  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.
