Impact of gaps on noise and signal parameter inference

Gravitational-wave data is never analysed without some conditioning: filtering and tapering act multiplicatively on the noise, destroying the stationarity on which the Whittle likelihood --- the backbone of nearly all gravitational-wave inference --- rests. The problem is particularly evident for LISA, where gaps are expected to overlap almost all signals and the noise must be estimated jointly with those astrophysical signals. We provide an analytical and numerical framework to assess the impact of noise mis-modelling when signal and noise parameters are inferred jointly in the presence of gaps. From a Fisher-matrix formalism verified against Bayesian inference, we obtain closed forms for the model uncertainties, their maximum-likelihood estimates and the true scatter in the latter, for various approximations to the likelihood. The Fisher-based linear formalism is extended through the Godambe--White curvature to the non-linear regime that severe mis-modelling produces. We survey gap families from long, well-tapered interruptions to short, high-rate ones, injecting massive black-hole binaries. Except in the case of frequent gaps, we show that the windowed covariance is well approximated by its leading diagonal --- the window convolved with the power spectral density --- for the estimation of both signal and noise parameters. By omitting the convolution, the signal sector remains unbiased but the noise parameters, in particular the optical-metrology noise which is dominant at high frequency, is displaced by over an order of magnitude in power. At the highest glitch rates the diagonal approximation stays accurate yet loses an order of magnitude in precision. To remedy this, we propose an exact time-domain likelihood of the surviving samples which can be efficiently (and accurately solved) via preconditioned conjugate gradient methods.