Measuring non-local lagrangian peak bias

We investigate non-local Lagrangian bias contributions involving gradients of the linear density field, forwhich we have predictions from the excursion set peak formalism.We begin bywriting down a bias expansion which includes all the bias terms, including the non-local ones. Having checked that the model furnishes a reasonable fit to the halo mass function, we develop a one-point cross-correlation technique to measure bias factors associated with χ²-distributed quantities. We validate the method with numerical realizations of peaks of Gaussian random fields before we apply it to N-body simulations.We focus on the lowest (quadratic) order nonlocal contributions -2_χ10(k₁ · k₂) and χ₀₁[3(k₁ · k₂)²-k₁²k₂²], where k₁, k₂ are wave modes. We can reproduce our measurement of χ₁₀ if we allow for an offset between the Lagrangian halo centre-of-mass and the peak position. The sign and magnitude of χ10 is consistent with Lagrangian haloes sitting near linear density maxima. The resulting contribution to the halo bias can safely be ignored for M = 10¹³M_⊙ h^-1, but could become relevant at larger halo masses. For the second non-local bias χ01 however, we measure a much larger magnitude than predicted by our model.We speculate that some of this discrepancy might originate from non-local Lagrangian contributions induced by non-spherical collapse.