Shear and vorticity in the spherical collapse of dark matter haloes

Traditionally the spherical collapse of objects is studied with respect to a uniform background density, yielding the critical over-density $δ_\mathrm{c}$ as key ingredient to the mass function of virialized objects. Here we investigate the shear and rotation acting on a peak in a Gaussian random field. By assuming that collapsing objects mainly form at those peaks, we use this shear and rotation as external effects changing the dynamics of the spherical collapse, which is described by the Raychaudhuri equation. We therefore assume that the shear and rotation have no additional dynamics on top of their cosmological evolution and thus only appear as inhomogeneities in the differential equation.