Tracer-Field Cross-Correlations with $k$-Nearest Neighbor Distributions

In astronomy and cosmology, significant effort is devoted to characterizing and understanding spatial cross-correlations between points - e.g. galaxy positions, high energy neutrino arrival directions, X-ray and AGN sources, and continuous field - e.g. weak lensing and Cosmic Microwave Background (CMB) maps. Recently, we introduced the $k$-nearest neighbor formalism to better characterize the clustering of discrete (point) datasets. Here we extend it to the point-field cross-correlation analysis. It combines $k$NN measurements of the point data set with measurements of the field smoothed on many scales. The resulting statistics are sensitive to all orders in the joint clustering of the points and the field. We demonstrate that this approach, unlike the 2-pt cross-correlation, can measure the statistical dependence of two datasets even when there are no linear (Gaussian) correlations. We further demonstrate that this framework is far more effective than the two-point function in detecting cross-correlations when the continuous field is contaminated by high levels of noise. For a particularly high level of noise, the cross-correlations between halos and the underlying matter field in a cosmological simulation, between $10h^{-1}{\rm Mpc}$ and $30h^{-1}{\rm Mpc}$, is detected at $>5\sigma$ significance using the technique presented here, when the two-point cross-correlation significance is $\sim 1\sigma$. Finally, we show that the $k$NN cross-correlations of halos and the matter field can be well-modeled on quasilinear scales by the Hybrid Effective Field Theory (HEFT) framework, with the same set of bias parameters as are used for the two-point cross-correlations. The substantial improvement in the statistical power of detecting cross-correlations with this method makes it a promising tool for various cosmological applications.