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We use two cosmological simulations of structure formation in the LambdaCDMscenario to study the evolutionary histories of dark-matter haloes and tocharacterize the Lagrangian regions from which they form. We focus on haloesidentified at redshift z_id=0 and show that the classic ellipsoidal collapsemodel systematically overestimates their collapse times. If one imposes thathalo collapse takes place at z_id, this model requires starting from asignificantly lower linear density contrast than what is measured in thesimulations at the locations of halo formation. We attempt to explain thisdiscrepancy by testing two key assumptions of the model. First, we show thatthe tides felt by collapsing haloes due to the surrounding large-scalestructure evolve non-linearly. Although this effect becomes increasinglyimportant for low-mass haloes, accounting for it in the ellipsoidal collapsemodel only marginally improves the agreement with N-body simulations. Second,we track the time evolution of the physical volume occupied by forming haloesand show that, after turnaround, it generally stabilizes at a well-definedredshift, z_c>z_id, contrary to the basic assumption of extendedPress-Schechter theory based on excursion sets. We discuss the implications ofthis result for understanding the origin of the mass-dependence and scatter inthe linear threshold for halo formation. Finally, we show that, when tuned forcollapse at z_c, a modified version of the ellipsoidal collapse model that alsoaccounts for the triaxial nature of protohaloes predicts their linear densitycontrast in an unbiased way.

The formation of CDM haloes II: collapse time and tides