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Recent results of N-body simulations have shown that current theoreticalmodels are not able to correctly predict the amplitude of the scale-dependenthalo bias induced by primordial non-Gaussianity, for models going beyond thesimplest, local quadratic case. Motivated by these discrepancies, we carefullyexamine three theoretical approaches based on (1) the statistics of thresholdedregions, (2) a peak-background split method based on separation of scales, and(3) a peak-background split method using the conditional mass function. Wefirst demonstrate that the statistics of thresholded regions, which is shown tobe equivalent at leading order to a local bias expansion, cannot explain themass-dependent deviation between theory and N-body simulations. In the twoformulations of the peak-background split on the other hand, we identify animportant, but previously overlooked, correction to the non-Gaussian bias thatstrongly depends on halo mass. This new term is in general significant for anyprimordial non-Gaussianity going beyond the simplest local fNL model. In aseparate paper, we compare these new theoretical predictions with N-bodysimulations, showing good agreement for all simulated types of non-Gaussianity.

Non-Gaussian Halo Bias Re-examined: Mass-dependent Amplitude from the Peak-Background Split and Thresholding