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

Understand

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.