A revisited Correction to the Halo Mass Function for local-type Primordial non-Gaussianity

We investigate the effect of primordial non-Gaussianities on halo number counts using N-body simulations with different values of $f_{\rm NL}^{\rm loc}$. We show how current theoretical models fail to adequately describe the non-Gaussian mass function of halos identified with different overdensity thresholds, $\Delta_{\rm b}$. We explain how these discrepancies are related to a variation in the density profile of dark matter halos, finding that the internal steepness (i.e. the compactness) of halos depends on the value of $f_{\rm NL}^{\rm loc}$. We then parametrize these deviations in halo number counts with a factor $\kappa(\Delta_{\rm b})$ that modifies the linear density threshold for collapse according to the halo identification threshold used, defined with respect to the Universe background density. We rely on a second-degree polynomial to describe $\kappa$ and employ a Bayesian analysis to determine the coefficients of this polynomial. In addition, we verify the independence of the latter on the sign and absolute value of $f_{\rm NL}^{\rm loc}$. Finally, we show how this re-parametrization prevents the extraction of biased constraints on $f_{\rm NL}^{\rm loc}$, correcting for large systematic errors especially in the case of halos identified with high density thresholds. This improvement is crucial in the perspective of deriving cosmological constraints with the non-Gaussian mass function from real data, as different mass definitions can be employed depending on the properties of the survey.