Density profiles

A commonly used model for galactic dark matter halos is the pseudo-isothermal halo:^[17]

${\displaystyle \rho (r)=\rho _{o}\left[1+\left({\frac {r}{r_{c}}}\right)^{2}\right]^{-1}}$

where ${\displaystyle \rho _{o}}$ denotes the finite central density and ${\displaystyle r_{c}}$ the core radius. This provides a good fit to most rotation curve data. However, it cannot be a complete description, as the enclosed mass fails to converge to a finite value as the radius tends to infinity. The isothermal model is, at best, an approximation. Many effects may cause deviations from the profile predicted by this simple model. For example, (i) collapse may never reach an equilibrium state in the outer region of a dark matter halo, (ii) non-radial motion may be important, and (iii) mergers associated with the (hierarchical) formation of a halo may render the spherical-collapse model invalid.^[18]

Numerical simulations of structure formation in an expanding universe lead to the empirical NFW (Navarro–Frenk–White) profile:^[19]

${\displaystyle \rho (r)={\frac {\rho _{crit}\delta _{c}}{\left({\frac {r}{r_{s}}}\right)\left(1+{\frac {r}{r_{s}}}\right)^{2}}}}$

where ${\displaystyle r_{s}}$ is a scale radius, ${\displaystyle \delta _{c}}$ is a characteristic (dimensionless) density, and ${\displaystyle \rho _{crit}}$ = ${\displaystyle 3H^{2}/8\pi G}$ is the critical density for closure. The NFW profile is called 'universal' because it works for a large variety of halo masses, spanning four orders of magnitude, from individual galaxies to the halos of galaxy clusters. This profile has a finite gravitational potential even though the integrated mass still diverges logarithmically. It has become conventional to refer to the mass of a halo at a fiducial point that encloses an overdensity 200 times greater than the critical density of the universe, though mathematically the profile extends beyond this notational point. It was later deduced that the density profile depends on the environment, with the NFW appropriate only for isolated halos.^[20] NFW halos generally provide a worse description of galaxy data than does the pseudo-isothermal profile, leading to the cuspy halo problem.

Higher resolution computer simulations are better described by the Einasto profile:^[21]

${\displaystyle \rho (r)=\rho _{e}\exp \left[-d_{n}\left(\left({\frac {r}{r_{e}}}\right)^{\frac {1}{n}}-1\right)\right]}$

where r is the spatial (i.e., not projected) radius. The term ${\displaystyle d_{n}}$ is a function of n such that ${\displaystyle \rho _{e}}$ is the density at the radius ${\displaystyle r_{e}}$ that defines a volume containing half of the total mass. While the addition of a third parameter provides a slightly improved description of the results from numerical simulations, it is not observationally distinguishable from the 2 parameter NFW halo,^[22] and does nothing to alleviate the cuspy halo problem.

Shape

The collapse of overdensities in the cosmic density field is generally aspherical. So, there is no reason to expect the resulting halos to be spherical. Even the earliest simulations of structure formation in a CDM universe emphasized that the halos are substantially flattened.^[23] Subsequent work has shown that halo equidensity surfaces can be described by ellipsoids characterized by the lengths of their axes.^[24]

Because of uncertainties in both the data and the model predictions, it is still unclear whether the halo shapes inferred from observations are consistent with the predictions of ΛCDM cosmology.

Halo substructure

Up until the end of the 1990s, numerical simulations of halo formation revealed little substructure. With increasing computing power and better algorithms, it became possible to use greater numbers of particles and obtain better resolution. Substantial amounts of substructure are now expected.^[25][26][27] When a small halo merges with a significantly larger halo it becomes a subhalo orbiting within the potential well of its host. As it orbits, it is subjected to strong tidal forces from the host, which cause it to lose mass. In addition the orbit itself evolves as the subhalo is subjected to dynamical friction which causes it to lose energy and angular momentum to the dark matter particles of its host. Whether a subhalo survives as a self-bound entity depends on its mass, density profile, and its orbit.^[18]

Angular momentum

As originally pointed out by Hoyle^[28] and first demonstrated using numerical simulations by Efstathiou & Jones,^[29] asymmetric collapse in an expanding universe produces objects with significant angular momentum.

Numerical simulations have shown that the spin parameter distribution for halos formed by dissipation-less hierarchical clustering is well fit by a log-normal distribution, the median and width of which depend only weakly on halo mass, redshift, and cosmology:^[30]

${\displaystyle \rho (\lambda )d\lambda ={\frac {1}{{\sqrt {2\pi }}\sigma _{ln\lambda }}}\exp \left[-{\frac {\ln \left({\frac {\lambda }{\bar {\lambda }}}\right)^{2}}{2\sigma _{\ln \lambda }^{2}}}\right]{\frac {d\lambda }{\lambda }}}$

with ${\displaystyle {\bar {\lambda }}\approx 0.035}$ and ${\displaystyle \sigma _{ln\lambda }\approx 0.5}$. At all halo masses, there is a marked tendency for halos with higher spin to be in denser regions and thus to be more strongly clustered.^[31]