Halo statistics in non-gaussian cosmologies: The collapsed fraction, conditional mass function and halo bias from the path-integral excursion set method

Characterizing the level of primordial non-Gaussianity (PNG) in the initial conditions for structure formation is one of the most promising ways to test inflation and differentiate among different scenarios. The scale-dependent imprint of PNG on the large-scale clustering of galaxies and quasars has already been used to place significant constraints on the level of PNG in our observed Universe. Such measurements depend upon an accurate and robust theory of how PNG affects the bias of galactic haloes relative to the underlying matter density field. We improve upon previous work by employing a more general analytical method - the path-integral extension of the excursion set formalism - which is able to account for the non-Markovianity caused by PNG in the random-walk model used to identify haloes in the initial density field. This non-Markovianity encodes information about environmental effects on halo formation which have so far not been taken into account in analytical bias calculations. We compute both scale-dependent and scale-independent corrections to the halo bias, along the way presenting an expression for the conditional collapsed fraction for the first time, and a new expression for the conditional halo mass function. To leading order in our perturbative calculation, we recover the halo bias results of Desjacques et al., including the new scale-dependent correction reported there. However, we show that the non-Markovian dynamics from PNG can lead to marked differences in halo bias when next-to-leading-order terms are included. We quantify these differences here. We find that the next-to-leading-order corrections suppress the amplitudes of both the scale-dependent and scale-independent bias by ~5-10 per cent for massive haloes with M ~ 10¹⁵M_⊙h^-1, and ~30-40 per cent for haloes with M ~ 10¹⁴M_⊙ h^-1. The corrections appear to be more significant as the halo mass is lowered, though we caution that the apparently large effects we observe in the low-mass regime likely signal a breakdown of the perturbative approach taken here.