The Navarro–Frenk–White (NFW) profile is a spatial mass distribution of dark matter fitted to dark matter haloes identified in N-body simulations by Julio Navarro, Carlos Frenk and Simon White. The NFW profile is one of the most commonly used model profiles for dark matter halos.
In the NFW profile, the density of dark matter as a function of radius is given by:
where ρ0 and the "scale radius", Rs, are parameters which vary from halo to halo.
The integrated mass within some radius Rmax is
The total mass is divergent, but it is often useful to take the edge of the halo to be the virial radius, Rvir, which is related to the "concentration parameter", c, and scale radius via
The virial radius is often referred to as , and is defined as the radius at which the average density within this radius is 200 times the critical density. In this case, the total mass in the halo is
The value of c is roughly 10 or 15 for the Milky Way, and may range from 4 to 40 for halos of various sizes.
The integral of the squared density is
so that the mean squared density inside of Rmax is
which for the virial radius simplifies to
and the mean squared density inside the scale radius is simply
Solving Poisson's equation gives the gravitational potential
with the limits and .
Radius of the maximum circular velocity
The radius of the maximum circular velocity (confusingly sometimes also referred to as ) can be found from the maximum of as
where is the positive root of
Dark matter simulations
The NFW profile is an approximation to the equilibrium configuration of dark matter produced in simulations of collisionless dark matter particles by numerous groups of scientists. Before the dark matter virializes, the distribution of dark matter deviates from an NFW profile, and significant substructure is observed in simulations both during and after the collapse of the halos.
Alternative models, in particular the Einasto profile, have been shown to represent the dark matter profiles of simulated halos as well as or better than the NFW profile. The Einasto profile has a finite (zero) central slope, unlike the NFW profile which has a divergent (infinite) central density. Because of the limited resolution of N-body simulations, it is not yet known which model provides the best description of the central densities of simulated dark-matter halos.
Observations of halos
The observations of bright galaxies like the Milky Way and M31 may be compatible with the NFW profile, but this is open to debate. The NFW dark matter profile is not consistent with observations of low surface brightness galaxies, which have less central mass than predicted. This is known as the cusp-core or cuspy halo problem.
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