4.3 Density peaks and dark matter halos as toy models for galaxy biasing

Let us illustrate the biasing from numerical simulations by considering two specific and popular models: primordial density peaks and dark matter halos [86Jump To The Next Citation Point]. We use the N-body simulation data of L = 100h− 1 Mpc again for this purpose [36]. We select density peaks with the threshold of the peak height νth = 1.0, 2.0, and 3.0. As for the dark matter halos, these are identified using the standard friend-of-friend algorithm with a linking length of 0.2 in units of the mean particle separation. We select halos of mass larger than the threshold Mth = 2.0 × 1012h−1M ⊙, Mth = 5.0 × 1012h−1M ⊙, and Mth = 1.0 × 1013h−1M ⊙.

Figures 5View Image and 6View Image depict the distribution of dark matter particles (upper panel), peaks (middle panel), and halos (lower panel) in the LCDM model at z = 0 and z = 2.2 within a circular slice (comoving radius of 150h −1 Mpc and thickness of 15h −1 Mpc). We locate a fiducial observer in the center of the circle. Then the comoving position vector r for a particle with a comoving peculiar velocity v at a redshift z is observed at the position s in redshift space:

--1-- r ⋅-v-r s = r + H (z) |r| |r|, (123 )
where H (z) is the Hubble parameter at z. The right panels in Figures 5View Image and 6View Image plot the observed distribution in redshift space, where the redshift-space distortion is quite visible: The coherent velocity field enhances the structure perpendicular to the line-of-sight of the observer (squashing) while the virialized clump becomes elongated along the line-of-sight (finger-of-God).
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Figure 5: Top-view of distribution of objects at z = 0 in real (left panels) and redshift (right panels) spaces around the fiducial observer at the center: dark matter particles (top panels), peaks with ν > 2 (middle panels), and halos with M > 1.3 × 1012M ⊙ (bottom panels) in the LCDM model. The thickness of those slices is 15h −1 Mpc. (Figure taken from [86Jump To The Next Citation Point].)
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Figure 6: Same as Figure 5View Image, but at z = 2.2. (Figure taken from [86Jump To The Next Citation Point].)

We use two-point correlation functions to quantify stochasticity and nonlinearity in biasing of peaks and halos, and explore the signature of the redshift-space distortion. Since we are interested in the relation of the biased objects and the dark matter, we introduce three different correlation functions: the auto-correlation functions of dark matter and the objects, ξmm and ξoo, and their cross-correlation function ξom. In the present case, the subscript o refers to either h (halos) or ν (peaks). We also use the superscripts R and S to distinguish quantities defined in real and redshift spaces, respectively. We estimate those correlation functions using the standard pair-count method. The correlation function ξ(S) is evaluated under the distant-observer approximation.

Those correlation functions are plotted in Figures 7View Image and 8View Image for peaks and halos, respectively. The correlation functions of biased objects generally have larger amplitudes than those of mass. In nonlinear regimes (ξ > 1) the finger-of-God effect suppresses the amplitude of ξ(S) relative to ξ(R), while ξ(S) is larger than ξ(R ) in linear regimes (ξ < 1) due to the coherent velocity field.

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Figure 7: Auto- and cross-correlation functions of dark matter and peaks in SCDM (left panels), LCDM (middle panels), and OCDM (right panels) for (a) z = 0 (upper panels) and (b) z = 2.2 (lower panels). Different symbols indicate the results in real space (open squares for ν > 3, filled triangles for ν > 2, open circles for ν > 1, and crosses for dark matter), while different curves indicate those in redshift space (dashed for ν > 3, dot-dashed for ν > 2, solid for ν > 1, and dotted for dark matter). (Figure taken from [86Jump To The Next Citation Point].)
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Figure 8: Same as Figure 7View Image, but for a halo model, again for (a) z = 0 (upper panels) and (b) z = 2.2 (lower panels): Open squares and dashed lines for 13 − 1 M > 10 h M ⊙, filled triangles and dot-dashed lines for 12 −1 M > 5 × 10 h M ⊙, open circles and solid lines for 12 − 1 M > 2 × 10 h M ⊙, and crosses and dotted lines for dark matter. For the SCDM model, we only plot the correlation functions with Mth = 5 × 1012h−1M ⊙ and Mth = 1013h−1M ⊙. (Figure taken from [86Jump To The Next Citation Point].)

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