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 overdensity δc as a key ingredient to themass 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. We find that the shear will always be larger than the rotation at peaks of the random field, which automatically results into a lower critical overdensity δc, since the shear always supports the collapse, while the rotation acts against it. Within this model, δc naturally inherits a mass dependence from the Gaussian random field, since smaller objects are exposed to more modes of the field. The overall effect on δc is approximately of the order of a few per cent with a decreasing trend to high masses.