Cross-Correlating the Universe: The Gravitational Wave Background and Large-Scale Structure

The nature of the gravitational wave background (GWB) is a key question in modern astrophysics and cosmology, with significant implications for understanding the structure and evolution of the Universe. We demonstrate how cross-correlating large-scale structure (LSS) tracers with the GWB spatial anisotropies can extract a clear astrophysical imprint from the GWB signal. Focusing on the unresolved population of supermassive black hole binaries (SMBHBs) as the primary source for the GWB at nHz frequencies, we construct full-sky maps of galaxy distributions and characteristic strain of the GWB to explore the relationship between GWB anisotropies and the LSS. We find that at current pulsar timing array (PTA) sensitivities, few loud SMBHBs act as Poisson-like noise. This results in anisotropies dominated by a small number of sources, making GWB maps where SMBHBs trace the LSS indistinguishable from a GWB from a uniform distribution of SMBHBs. In contrast, we find that the bulk of the unresolved SMBHBs produce anisotropies which mirror the spatial distribution of galaxies, and thus trace the LSS. Importantly, we show that cross-correlations are required to retrieve a clear LSS imprint in the GWB. Specifically, we forecast the distinguishability of this LSS signature at a $3\sigma$ level in near-future PTA experiments that probe angular scales of $\ell_{\text{max}} \geq 42$, and $5\sigma$ for $\ell_{\text{max}} \geq 72$ in optimistic settings. These values assume that GWB anisotropy maps can be reconstructed at these angular resolutions and that loud sources above a resolvability threshold can be identified and removed. Our approach opens new avenues to employ the GWB as an LSS tracer, providing unique insights into SMBHB population models and the nature of the GWB itself. Our results motivate further exploration of synergies between next-generation PTAs and LSS tracers.