A theoretical approach to density-split clustering
We present an analytical model for density-split correlation functions, that probe galaxy clustering in different density environments. Specifically, we focus on the cross-correlation between density-split regions and the tracer density field. We show that these correlation functions can be expressed in terms of the two-point probability density function (PDF) of the density field. We derive analytical predictions using three levels of approximation for the two-point PDF: a bivariate Gaussian distribution, a bivariate shifted log-normal distribution, and a prediction based on the Large Deviation Theory (LDT) framework. For count-in-cell densities, obtained through spherical top-hat smoothing, one can leverage spherical collapse dynamics and LDT to predict the density two-point PDF in the large-separation regime relative to the smoothing radius. We validate our model against dark matter N-body simulations in real space, incorporating Poisson shot noise and galaxy bias. Our results show that the LDT prediction outperforms the log-normal approximation, and agrees with simulations on large scales within the cosmic variance of a typical DESI DR1 sample, despite relying on only one degree of freedom.