  It is often assumed that the halo-patch fluctuation field can be written as a
Taylor series in the initial Lagrangian dark matter density fluctuation field.
We show that if this Lagrangian bias is local, and the initial conditions are
Gaussian, then the two-point cross-correlation between halos and mass should be
linearly proportional to the mass-mass auto-correlation function. This
statement is exact and valid on all scales; there are no higher order
contributions, e.g., from terms proportional to products or convolutions of
two-point functions, which one might have thought would appear upon truncating
the Taylor series of the halo bias function. In addition, the auto-correlation
function of locally biased tracers can be written as a Taylor series in the
auto-correlation function of the mass; there are no terms involving, e.g.,
derivatives or convolutions. Moreover, although the leading order coefficient,
the linear bias factor of the auto-correlation function is just the square of
that for the cross-correlation, it is the same as that obtained from expanding
the mean number of halos as a function of the local density only in the
large-scale limit. In principle, these relations allow simple tests of whether
or not halo bias is indeed local in Lagrangian space. We discuss why things are
more complicated in practice. We also discuss our results in light of recent
work on the renormalizability of halo bias, demonstrating that it is better to
renormalize than not. We use the Lognormal model to illustrate many of our
findings.
