Stochasticity of Bias and Nonlocality of Galaxy Formation: Linear Scales

If one wants to represent the galaxy number density at some point in terms of only the mass density at the same point, there appears the stochasticity in such a relation, which is referred to as ``stochastic bias''. The stochasticity is there because the galaxy number density is not merely a local function of a mass density field, but it is a nonlocal functional, instead. Thus, the phenomenological stochasticity of the bias should be accounted for by nonlocal features of galaxy formation processes. Based on mathematical arguments, we show that there are simple relations between biasing and nonlocality on linear scales of density fluctuations, and that the stochasticity in Fourier space does not exist on linear scales under a certain condition, even if the galaxy formation itself is a complex nonlinear and nonlocal precess. The stochasticity in real space, however, arise from the scale-dependence of bias parameter, $b$. As examples, we derive the stochastic bias parameters of simple nonlocal models of galaxy formation, i.e., the local Lagrangian bias models, the cooperative model, and the peak model. We show that the stochasticity in real space is also weak, 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 galaxy formation process.