The large-scale gravitational bias from the quasi-linear regime

It is known that in gravitational instability scenarios the nonlinear dynamics induces non-Gaussian features in cosmological density fields that can be investigated with perturbation theory. Here, I derive the expression of the joint moments of cosmological density fields taken at two different locations. The results are valid when the density fields are filtered with a top-hat filter window function, and when the distance between the two cells is large compared to the smoothing length. In particular I show that it is possible to get the generating function of the coefficients C_p,q defined by 〈δ^p(x₁)δ^q(x ₂)〉_c = C_p,q〈δ²(x)〉 ^p+q-2〈δ(x₁)δ(x₂)〉 where δ(x) is the local smoothed density field. It is then possible to reconstruct the joint density probability distribution function (PDF), generalizing for two points what has been obtained previously for the one-point density PDF. I discuss the validity of the large separation approximation in an explicit numerical Monte Carlo integration of the C_2,1 parameter as a function of |x₁ - x₂|. A straightforward application is the calculation of the large-scale "bias" properties of the over-dense (or under-dense) regions. The properties and the shape of the bias function are presented in details and successfully compared with numerical results obtained in an N-body simulation with CDM initial conditions.