Tidal shear and the consistency of microscopic Lagrangian halo approaches

We delineate the conditions under which the consistency relation for the non-Gaussian bias and the universality of the halo mass function hold in the context of microscopic Lagrangian descriptions of halos. The former is valid provided that the collapse barrier depends only on the physical fields (instead of fields normalized by their variance for example) and explicitly includes the effect of {\it all} physical fields such as the tidal shear. The latter holds provided that the response of the halo number density to a long-wavelength density fluctuation is equivalent to the response induced by shifting the spherical collapse threshold. Our results apply to any Lagrangian halo bias prescription. Effective "moving" barriers, which are ubiquitous in the literature, do not generally satisfy the consistency relation. Microscopic barriers including the tidal shear lead to two additional, second-order Lagrangian bias parameters which ensure that the consistency relation is satisfied. We provide analytic expressions for them.