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Let delta(x) be the field of mass-density fluctuations and g (x) the corresponding field of galaxy-density fluctuations, at a given time and for a given type of object. The fields are both smoothed with a fixed window which defines the term ``local''. The local biasing relation is considered to be a random process, specified by the biasing conditional distribution P(g | delta). Let the one-point probability distribution functions (PDF) P(delta) and P(g) be of zero means and standard deviations sigma2 ident < delta2 > and sigmag2 ident < g2 >.

Define the mean biasing function b(delta) by the conditional mean,

Equation 1 (1)

This function is plotted in Figure 1. It is a natural generalization of the deterministic linear biasing relation, g = b1 delta. The function b(delta) allows for any possible non-linear biasing. We find it useful to characterize the function b(delta) by its moments bhat and btilde defined by

Equation 2 (2)

It will become clear that bhat is the natural extension of b1 and that btilde / bhat is the relevant measure of non-linearity, independent of stochasticity.

The local statistical character of the biasing can be expressed by the conditional moments of higher order about the mean at a given delta Define the random biasing field epsilon by epsilon ident g - < g | delta >, with < epsilon | delta > = 0. The local variance of epsilon at a given delta defines the biasing scatter function sigmab(delta) and by averaging over delta one obtains the local biasing scatter parameter:

Equation 3 (3)

The scaling by sigma2 is for convenience. The function < epsilon2 | delta >1/2 is marked by error bars in Figure 1. Here and below we make use of a straightforward lemma, valid for any functions p(g) and q (delta):

Equation 4 (4)

From the three basic parameters defined above one can derive other biasing parameters. A common one is the ratio of variances,

Equation 5 (5)

The second equality is a result of Eq. (4). It immediately shows that bvar is sensitive both to non-linearity and to stochasticity, with bvar geq btilde. This makes bvar biased compared to bhat,

Equation 6 (6)

Using Eq. (4), the mean parameter bhat is related to the covariance,

Equation 7 (7)

Thus, bhat is the slope of the linear regression of g on delta, which makes it a natural generalization of b1. Unlike the variance sigmag2 in Eq. (5), the covariance in Eq. (7) has no contribution from sigmab. A complementary parameter to bvar is the linear correlation coefficient,

Equation 8 (8)

The ``inverse" regression, of delta on g, yields another biasing parameter:

Equation 9 (9)

Thus, binv is biased relative to bhat, even more than bvar. The parameter binv is close to what is measured in practice by two-dimensional linear regression [52], because the errors in delta are larger than in g. Note that btilde and sigmab nicely separate the non-linearity and stochasticity, while bvar, r and binv mix them.

In the case of linear, stochastic biasing, the above parameters reduce to

Equation 10a
Equation 10b (10)

Thus, b1 leq bvar leq binv. In the case of non-linear deterministic biasing:

Equation 11 (11)

In the fully degenerate case of linear and deterministic biasing, all the b parameters are the same, and only then r = 1.

In actual applications, the above local biasing parameters are involved when the parameter ``beta'' is measured from observational data. For linear and deterministic biasing this parameter is defined unambiguously as beta1 ident f (Omega) / b1, but any deviation from this model causes us to measure different beta's by the different methods. For example, it is betavar ident f (Omega) / bvar which is determined from sigmag and sigma f (Omega). The former is typically determined from a redshift survey, and the latter either from an analysis of peculiar velocity data, from the abundance of rich clusters, or by COBE normalization of a specific power-spectrum shape. In the case of stochastic biasing bvar is always an overestimate of btilde, Eq. (5), and when the biasing is linear bvar is an overestimate of b1. Therefore betavar is underestimated accordingly.

Another useful way of estimate beta is via the linear regression of the fields in our cosmological neighborhood, e.g., -del . v(x) on g (x) [17, 33, 52]. In the mildly-non-linear regime, -del . v(x) is actually replaced by another function of the first spatial derivatives of the velocity field, which better approximates the scaled mass-density field f (Omega)delta(x) [47]. The regression is effectively delta on g, because the errors in del . v (or f delta) are typically more than twice as large as the errors in g. Hence, the measured parameter is close to betainv ident f (Omega) / binv. In the case of linear and stochastic biasing, Eq. (10), binv is an overestimate of b1 so the corresponding beta is underestimated accordingly.

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