  Spatial variations in survey properties due to selection effects generate
substantial systematic errors in large-scale structure measurements in optical
galaxy surveys on very large scales. On such scales, the statistical
sensitivity of optical surveys is also limited by their finite sky coverage. By
contrast, gravitational wave (GW) sources appear to be relatively free of these
issues, provided the angular sensitivity of GW experiments can be accurately
characterized. We quantify the expected cosmological information gain from
combining the forecast LSST 3$\times$2pt analysis (combination of three 2-point
correlations of galaxy density and weak lensing shear fields) with the
large-scale auto-correlation of GW sources from proposed next-generation GW
experiments. We find that in $\Lambda$CDM and $w$CDM models, there is no
significant improvement in cosmological constraints from combining GW with LSST
3$\times$2pt over LSST alone, due to the large shot noise for the former;
however, this combination does enable a $\sim6\%$ constraint on the linear
galaxy bias of GW sources. More interestingly, the optical-GW data combination
provides tight constraints on models with primordial non-Gaussianity (PNG), due
to the predicted scale-dependent bias in PNG models on large scales. Assuming
that the largest angular scales that LSST will probe are comparable to those in
Stage III surveys ($\ell_{\rm min}\sim50$), the inclusion of next-generation GW
measurements could improve constraints on the PNG parameter $f_{\rm NL}$ by up
to a factor of $\simeq6.6$ compared to LSST alone, yielding $\sigma(f_{\rm
NL})=8.5$. These results assume the expected capability of a network of
Einstein Telescope-like GW observatories, with a detection rate of $10^6$
events/year. We investigate the sensitivity of our results to different
assumptions about future GW detectors as well as different LSST analysis
choices.
