Scale-dependent halo bias in the excursion set approach

If one accounts for correlations between scales, then non-local, k-dependent halo bias is part and parcel of the excursion set approach, and hence of halo model predictions for galaxy bias. We present an analysis that distinguishes between a number of different effects, each of which contributes to scale-dependent bias in real space. We show how to isolate these effects and remove the scale dependence, order by order, by cross-correlating the halo field with suitably transformed versions of the mass field. These transformations may be thought of as simple one-point, two-scale measurements that allow one to estimate quantities which are usually constrained using n-point statistics. As part of our analysis, we present a simple analytic approximation for the first-crossing distribution of walks with correlated steps which are constrained to pass through a specified point, and demonstrate its accuracy. Although we concentrate on non-linear, non-local bias with respect to a Gaussian random field, we show how to generalize our analysis to more general fields.