Constraints on halo formation from cross-correlations with correlated variables

Cross-correlations between biased tracers and the dark matter field encode information about the physical variables which characterize these tracers. However, if the physical variables of interest are correlated with one another, then extracting this information is not as straightforward as one might naively have thought. We show how to exploit these correlations so as to estimate scale-independent bias factors of all orders in a model-independent way. We also show that failure to account for this will lead to incorrect conclusions about which variables matter and which do not. Morever, accounting for this allows one to use the scale dependence of bias to constrain the physics of halo formation; to date the argument has been phrased the other way around. We illustrate by showing that the scale dependence of linear and nonlinear bias, measured on nonlinear scales, can be used to provide consistent estimates of how the critical density for halo formation depends on halo mass. Our methods work even when the bias is nonlocal and stochastic, such as when, in addition to the spherically averaged density field and its derivatives, the quadrupolar shear field also matters for halo formation. In such models, the nonlocal bias factors are closely related to the more familiar local nonlinear bias factors, which are much easier to measure. Our analysis emphasizes the fact that biased tracers are biased because they do not sample fields (density, velocity, shear, etc.) at all positions in space in the same way that the dark matter does.