The formation of CDM haloes II: collapse time and tides
  We use two cosmological simulations of structure formation in the LambdaCDM
scenario to study the evolutionary histories of dark-matter haloes and to
characterize the Lagrangian regions from which they form. We focus on haloes
identified at redshift z_id=0 and show that the classic ellipsoidal collapse
model systematically overestimates their collapse times. If one imposes that
halo collapse takes place at z_id, this model requires starting from a
significantly lower linear density contrast than what is measured in the
simulations at the locations of halo formation. We attempt to explain this
discrepancy by testing two key assumptions of the model. First, we show that
the tides felt by collapsing haloes due to the surrounding large-scale
structure evolve non-linearly. Although this effect becomes increasingly
important for low-mass haloes, accounting for it in the ellipsoidal collapse
model only marginally improves the agreement with N-body simulations. Second,
we track the time evolution of the physical volume occupied by forming haloes
and show that, after turnaround, it generally stabilizes at a well-defined
redshift, z_c>z_id, contrary to the basic assumption of extended
Press-Schechter theory based on excursion sets. We discuss the implications of
this result for understanding the origin of the mass-dependence and scatter in
the linear threshold for halo formation. Finally, we show that, when tuned for
collapse at z_c, a modified version of the ellipsoidal collapse model that also
accounts for the triaxial nature of protohaloes predicts their linear density
contrast in an unbiased way.
