Precision spectral estimation at sub-Hz frequencies: closed-form posteriors and Bayesian noise projection

We consider the problem of estimating cross-spectral quantities in the low-frequency regime, where long observation times limit averaging over large ensembles of periodograms, thereby preventing the use of approximate Gaussian statistics. This case is relevant for precision low-frequency gravitational experiments such as LISA and LISA Pathfinder. We present a Bayesian method for estimating spectral quantities in multivariate Gaussian time series. The approach, based on periodograms and Wishart statistics, yields closed-form expressions at any given frequency for the marginal posterior distributions of the individual power spectral densities, the pairwise coherence, and the multiple coherence, as well as for the joint posterior distribution of the full cross-spectral density matrix. In the context of noise projection -- where one series is modeled as a linear combination of filtered versions of the others, plus a background component -- the method also provides closed-form posteriors for both the susceptibilities, i.e., the filter transfer functions, and the power spectral density of the background. We apply the method to data from the LISA Pathfinder mission, showing effective decorrelation of temperature-induced acceleration noise and reliable estimation of its coupling coefficient.