arXiv:1111.4211v1 [astro-ph.CO] 17 Nov 2011 Mon. Not. R. Astron. Soc. 000, 1–10 (2011) Printed 21 November 2011 (MN L A T E X style file v2.2) The spatial and velocity bias of linear density peaks and proto-haloes in the LCDM cosmology Anna Elia ⋆ , Aaron D. Ludlow and Cristiano Porciani Argelander Institut f¨ ur Astronomie der Universit¨ at Bonn, Auf dem H¨ ugel 71, D-53121 Bonn, Germany 21 November 2011 ABSTRACT We use high resolution N-body simulations to investigate the Lagrangian bias of cold dark matter haloes within the LCDM cosmology. Our analysis focuses on “proto- haloes”, which we identify in the simulation initial conditions with the subsets of particles belonging to individual redshift-zero haloes. We then calculate the number- density and velocity-divergence fields of proto-haloes and estimate their auto spectral densities. We also measure the corresponding cross spectral densities with the linear matter distribution. We use our results to test a Lagrangian-bias model presented by Desjacques and Sheth which is based on the assumption that haloes form out of local density maxima of a specific height. Our comparison validates the predicted functional form for the scale-dependence of the bias for both the density and velocity fields. We also show that the bias coefficients are accurately predicted for the velocity divergence. On the contrary, the theoretical values for the density bias parameters do not accurately match the numerical results as a function of halo mass. This is likely due to the simplistic assumptions that relate virialized haloes to density peaks of a given height in the model. We also detect appreciable stochasticity for the Lagrangian density bias, even on very large scales. These are not included in the model at leading order but correspond to higher order corrections. Key words: cosmology: theory, large-scale structure – galaxies: haloes – methods: analytical, N-body simulation. 1 INTRODUCTION Galaxy redshift surveys are powerful probes of cosmology. The main observables able to constrain cosmological param- eters are the overall shape of the galaxy power spectrum at wavenumbers k< 0.1 Mpc −1 and the baryonic acoustic oscillations within it. These are treated as proxies for the matter power spectrum for which we can make robust the- oretical predictions. Galaxies, however, are biased tracers of the cosmic mass distribution and many features appear- ing in their power spectrum depend on how a specific ob- servational sample was selected. To reconstruct the matter power spectrum we thus need an accurate bias model whose free coefficients should be used as nuisance parameters and marginalized over. In the era of precision cosmology, where measurements of the matter power spectrum with per cent accuracy are required, this task is particularly demanding. Bias models can be divided into two broad classes. Eu- lerian biasing schemes relate the galaxy density contrast, δg(x,t), to the matter density distribution, δ, evaluated at the same time t (but not necessarily at the same spa- ⋆ E-mail: elia@astro.uni-bonn.de tial location). After smoothing the fields on large scales, so that |δ| is typically much smaller than unity, one can write (Fry & Gazta˜ naga 1993) δg(x) = B0 +  d 3 x1 B1(x − x1) δ(x1)+ (1) + 1 2  d 3 x1 d 3 x2 B2(x − x1, x − x2) δ(x1) δ(x2)+ + ..., where all fields are evaluated at the same time t and the details of the bias model are specified by the kernel func- tions, Bi . If one further assumes that biasing is local (i.e. that all kernels can be written as products of Dirac delta distributions), this reduces to δg(x)= b0 + b1 δ(x)+ b2 2 δ 2 (x)+ ..., (2) where now the bias coefficients bi are real numbers. In Lagrangian bias models, on the other hand, one con- siders the regions in the initial conditions that will collapse to form galaxies (or their hosting dark-matter haloes) at time t and writes their density contrast, δ L g (q), in terms of the linear density contrast, δ0(q). Large-scale expansions c  2011 RAS