Phase Space dynamics of triaxial collapse: Joint density-velocity
  evolution
  We investigate the dynamics of triaxial collapse in terms of eigenvalues of
the deformation tensor, the velocity derivative tensor and the gravity Hessian.
Using the Bond-Myers model of ellipsoidal collapse, we derive a new set of
equations for the nine eigenvalues and examine their dynamics in phase space.
The main advantage of this form is that it eliminates the complicated elliptic
integrals that appear in the axes evolution equations and is more natural way
to understand the interplay between the perturbations.
  This paper focuses on the density-velocity dynamics. The Zeldovich
approximation implies that the three tensors are proportional; the
proportionality constant is set by demanding `no decaying modes'. We extend
this condition into the non-linear regime and find that the eigenvalues of the
gravity Hessian and the velocity derivative tensor are related as ${\tilde q}_d
+ {\tilde q}_v=1$, where the triaxiality parameter ${\tilde q} =
(\lambda_{\mathrm{max}} - \lambda_{\mathrm{inter}})/(\lambda_{\mathrm{max}} -
\lambda_{\mathrm{min}})$. This is a {\it new universal relation} holding true
over all redshifts and a range of mass scales to within a few percent accuracy.
The mean density-velocity divergence relation at late times is close to linear,
indicating that the dynamics is dictated by collapse along the largest
eigendirection. This relation has a scatter, which we show, is intimately
connected to the velocity shear. Finally, as an application, we compute the
PDFs of the two variables and compare with other forms in the literature.
