  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 (2002; SB02) at all masses,
redshifts, and separations, while the model of Porciani et al. (1998) 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 SB02 model 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 (1993), we conclude that a more complete understanding of the
progenitor distribution is the most important outstanding issue in the analytic
modeling of Lagrangian bias.
