The formation of CDM haloes I: Collapse thresholds and the ellipsoidal collapse model

TL;DR

This paper investigates the collapse thresholds of dark matter haloes using simulations, confirming the ellipsoidal collapse model's accuracy and proposing a modified model accounting for halo shapes to better match observed formation conditions.

Contribution

It quantifies the dependence of collapse thresholds on tidal fields and halo shapes, and introduces a modified model incorporating triaxial protohalo shapes for improved accuracy.

Findings

Ellipsoidal collapse model accurately predicts mean collapse thresholds.

Halo formation redshift influences initial density contrast.

Modified model with triaxial shapes better matches simulation data.

Abstract

In the excursion set approach to structure formation initially spherical regions of the linear density field collapse to form haloes of mass $M$ at redshift $z_{\rm id}$ if their linearly extrapolated density contrast, averaged on that scale, exceeds some critical threshold, $\delta_{\rm c}(z_{\rm id})$. The value of $\delta_{\rm c}(z_{\rm id})$ is often calculated from the spherical or ellipsoidal collapse model, which provide well-defined predictions given auxiliary properties of the tidal field at a given location. We use two cosmological simulations of structure growth in a $\Lambda$ cold dark matter scenario to quantify $\delta_{\rm c}(z_{\rm id})$, its dependence on the surrounding tidal field, as well as on the shapes of the Lagrangian regions that collapse to form haloes at $z_{\rm id}$. Our results indicate that the ellipsoidal collapse model provides an accurate description of the mean dependence of $\delta_{\rm c}(z_{\rm id})$ on both the strength of the tidal field and on halo mass. However, for a given $z_{\rm id}$, $\delta_{\rm c}(z_{\rm id})$ depends strongly on the halo's characteristic formation redshift: the earlier a halo forms, the higher its initial density contrast. Surprisingly, the majority of haloes forming $today$ fall below the ellipsoidal collapse barrier, contradicting the model predictions. We trace the origin of this effect to the non-spherical shapes of Lagrangian haloes, which arise naturally due to the asymmetry of the linear tidal field. We show that a modified collapse model, that accounts for the triaxial shape of protohaloes, provides a more accurate description of the measured minimum overdensities of recently collapsed objects.