Nonlocal Lagrangian bias

Ravi K. Sheth, Kwan Chuen Chan, and Román Scoccimarro

Phys. Rev. D 87, 083002 – Published 3 April 2013

Abstract

Halos are biased tracers of the dark matter distribution. It is often assumed that the initial 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 the dark matter density fluctuation field. 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 that 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 type of nonlocality 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, e.g., analyses of higher-order clustering in Lagrangian or Eulerian space. Comparison of our predictions with measurements of the halo bispectrum in simulations is encouraging. 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.

Received 6 August 2012

DOI:https://doi.org/10.1103/PhysRevD.87.083002

Authors & Affiliations

Ravi K. Sheth^*

Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy and Center for Particle Cosmology, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA

Kwan Chuen Chan^† and Román Scoccimarro^‡

Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, New York 10003, USA

^*sheth@ictp.it
^†kcc274@nyu.edu
^‡rs123@nyu.edu

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Vol. 87, Iss. 8 — 15 April 2013

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Images

Figure 1
Difference between the actual overdensity $δ_{h}$ within a protohalo in the GIF2 simulations of Ref. 23 and the expected overdensity given the value of the shear field [i.e., $B (q)$ of Eq. (25)], 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 $σ (m) / δ_{c}$ (large masses are on the left), and the overdensity difference has been scaled by $σ (m)$ , as this removes most of the mass dependence of the scatter around the median relation.Reuse & Permissions
Figure 2
Excursion set description of the nonlocal Lagrangian bias which results from the local shear affecting the collapse threshold (Eq. (25) with $q_{c}^{2} = 8 δ_{c}^{2}$ ). Left: Distribution of first crossing scales for unconditioned walks, walks which began from $δ_{0} / δ_{c} = 0.2$ but $q_{0} = 0$ , and walks which began from $(q_{0} / q_{c})^{2} = 0.4$ but $δ_{0} = 0$ (locii traced by symbols which are in the middle, highest and lowest at $\ln (y) = 2$ ). Dotted curves show the Press-Schechter and Sheth-Tormen distributions, and solid curve shows our Eq. (27) which follows from approximating the collapse barrier following Eq. (26). Right: Associated large scale bias factors; dotted curves show our predictions [Eqs. (29, 30)].Reuse & Permissions
Figure 3
Comparison of excursion set predictions for the first crossing distribution (left) and associated nonlocal Lagrangian bias factors (right) for a flat $Λ CDM$ 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. (27, 31), respectively (solid lines). (The curve showing Eq. (a4) is almost indistinguishable from that for (31), 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. (33, 34).Reuse & Permissions
Figure 4
Dependence of excursion set prediction for the relation between $c_{2}^{L}$ and $b_{1}^{L}$ on the shape of the power spectrum. Solid and dashed lines are for correlated steps with $Γ^{2} = 1 / 3$ (similar to TopHat smoothing of a $Λ CDM$ power spectrum) and $2 / 3$ , respectively; dotted line is for uncorrelated steps and is independent of $P (k)$ .Reuse & Permissions
Figure 5
Comparison of the predicted relation between $c_{2} = c_{2}^{L} - 8 b_{1}^{L} / 21$ 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 $c_{2}^{L} = 0$ , respectively.Reuse & Permissions
Figure 6
Dependence of excursion set prediction for the first crossing distribution of the barrier $δ_{c} (1 + q / q_{c})$ by six-dimensional walks, on $q_{c}$ . Solid curves show Eq. (a4), and dashed curves show the approximation Eq. (31). Curve labeled MS shows the $q_{c} \to \infty$ limit, and dotted curve labeled BCEK shows this limit for one-dimensional walks with uncorrelated steps. For the walks with correlated steps, the underlying power spectrum is $Λ CDM$ , and the correlations are due to smoothing with a real space TopHat.Reuse & Permissions

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