Evolution of axis ratios from phase space dynamics of triaxial collapse
  We investigate the evolution of axis ratios of triaxial haloes using the
phase space description of triaxial collapse. In this formulation, the
evolution of the triaxial ellipsoid is described in terms of the dynamics of
eigenvalues of three important tensors: the Hessian of the gravitational
potential, the tensor of velocity derivatives and the deformation tensor. The
eigenvalues of the deformation tensor are directly related to the parameters
that describe triaxiality, namely, the minor to major and intermediate to major
axes ratios ($s$ and $q$) and the triaxiality parameter $T$. Using the phase
space equations, we evolve the eigenvalues and examine the evolution of the PDF
(probability distribution function) of the axes ratios as a function of mass
scale and redshift for Gaussian initial conditions. We find that the
ellipticity and prolateness increase with decreasing mass scale and decreasing
redshift. These trends agree with previous analytic studies but differ from
numerical simulations. However, the PDF of the scaled parameter ${\tilde q} =
(q-s)/(1-s)$ follows a universal distribution over two decades in mass range
and redshifts which is in qualitative agreement with the universality for
conditional PDF reported in simulations. We further show using the phase space
dynamics that, in fact, ${\tilde q}$ is a phase space invariant and is
conserved individually for each halo. These results, demonstrate that the phase
space analysis is a useful tool that provides a different perspective on the
evolution of perturbations and can be applied to more sophisticated models in
the future.
