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In the scheme outlined above, the function b(delta) contains the information about the mean biasing (via the parameter bhat) and its non-linear features (e.g., via bhat / btilde). The next quantity of interest in the case of stochastic biasing is the conditional standard deviation, the function sigmab(delta), and its variance over delta, sigmab2. In order to evaluate the actual effects of non-linear and stochastic biasing on the various measurements of beta, 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: Omega = 1 with a tauCDM power spectrum which roughly obeys the constraints from large-scale structure. The simulation mass resolution is 2 x 1010 Msun inside a box of comoving side 85 h-1 Mpc. The present epoch is identified with sigma8 = 0.6. Figure 1 demonstrates the qualitative features of the biasing scheme. The non-linear behavior at delta < 1 is characteristic of all masses, times, and smoothing scales: b(delta) << 1 near delta= -1 and it steepens to b(delta) > 1 towards delta = 0. At delta > 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 btilde2 / bhat2 = 1.08, and the scatter parameter is sigmab2 / bhat2 = 0.15. The effects of stochasticity and non-linearity in this specific case thus lead to moderate differences in the various measures of beta, 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(delta) of the matter fluctuations, the biasing function b(delta) should obey by definition at least the following two constrains. First, g geq -1 everywhere, because the galaxy density rhog cannot be negative, with g = -1 at delta = -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 delta = -1 and reduces to the linear relation near delta = 0 is [17]

Equation 18 (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 delta < 0 regime. However, the same value of b does not necessarily fit the biasing relation in the delta > 0 regime. A better approximation could thus be provided by a combination of two functions like Eq. (18) with two different parameters bn and bp in the regimes delta leq 0 and delta > 0 respectively. The parameter bn is always larger than unity while bp ranges from slightly below unity to much above unity. The best fit to Fig. 1 at z = 0 has bn ~ 2 and bp ~ 1. At high redshift both bn and bp become significantly larger.

The non-linear biasing function can alternatively be parameterized by

Equation 19 (19)

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

Equation 20 (20)

where sigma2 ident <delta2>, S ident <delta3>, 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 bhat and btilde for these non-linear toy models, we approximate the distribution P(delta) as log-normal in rho / rhobar = 1 + delta [12, 39], where sigma is the single free parameter. With Eq. (20), Assuming b2 << b1 and sigma << 1, one obtains btilde2 / bhat2 appeq 1 + (1/2) (b2 / b1)2 sigma2. This is always larger than unity, but the deviation is small. Alternatively, using the functional form of Eq. (18), with bn ranging from 1 to 5 and bp ranging from 0.5 to 3, and with sigma = 0.7, we find numerically that btilde / bhat 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 btilde / bhat is typically only slightly larger than unity. This means that the effects of non-linear biasing on measurements of beta are likely to be relatively small.

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