The dynamical structure of broken power-law and double power-law models for dark matter haloes

TL;DR

This paper analyzes the dynamical properties of broken-power-law models for dark matter haloes, revealing limitations in their support by isotropic velocity distributions, which impacts their use in lensing studies.

Contribution

It provides analytical expressions for the dynamical properties of BPL models and shows that realistic parameters cannot support positive distribution functions with isotropic velocities.

Findings

BPL models with realistic parameters lack positive isotropic distribution functions.

Not all double power-law models can be supported by isotropic or radially anisotropic distributions.

Some models have negative distribution functions despite well-behaved density profiles.

Abstract

Galaxy kinematics and gravitational lensing are two complementary ways to constrain the distribution of dark matter on galaxy scales. The typical dark matter density profiles adopted in dynamical studies cannot easily be adopted in lensing studies. Ideally, a mass model should be used that has the global characteristics of realistic dark matter distributions, and that allows for an analytical calculation of the magnifications and deflection angles. A simple model with these properties, the broken-power-law (BPL) model, has very recently been introduced. We examine the dynamical structure of the family of BPL models. We derive simple closed expressions for basic dynamical properties, and study the distribution function under the assumption of velocity isotropy. We find that none of the BPL models with realistic parameters has an isotropic distribution function that is positive over the entire phase space, implying that the BPL models cannot be supported by an isotropic velocity distribution, or models with a more radially anisotropic orbital structure. This result limits the attractiveness of the BPL family as a tool for lensing studies to some degree. More generally, we find that not all members of the general family of double power-law or Zhao models, often used to model dark matter haloes, can be supported by an isotropic or radially anisotropic distribution function. In other words, the distribution function may become negative even for spherically symmetric models with a well-behaved density profile.