Ellipticity and prolaticity of the initial gravitational-shear field at the position of density maxima

Abstract

Dark-matter haloes are supposed to form at the positions of maxima in the initial matter density field. The gravitational-shear field's ellipticity and prolaticity that serve as input for the ellipsoidal-collapse model, however, are derived from a distribution that does not take the additional maximum constraint into account. In this article, I quantify the variations of the most probable and the expected values of the ellipticity and the prolaticity when considering this additional constraint as well as the implications for the ellipsoidal-collapse model. Based on the statistics of Gaussian random fields, it is possible to set up a joint distribution for the eigenvalues of the gravitational-shear tensor and the matter density that incorporates the maximum constraint by invoking a vanishing first derivative and a negative definite second derivative of the density field into the calculation. In the density range relevant for cosmological structure formation, both the most probable and the expected value of the ellipticity calculated from the standard distribution used in the literature are about 3-8 per cent higher compared to the ones calculated under the additional assumption of a density maximum. Additionally, the analogous quantities for the prolaticity do not vanish but acquire slightly positive values in the range of $10^{-3}$-$10^{-2}$. For large overdensities, the predictions from both distributions converge. The values for $\delta_\mathrm{c}$ and $\Delta_\mathrm{v}$ derived from the ellipsoidal-collapse model using the standard distribution for the initial ellipticity and prolaticity are up to 4 and 6 per cent higher, respectively, than those obtained taking the additional maximum constraint into account in the range of $10^{13}$-$10^{15}~h^{-1}~\mathrm{M}_\odot$ in mass and 0-2 in redshift.