4. BIASING IN SIMULATIONS AND TOY MODELS

In the scheme outlined above, the function b() contains the information about the mean biasing (via the parameter ) and its non-linear features (e.g., via / ). The next quantity of interest in the case of stochastic biasing is the conditional standard deviation, the function _b(), and its variance over , _b². In order to evaluate the actual effects of non-linear and stochastic biasing on the various measurements of , one should try to evaluate these functions or parameters from simulations, theoretical approximations and observations.

In an ongoing study that generalizes earlier investigations [10, 46], we are investigating the biasing in high-resolution N-body simulations of several cosmological scenarios, both for galactic halos and for galaxies as identified using semi-analytic models [54]. We refer here to a representative cosmological model: = 1 with a CDM power spectrum which roughly obeys the constraints from large-scale structure. The simulation mass resolution is 2 x 10¹⁰ M inside a box of comoving side 85 h^-1 Mpc. The present epoch is identified with ₈ = 0.6. Figure 1 demonstrates the qualitative features of the biasing scheme. The non-linear behavior at < 1 is characteristic of all masses, times, and smoothing scales: b() << 1 near = -1 and it steepens to b() > 1 towards = 0. At > 1 the behavior strongly depends on the mass, time and smoothing scale. The scatter in the figure includes both shot noise and physical scatter which are hard to separate properly. In the case shown at z = 0, the non-linear parameter is ² / ² = 1.08, and the scatter parameter is _b² / ² = 0.15. The effects of stochasticity and non-linearity in this specific case thus lead to moderate differences in the various measures of , on the order of 20-30%. Gas-dynamics and other non-gravitational processes may extend the range of estimates even further.

Given the distribution P() of the matter fluctuations, the biasing function b() should obey by definition at least the following two constrains. First, g -1 everywhere, because the galaxy density _g cannot be negative, with g = -1 at = -1, because there are no galaxies where there is no matter. Second, <g> = 0 because g describes fluctuations about the mean galaxy density. An example for a simple functional form that obeys the constraint at = -1 and reduces to the linear relation near = 0 is [17]

(18)

The constraint <g> = 0 is to be enforced by a specific choice of the factor c for a given b. With b > 1, this functional form indeed provides a reasonable fit to the simulated halo biasing relation in the < 0 regime. However, the same value of b does not necessarily fit the biasing relation in the > 0 regime. A better approximation could thus be provided by a combination of two functions like Eq. (18) with two different parameters b_n and b_p in the regimes 0 and > 0 respectively. The parameter b_n is always larger than unity while b_p ranges from slightly below unity to much above unity. The best fit to Fig. 1 at z = 0 has b_n ~ 2 and b_p ~ 1. At high redshift both b_n and b_p become significantly larger.

The non-linear biasing function can alternatively be parameterized by

(19)

Since g must average to zero, this general power series can be written as

(20)

where ² <²>, S <³>, etc. This determines the constant term b₀. The constraint at -1 provides another relation between the parameters. Therefore, the expansion to third order contains only two free parameters out of four.

In order to evaluate the parameters and for these non-linear toy models, we approximate the distribution P() as log-normal in / = 1 + [12, 39], where is the single free parameter. With Eq. (20), Assuming b₂ << b₁ and << 1, one obtains ² / ² 1 + (1/2) (b₂ / b₁)^{2 2}. This is always larger than unity, but the deviation is small. Alternatively, using the functional form of Eq. (18), with b_n ranging from 1 to 5 and b_p ranging from 0.5 to 3, and with = 0.7, we find numerically that / is in the range 1.0 to 1.15. These two toy models, calibrated by the N-body simulations, indicate that despite the obvious non-linearity, especially in the negative regime, the non-linear parameter / is typically only slightly larger than unity. This means that the effects of non-linear biasing on measurements of are likely to be relatively small.