Non-Gaussian Halo Bias Re-examined: Mass-dependent Amplitude from the
  Peak-Background Split and Thresholding
  Recent results of N-body simulations have shown that current theoretical
models are not able to correctly predict the amplitude of the scale-dependent
halo bias induced by primordial non-Gaussianity, for models going beyond the
simplest, local quadratic case. Motivated by these discrepancies, we carefully
examine three theoretical approaches based on (1) the statistics of thresholded
regions, (2) a peak-background split method based on separation of scales, and
(3) a peak-background split method using the conditional mass function. We
first demonstrate that the statistics of thresholded regions, which is shown to
be equivalent at leading order to a local bias expansion, cannot explain the
mass-dependent deviation between theory and N-body simulations. In the two
formulations of the peak-background split on the other hand, we identify an
important, but previously overlooked, correction to the non-Gaussian bias that
strongly depends on halo mass. This new term is in general significant for any
primordial non-Gaussianity going beyond the simplest local fNL model. In a
separate paper, we compare these new theoretical predictions with N-body
simulations, showing good agreement for all simulated types of non-Gaussianity.
