Measuring anisotropies in the PTA band with cross-correlations

The astrophysical gravitational wave background in the nanohertz (nHz) band is expected to be primarily composed of the superposition of signals from binaries of supermassive black holes. The spatial discreteness of these sources introduces shot noise, which, in certain regimes, would overwhelm efforts to measure the anisotropy of the gravitational wave background. Moreover, the fact the time-residual map has a finite angular resolution and the presence of pulsar noise, affects our ability to construct the angular power spectrum of the anisotropy from a time-residual map (finite resolution noise). In this work, we explicitly demonstrate, starting from first principles, that cross-correlating a gravitational wave background map with a sufficiently dense galaxy survey can mitigate this issue. This approach could potentially reveal underlying properties of the gravitational wave background that would otherwise remain obscured. We quantify both the shot noise and the finite resolution noise level and show that cross-correlating the gravitational wave background with a galaxy catalog improves by more than one order of magnitude the prospects for a first detection of the background anisotropy by a gravitational wave observatory operating in the nHz frequency range. In particular, we find that with a futuristic scenario with an effective number of frequencies equal to $N_f=10$, the detection of the spectral amplitude can be achieved combining the first $20$ multipoles, with a threshold to resolve single events SKA-like. Increasing observation time, pulsar number or reducing the pulsar white noise considerably improves the detection significance.