Clock-noise subtraction in geometric time-delay interferometry for space-based gravitational-wave parameter estimation

Abstract: Millihertz gravitational-wave observations with space-based interferometers require time-delay interferometry (TDI) observables whose residual instrumental noise is sufficiently controlled for both detection and parameter inference. Although TDI suppresses laser phase noise in unequal and time-dependent arms, clock jitter from onboard ultra-stable oscillators can remain above the secondary-noise floor and bias the effective noise weighting used in data analysis. We formulate a clock-noise subtraction scheme directly in the geometric-TDI framework. The construction introduces generalized clock-noise observables for the four space-time link structures that arise when both delay and time-advance operators are allowed. This makes the clock-noise residual algebraically parallel to the laser-noise residual and yields explicit subtraction terms for arbitrary two-path geometric TDI observables. We illustrate the method with representative first- and second-generation geometric TDI combinations, and test it with time-domain simulations using LISA-like orbits and noise levels. For a modified second-generation U-type observable, the subtraction suppresses the clock-noise residual below the signal region, restores the expected sensitivity to a monochromatic source, and improves the Fisher and Markov-chain Monte Carlo parameter constraints on the source amplitude, frequency and phase. These results show that clock-noise calibration is a necessary component of precision data analysis for future space-based gravitational-wave detectors.

• The paper rigorously formulates a clock-noise subtraction scheme in geometric TDI that reduces residual noise in space-based gravitational wave data.
• The study develops a recursive algorithm using time-shifted observables to efficiently cancel clock noise across various TDI combinations.
• Numerical simulations confirm that the subtraction scheme improves parameter estimation by suppressing clock jitter and restoring gravitational wave signal sensitivity.

Clock-Noise Subtraction in Geometric TDI for Space-Based GW Parameter Estimation

Problem Statement and Motivation

Millihertz-band gravitational wave astronomy requires high-fidelity parameter estimation using space-based interferometer arrays such as LISA, TianQin, and Taiji. Achieving optimal sensitivity and robust inference hinges on controlling instrumental noise sources, especially after laser phase noise cancellation via time-delay interferometry (TDI). However, clock jitter from the ultra-stable oscillators (USOs) onboard spacecraft persists as a significant residual contaminant, often exceeding the secondary-noise floor and distorting likelihood-based parameter estimation. This work rigorously formalizes clock-noise subtraction in the geometric-TDI framework, extending noise mitigation to arbitrary TDI combinations, including those involving both delay and advance operators.

Geometric TDI Framework and Clock Noise Analogy

The geometric TDI formalism represents observables as interference patterns synthesized from two virtual optical paths in a space-time diagram. Standard geometric TDI removes laser phase noise by constructing delayed combinations of inter-spacecraft data streams, but does not generically treat clock noise algebraically parallel to laser noise.

This paper systematically introduces generalized clock-noise observables relevant to the four fundamental link structures that arise—delayed and advanced links in both forward and backward time directions. By expressing clock residuals using these observables, the subtraction equations adopt an explicit algebraic structure mirroring the geometric TDI treatment of laser noise. The resulting scheme enables algorithmic clock-noise cancellation for any two-path geometric TDI observable.

Mathematical Construction and Algorithmic Implementation

The core construction is a double-sum expression for the residual clock noise in geometric TDI combinations. Intermediate variables (e.g., $\eta_{ij}$, $\eta_{ik}$) are built from measured carrier, sideband, test-mass, and reference data streams, combining delay and advance operations to propagate clock noise analogously to the laser phase noise terms. Generalized clock-noise observables $r_{ij}$, $r_{-i,j}$, $r_{i,-j}$, and $r_{-i,-j}$ encapsulate the clock jitter paths, with explicit formulas given for each combination scenario.

The paper details a recursive, exhaustive algorithm for evaluating clock-noise contributions across arbitrary geometric TDI routes. By precomputing all possible time-shifted observables (cataloged by link structure), efficient numerical subtraction is achievable across first-generation (e.g., Monitor-E), second-generation (e.g., $[X]_1^{16}$), and modified combinations (e.g., $[U]_3^{16}$). Tables enumerate the mapping from link types to clock-noise terms, facilitating implementation.

Numerical Simulations and Sensitivity Recovery

Simulations are performed using LISA-like orbital parameters and realistic noise budgets, including GW signal injection and synthetic laser, clock, test-mass, and optical path noise. Data streams are generated and processed according to the geometric TDI formalism, with subsequent application of the clock-noise subtraction algorithms. For the $[U]_3^{16}$ observable, clock-noise subtraction reliably suppresses residuals below the expected GW signal region.

The agreement between theoretical and simulated power spectral densities demonstrates that the subtraction scheme restores sensitivity and visibility of monochromatic GW sources, which would otherwise be compromised due to clock noise residuals despite laser noise suppression.

Impact on Parameter Estimation

Precision parameter estimation (PE) in TDI channels is directly limited by the effective noise spectral density entering the likelihood. Residual clock noise broadens posterior distributions of amplitude, frequency, and phase, degrading scientific inference. Bayesian PE experiments using simulated TDI $[U]_3^{16}$ data confirm that applying the clock-noise subtraction scheme contracts posteriors, reduces SNR-normalized uncertainties, and recovers signal parameters without bias. The improvement is frequency-dependent due to the spectral structure of the residual clock noise.

These results underline that clock-noise calibration is not merely an ancillary aspect but a critical component of the noise model used in GW parameter inference for millihertz-band observatories.

Practical and Theoretical Implications

The formalism unifies clock-noise subtraction with geometric TDI, supporting arbitrary delay/advance link structures and establishing a clear operational prescription. Practically, this enables robust data analysis pipelines for future space-based GW missions, guaranteeing that sensitivity losses and parameter uncertainty inflation due to clock jitter are avoided. Theoretically, the geometric analog between laser and clock noise residuals invites further refinement in noise modeling and opens prospects for more sophisticated calibration approaches (e.g., optimal frequency combs, advanced sideband schemes).

Future developments will likely focus on extending the framework to more complex mission configurations (multiple spacecraft, variable armlengths), incorporating non-stationary clock errors, and integrating clock-noise calibration into full Bayesian PE chains, possibly leveraging joint modeling with laser-noise suppression and secondary instrument noise contributions.

Conclusion

This paper rigorously formulates clock-noise subtraction within the geometric TDI framework, providing explicit algebraic and algorithmic prescriptions for noise mitigation in any TDI observable. Numerical simulation and parameter estimation studies show strong suppression of clock jitter residuals and recovery of GW signal sensitivity and inference precision. Clock-noise calibration is thus essential for the data analysis chain in millihertz-band space-based GW detectors and is formally and practically unified with geometric TDI (2607.09335).