic mapping of TDI combinations into round-trip and non-round-trip link combinations enables concise, computationally efficient, and physically transparent modeling of GW signal responses, underpinning both practical data processing and theoretical exploration in ongoing and future space-based GW missions (2606.28173).