Noisy Pipelines I: An end-to-end pipeline to perform multivariate noise inference for LISA

Technical Summary: Noisy Pipelines I: An end-to-end pipeline to perform multivariate noise inference for LISA

Problem Statement
The Laser Interferometer Space Antenna (LISA) requires rigorous post-processing of instrumental data prior to global-fit analysis. Raw time-domain data must be filtered, downsampled, segmented, and windowed to enable frequency-domain inference. These operations alter the underlying noise process, introducing spectral artifacts such as leakage, aliasing, and non-stationarity due to the evolution of arm lengths. Existing global-fit pipelines often assume stationary, Gaussian noise with constant arm lengths or fail to account for how conditioning steps (filtering, decimation) modify the noise covariance matrix. Furthermore, the drift of the zeros in Time Delay Interferometry (TDI) transfer functions—caused by changing arm lengths—creates mismatches between static covariance models and the actual data, particularly near the spectral zeros where the instrument covariance drops to zero. Without correcting for these effects, noise parameter estimates become biased, propagating errors into astrophysical parameter inference.

Methodology
The authors present lisa-noise, a JAX-based pipeline designed to infer the multivariate noise covariance of LISA's Michelson channels ($X, Y, Z$) directly from conditioned data. The methodology integrates the following components:

1. Noise Modeling: The pipeline models the two fundamental secondary noise sources: Test Mass (TM) acceleration and Optical Metrology System (OMS) readout noise. These are represented as cubic splines in the log-log domain. The model assumes unequal and time-varying arm lengths throughout the observation.
2. TDI Propagation: Unlike approaches that fit TDI spectra directly, this method models the single-link noises and propagates them through known TDI transfer functions (mixing matrices). This requires explicitly modeling the conditioning effects on the covariance.
3. Handling Non-Stationarity (Drifting Zeros): To address the drift of TDI transfer function zeros over a data segment, the pipeline averages the numerical TDI transfer functions over multiple epochs ($n_e = 8$) within each segment. This accounts for the movement of zeros across frequency bins without assuming a static geometry.
4. Conditioning Corrections: The likelihood function incorporates exact corrections for all post-processing steps applied to the data:
   • Filtering: The response of zero-phase band-pass and anti-alias filters is applied to the covariance.
   • Decimation: Aliased power from frequencies above the new Nyquist limit is folded into the band and added to the covariance matrix.
   • Windowing: The convolution of the taper (e.g., Tukey window) with the spectral model is applied to account for spectral leakage and bin correlations.
5. Inference Schemes: The pipeline utilizes a Whittle-based likelihood. Two inference strategies are implemented:
   • Maximum Likelihood Estimation (MLE): Uses an L-BFGS-B optimizer with analytic gradients. A "zoom" scheme is employed for large priors, performing a multi-start search that iteratively narrows the parameter space to avoid local maxima.
   • Bayesian Inference: Uses the parallel-tempered ensemble sampler Eryn to sample the posterior distribution, initialized near the MLE solution.

Key Contributions

• End-to-End Conditioning: The paper introduces a framework where the effects of filtering, decimation, segmentation, and windowing are mathematically folded into the noise covariance matrix (Eq. 20), rather than treating the data as raw instrument output.
• Drifting Zero Correction: The authors demonstrate that averaging transfer functions over epochs within a segment is necessary and sufficient to model the drift of TDI zeros, ensuring the covariance model remains consistent with the data even at spectral nulls.
• Multivariate Inference: The pipeline estimates the full $3 \times 3$ complex covariance matrix of the $X, Y, Z$ channels, capturing inter-channel correlations, rather than treating channels as independent.
• Computational Efficiency: Implemented in JAX, the pipeline achieves a likelihood evaluation cost of approximately 3 ms per 7-day segment on a standard laptop CPU, enabling rapid gradient-based optimization and Bayesian sampling.

Results
The pipeline was tested on 30 days of simulated LISA data (four 7-day segments) with unequal, time-varying arm lengths and injected noise spectra.

• Recovery Accuracy: A single-start MLE from a tight prior recovers the injected spectra within 0.033 in $\log_{10}$ at every knot in a fraction of a second. A multi-start search over a wide prior ($\pm 10$ in $\log_{10}$) converges to the global maximum in approximately 5 minutes per segment.
• Posterior Consistency: The MLE results align closely with the posterior mean from MCMC sampling (within 1.5 posterior widths). The Fisher information matrix provides a reasonable approximation of parameter uncertainties, except for high-frequency TM knots where the noise is sub-dominant to OMS noise.
• Whitening Verification: The fitted covariance successfully whitens the conditioned data. The whitened residuals ($q_j$) follow the expected $\Gamma(3, 1)$ distribution across the entire frequency band, including through the drifting zeros of the transfer functions. This confirms that the model correctly accounts for the conditioning artifacts and the geometric evolution of the constellation.
• Performance: The full analysis of a 30-day dataset (four segments) takes approximately 20 minutes on a laptop, with segments processed in parallel.

Significance and Claims
The paper claims that lisa-noise provides a necessary correction for the "spectral artifacts" introduced by standard data conditioning, which previous global-fit pipelines often neglected. By explicitly modeling the conditioning steps and the drift of TDI zeros, the pipeline produces a noise covariance matrix that is consistent with the data in every frequency band and through every zero. The authors state that this approach yields unbiased noise estimates, which are critical for accurate global-fit analyses of gravitational wave sources. The work is presented as a complementary approach to direct TDI spectral fitting, with the authors suggesting a future hybrid solution that combines instrumental input with agnostic spline modeling. The code is modular, allowing for future extensions to include data gaps, glitches, and time-frequency representations.