Toward an improved analytical description of Lagrangian bias

We carry out a detailed numerical investigation of the spatial correlation function of the initial positions of cosmological dark matter halos. In this Lagrangian coordinate system, which is especially useful for analytic studies of cosmological feedback, we are able to construct cross-correlation functions of objects with varying masses and formation redshifts and compare them with a variety of analytical approaches. For the case in which both formation redshifts are equal, we find good agreement between our numerical results and the bivariate model of Scannapieco & Barkana at all masses, redshifts, and separations, while the model of Porciani and coworkers does well for all parameters except for objects with different masses at small separations. We find that the standard mapping between Lagrangian and Eulerian bias performs well for rare objects at all separations, but fails if the objects are highly nonlinear (low-sigma) peaks. In the Lagrangian case, in which the formation redshifts differ, the model of Scannapieco & Barkana does well for all separations and combinations of masses, apart from a discrepancy at small separations in situations in which the smaller object is formed earlier, and the difference between redshifts or masses is large. As this same limitation arises in the standard approach to the single-point progenitor distribution developed by Lacey & Cole, we conclude that a more complete understanding of the progenitor distribution is the most important outstanding issue in the analytic modeling of Lagrangian bias.