Spherical Infall Model in a Cosmological Background Density Field

We discuss the influence of the cosmological background density field on the spherical infall model. The spherical infall model has been used in the Press-Schechter formalism to evaluate the number abundance of clusters of galaxies, as well as to determine the density parameter of the universe from the infalling flow. Therefore, the understanding of collapse dynamics play a key role for extracting the cosmological information. Here, we consider the modified version of the spherical infall model. We derive the mean field equations from the Newtonian fluid equations, in which the influence of cosmological background inhomogeneity is incorporated into the averaged quantities as the {\it backreaction}. By calculating the averaged quantities explicitly, we obtain the simple expressions and find that in case of the scale-free power spectrum, the density fluctuations with the negative spectral index make the infalling velocities slow. This suggests that we underestimate the density parameter $\Omega$ when using the simple spherical infall model. In cases with the index $n>0$, the effect of background inhomogeneity could be negligible and the spherical infall model becomes the good approximation for the infalling flows. We also present a realistic example with the cold dark matter power spectrum. There, the anisotropic random velocity leads to slowing down the mean infalling velocities.