Bye bye, local bias: the statistics of the halo field are not determined by the local mass density

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Abstract

Context: Bias models which relate the dark matter field to the spatial distribution of halos are widely used in current cosmological analyses. Many such models predict halos purely from the local matter density, an assumption which has not been verified in a model-agnostic setting. Bias models in perturbation theory require the inclusion of other local properties, but it is not clear whether this extends to non-perturbative approaches.

Aims: We assess the validity of the assumption that only the local dark matter density can be used to predict the number density of halos in a model-independent way and in the non-perturbative regime.

Methods: Utilising $N$-body simulations, we introduce a test wherein we study the properties of the halo counts field after spatial voxels with near-equal dark matter density have been permuted. If local-in-matter-density biasing were valid, the statistical properties of the permuted and un-permuted fields would be indistinguishable since both represent equally fair draws of the stochastic biasing model.

Results: For voxels of side length $\sim4-60\,h^{-1}{\rm\,Mpc}$ and for halos less massive than $\sim10^{15}\,h^{-1}{\rm\,M_\odot}$, we find that the permuted halo field has significantly too much power on large scales compared to the un-permuted field. We interpret this as due to these bias models removing small-scale power by not modelling correlations between neighbouring voxels. Since the permutation conserves the total variance of the halo counts field, large-scale power is substantially boosted to compensate for the missing small-scale power. This conclusion is robust to the choice of initial conditions and cosmology.

Conclusions: The assumption of local-in-matter-density halo biasing cannot, therefore, reproduce the distribution of halos across a large range of scales and halo masses, no matter how complex the model. To reproduce this distribution accurately, one must either allow the biasing to be a function of other quantities than the local matter density and/or remove the assumption that neighbouring voxels are statistically independent.

Power spectrum of the halo counts field before and after permuting voxels with approximately equal density, for halos of mass $10^{13.7}-10^{14.0} \, h^{-1} {\rm \, M}_\odot$ . The permuted fields have significantly greater power on large scales, thus LIMD halo biasing cannot reproduce the distribution of halos. The mean and standard deviation are computed over 100 permuations. The vertical coloured lines correspond to the wavenumbers used to characterise this discrepancy in later plots.