Modelling the shapes of the largest gravitationally bound objects

We combine the physics of the ellipsoidal collapse model with the excursion set theory to study the shapes of dark matter haloes. In particular, we develop an analytic approximation to the non-linear evolution that is more accurate than the Zeldovich approximation; we introduce a planar representation of halo axial ratios, which allows a concise and intuitive description of the dynamics of collapsing regions and allows one to relate the final shape of a halo to its initial shape; we provide simple physical explanations for some empirical fitting formulae obtained from numerical studies. Comparison with simulations is challenging, as there is no agreement about how to define a non-spherical gravitationally bound object. Nevertheless, we find that our model matches the conditional minor-to-intermediate axial ratio distribution rather well, although it disagrees with the numerical results in reproducing the minor-to-major axial ratio distribution. In particular, the mass dependence of the minor-to-major axis distribution appears to be the opposite to what is found in many previous numerical studies, where low-mass haloes are preferentially more spherical than high-mass haloes. In our model, the high-mass haloes are predicted to be more spherical, consistent with results based on a more recent and elaborate halo finding algorithm, and with observations of the mass dependence of the shapes of early-type galaxies. We suggest that some of the disagreement with some previous numerical studies may be alleviated if we consider only isolated haloes.