Parameter estimation of the gravitational-wave angular power spectrum in the dirty-map space

We consider a search for the anisotropic stochastic gravitational-wave background (SGWB) that decomposes the sky map into its spherical harmonics components in order to obtain estimators of the angular power spectrum. Such a search often requires the inversion of a Fisher information matrix which contains small singular values. Rather than dealing with biases induced by regularization methods used to facilitate this matrix inversion, we opt to avoid this inversion step entirely by working in the so-called ``dirty map" space, and we introduce methodology for statistical model inference in this space. We apply our methodology to simulated model signals added to detector noise characterized by Advanced LIGO's third observing run and consider angular power spectra for both the SGWB auto-correlation search as well as a cross-correlation search between the SGWB and electromagnetic tracers of matter structure in the universe. In both cases we are able to reliably recover simulated model parameters for sufficiently strong signals up to maximum order spherical harmonic modes of $\ell_{max}=10$. We find the limitations of our methodology to arise from the computational cost of testing complex models, the assumption of Gaussianity of the angular power spectrum, and a cosmic variance-like source of uncertainty which scales with the strength of the underlying signal.