Evolution of axis ratios from phase space dynamics of triaxial collapse

TL;DR

This paper uses phase space dynamics to analyze how the shape ratios of triaxial haloes evolve over time, revealing universal behaviors and invariants that enhance understanding of structure formation.

Contribution

It introduces a phase space framework to study the evolution of halo axis ratios, identifying a universal distribution and invariants in their dynamics.

Findings

Ellipticity and prolateness increase with decreasing mass and redshift.

The scaled axis ratio parameter ${ ilde q}$ follows a universal distribution.

${ ilde q}$ is a phase space invariant for individual haloes.

Abstract

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.