Non-local Lagrangian bias

TL;DR

This work addresses the nonlocal Lagrangian bias of cosmic structures by showing that halo formation influenced by anisotropic collapse and the local shear leads to nonlocal bias terms that affect higher-order clustering. It develops an excursion-set framework extended to multidimensional walks with a barrier depending on shear quantities like q^2, providing analytic first-crossing distributions for both uncorrelated and correlated steps and deriving how Lagrangian bias factors b1^L, b2^L, c2^L translate into Eulerian bias via G2 and related invariants. The results reveal that the nonlocal term c2 in Eulerian space can be as large as or larger than the local bias terms, especially for massive halos, and that correlations between steps moderately modify universality but preserve good agreement with N-body measurements of the halo bispectrum. Comparison with LasDamas simulated protohalos confirms the presence and magnitude of nonlocal Lagrangian bias, highlighting the necessity of including nonlocal terms in analyses of higher-order statistics such as the halo bispectrum. The framework is general and extendable to other cosmic web components, offering a principled approach to modeling the abundance and spatial distribution of halos, filaments, sheets, and voids alike.

Abstract

Halos are biased tracers of the dark matter distribution. It is often assumed that the patches from which halos formed are locally biased with respect to the initial fluctuation field, meaning that the halo-patch fluctuation field can be written as a Taylor series in that of the dark matter. If quantities other than the local density influence halo formation, then this Lagrangian bias will generically be nonlocal; the Taylor series must be performed with respect to these other variables as well. We illustrate the effect with Monte-Carlo simulations of a model in which halo formation depends on the local shear (the quadrupole of perturbation theory), and provide an analytic model which provides a good description of our results. Our model, which extends the excursion set approach to walks in more than one dimension, works both when steps in the walk are uncorrelated, as well as when there are correlations between steps. For walks with correlated steps, our model includes two distinct types of nonlocality: one is due to the fact that the initial density profile around a patch which is destined to form a halo must fall sufficiently steeply around it -- this introduces k-dependence to even the linear bias factor, but otherwise only affects the monopole of the clustering signal. The other is due to the surrounding shear field; this affects the quadratic and higher order bias factors, and introduces an angular dependence to the clustering signal. In both cases, our analysis shows that these nonlocal Lagrangian bias terms can be significant, particularly for massive halos; they must be accounted for in analyses of higher order clustering such as the halo bispectrum in Lagrangian or Eulerian space. Although we illustrate these effects using halos, our analysis and conclusions also apply to the other constituents of the cosmic web -- filaments, sheets and voids.

Figures (6)

• Figure 1: Difference between the actual overdensity $\delta_h$ within a protohalo in the GIF2 simulations of dts2013 and the expected overdensity given the value of the shear field (i.e. $B(q)$ of equation \ref{['Bq']}), shown as a function of halo mass, for two choices of the critical value $q_c$ (smaller $q_c$ means the shear matters more). Masses have been scaled to $\sigma(m)/\delta_c$ (large masses are on the left), and the overdensity difference has been scaled by $\sigma(m)$, as this removes most of the mass dependence of the scatter around the median relation.
• Figure 2: Excursion set description of the nonlocal Lagrangian bias which results from the local shear affecting the collapse threshold (Eq. \ref{['Bq']} with $q_c^2 = 8\delta_c^2$). Left: Distribution of first crossing scales for unconditioned walks (black); walks which began from $\delta_0/\delta_c = 0.2$ but $q_0=0$ (red); walks which began from $(q_0/q_c)^2 = 0.4$ but $\delta_0=0$ (magenta). Dotted curves show the Press-Schechter and Sheth-Tormen distributions, and solid curve shows our Eq. (\ref{['vfv']}) which follows from approximating the collapse barrier following Eq. (\ref{['Bqapprox']}). Right: Associated large scale bias factors; dotted curves show our predictions (Eqs. \ref{['b1b2']} and \ref{['c2']}).
• Figure 3: Comparison of excursion set predictions for the first crossing distribution (left) and associated nonlocal Lagrangian bias factors (right) for a flat $\Lambda$CDM cosmological model, when correlations between steps have been included in the analysis. Panel on the left shows results for walks with uncorrelated steps (labeled sharp-k), and for walks in which correlations arise from TopHat smoothing. The corresponding first crossing distributions are well-described by Eqs. (\ref{['vfv']}) and (\ref{['vfvMS']}), respectively (solid lines). (The curve showing Eq. (\ref{['vfvavq']}) is almost indistinguishable from that for (\ref{['vfvMS']}), so we have not bothered to show it.) We have only shown the bias factors for the TopHat smoothing filter; they are well-described by the solid lines which show Eqs. (\ref{['b1b2-corr']}) and (\ref{['c2-corr']}).
• Figure 4: Dependence of excursion set prediction for the relation between $c_2^{\rm L}$ and $b_1^{\rm L}$ on the shape of the power-spectrum. Solid and dashed lines are for correlated steps with $\Gamma^2 = 1/3$ (similar to tophat smoothing of a $\Lambda$CDM power spectrum) and 2/3, respectively; dotted line is for uncorrelated steps and is independent of $P(k)$.
• Figure 5: Comparison of the predicted relation between $c_2=c_2^{\rm L}-8b_1^{\rm L}/21$ and the Eulerian linear bias factor $b_1$ with the one estimated from bispectrum measurements of Lagrangian protohalos in N-body simulations (symbols). Solid lines show the predictions of the uncorrelated (top) and correlated (bottom) steps model; dotted and dashed lines show the local Eulerian and local Lagrangian bias models, $c_2=0$ and $c_2^{\rm L}=0$, respectively.