Stochasticity of Bias and Nonlocality of Galaxy Formation: Linear Scales

If one wants to represent the galaxy number density at some point in terms ofonly the mass density at the same point, there appears the stochasticity insuch a relation, which is referred to as ``stochastic bias''. The stochasticityis there because the galaxy number density is not merely a local function of amass density field, but it is a nonlocal functional, instead. Thus, thephenomenological stochasticity of the bias should be accounted for by nonlocalfeatures of galaxy formation processes. Based on mathematical arguments, weshow that there are simple relations between biasing and nonlocality on linearscales of density fluctuations, and that the stochasticity in Fourier spacedoes not exist on linear scales under a certain condition, even if the galaxyformation itself is a complex nonlinear and nonlocal precess. The stochasticityin real space, however, arise from the scale-dependence of bias parameter, $b$.As examples, we derive the stochastic bias parameters of simple nonlocal modelsof galaxy formation, i.e., the local Lagrangian bias models, the cooperativemodel, and the peak model. We show that the stochasticity in real space is alsoweak, except on the scales of nonlocality of the galaxy formation. Therefore,we do not have to worry too much about the stochasticity on linear scales,especially in Fourier space, even if we do not know the details of galaxyformation process.Comment: 24 pages, latex, including 2 figures, ApJ, in pres