Constraints on halo formation from cross-correlations with correlated variables

Abstract

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.

Halo bias in Lagrangian space: estimators and theoretical predictions

“…Similar plots for linear bias can be found for instance in Baldauf et al (2015). A convenient parametrization of linear bias is therefore Chan et al 2015;Castorina et al 2016) …”

“…Using N-body simulations, works such as Desjacques et al (2010); Paranjape et al (2013a); Biagetti et al (2014); Baldauf et al (2015); Castorina et al (2016) do clearly show the presence of a scale dependent bias term at linear level and its measured value agrees fairly well with analytical models of structure formation (Paranjape et al 2013a;Castorina et al 2016).…”

“…It can have arbitrary dependence on cosmology and redshift, and testing these possible dependences is one of the goals of this work. A first comparison of ESPτ with N-body simulations as well as explicit mathematical details on the model can be found in Castorina et al (2016). We fit β to the measurements at z = 0, finding that β = 0.3 provides a good fit to the data.…”

“…We actually do expect the shear to play an important role in the process of halo collapse, that is very likely to happen in an ellipsoidal fashion (Bond & Myers 1996;Sheth & Tormen 1999;Desjacques & Smith 2008). The work in Sheth et al (2013);Castorina et al (2016) have not shown clear evidence for tidal bias in Lagrangian space using N-body simulations. It is therefore interesting, and it is one of our main goals, to check whether this bias coefficient has non-zero value in a Lagrangian scheme and what would be the implications.…”

“…This also comes into play when we try to extend this method to b3 (see Appendix A). Furthermore, unlike b1, the functional form of the scale dependence of b s 2 is highly non-trivial (Matsubara 2008;Castorina et al 2016), and hence much more sensitive to the halo window, which is not exactly known, as well as kmax, up to which any such a fit is done.…”