Consistency relations for the Lagrangian halo bias and their
  implications
  The protohalo patches from which halos form are defined by a number of
constraints imposed on the Lagrangian dark matter density field. Each of these
constraints contributes to biasing the spatial distribution of the protohalos
relative to the matter. We show how measurements of this spatial distribution
-- linear combinations of protohalo bias factors -- can be used to make
inferences about the physics of halo formation. Our analysis exploits the fact
that halo bias factors satisfy consistency relations which encode this physics,
and that these relations are the same even for sub-populations in which
assembly bias has played a role. We illustrate our methods using a model in
which three parameters matter: a density threshold, the local slope and the
curvature of the smoothed density field. The latter two are nearly degenerate;
our approach naturally allows one to build an accurate effective two-parameter
model for which the consistency relations still apply. This, with an accurate
description of the smoothing window, allows one to describe the
protohalo-matter cross-correlation very well, both in Fourier and configuration
space. We then use our determination of the large scale bias parameters
together with the consistency relations, to estimate the enclosed density and
mean slope on the Lagrangian radius scale of the protohalos. Direct
measurements of these quantities, made on smaller scales than those on which
the bias parameters are typically measured, are in good agreement.
