Full-Covariance Bayesian Inference of Stochastic Gravitational Wave Backgrounds with Time-Domain Simulations for Taiji-like Missions

For Taiji-like missions, we implement a Bayesian spectral inference framework that combines second-generation time-domain (TD) simulations of time-delay interferometry (TDI) with a frequency-domain (FD) spectral likelihood for stochastic gravitational-wave background (SGWB) analyses. The $X,Y,Z$ Michelson streams generated with \trianglesim{} are divided into finite segments, Fourier transformed, and modeled with a segment-dependent complex $3\times 3$ covariance matrix. For each segment we evaluate the orbit-dependent response functions and noise transfer functions, allowing unequal-arm and time-evolving effects to enter through the full $XYZ$ covariance. Controlled simulations performed with \trianglesim{} show that the calculated functions reproduce the realization-averaged spectra at the few-percent level over the retained frequency band away from TDI nulls. We then compare parameter-estimation results for static equal-arm FD, equal-arm TD, and unequal-arm TD configurations, using in each case a full $XYZ$-covariance likelihood matched to the corresponding detector configuration. All three yield consistent uncertainty trends and Bayesian-evidence diagnostics for astrophysical-background recovery after marginalizing over instrumental noise and an effective Galactic double-white-dwarf foreground. Finally, in a ten-parameter model containing instrumental noise, an effective Galactic double-white-dwarf foreground, a stochastic astrophysical background, and a sound-wave spectrum from a cosmological first-order phase transition, we recover its peak amplitude and frequency and find Bayesian evidence favoring its inclusion in all three matched configurations.