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.[1] The NFW profile is one of the most commonly used model profiles for dark matter halos.[2]

Density distribution

In the NFW profile, the density of dark matter as a function of radius is given by:

\( \rho (r)=\frac{\rho_0}{\frac{r}{R_s}\left(1~+~\frac{r}{R_s}\right)^2} \)

where \( \rho_0 \) and the "scale radius", \( R_s \), are parameters which vary from halo to halo.

The integrated mass inside of some radius Rmax is

\( M=\int_0^{R_\max} 4\pi r^2 \rho (r) \, dr=4\pi \rho_0 R_s^3 \left[ \ln\left(\frac{R_s+R_\max}{R_s}\right)-\frac{R_\max}{R_s+R_\max}\right] \)

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

\( R_\mathrm{vir}=cR_s \)

The virial radius is often referred to as \( R_{200} , 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

\( M=\int_0^{R_\mathrm{vir}} 4\pi r^2 \rho (r) \, dr=4\pi \rho_0 R_s^3 \left[\ln(1+c) - \frac{c}{1+c}\right] \)

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

\( \int_0^{R_\max} 4\pi r^2 \rho (r)^2 \, dr=\frac{4\pi}{3} R_s^3 \rho_0^2 \left[1-\frac{R_s^3}{(R_s+R_\max)^3}\right] \)

so that the mean squared density inside of Rmax is

\(\langle \rho^2 \rangle_{R_\max}=\frac{R_s^3\rho_0^2}{R_\max^3} \left[1-\frac{R_s^3}{(R_s+R_\max)^3}\right] \)

which for the virial radius simplifies to

\( \langle \rho^2 \rangle_{R_\mathrm{vir}}=\frac{\rho_0^2}{c^3} \left[1-\frac{1}{(1+c)^3}\right] \approx \frac{\rho_0^2}{c^3} \)

and the mean squared density inside the scale radius is simply

\( \langle \rho^2 \rangle_{R_s}=\frac{7}{8}\rho_0^2 \)

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.[3] 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.[4][5] 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 both the Milky Way and M31 may be compatible with the NFW profile for the dark matter halo.[6] The dark matter profile of smaller galaxies tend to have flatter distributions of dark matter in the central region, known as the cuspy halo problem.


Navarro, Julio F.; Frenk, Carlos S.; White, Simon D. M. (May 10, 1996). "The Structure of Cold Dark Matter Halos". The Astrophysical Journal 462: 563. arXiv:astro-ph/9508025. Bibcode:1996ApJ...462..563N. doi:10.1086/177173.
Bertone, Gianfranco (2010). Particle Dark Matter: Observations, Models and Searches. Cambridge University Press. p. 762. ISBN 978-0-521-76368-4.
Y. P. Jing (20 May 2000). "The Density Profile of Equilibrium and Nonequilibrium Dark Matter Halos". The Astrophysical Journal 535 (1): 30–36. arXiv:astro-ph/9901340. Bibcode:2000ApJ...535...30J. doi:10.1086/308809.
Merritt, David; Graham, Alister; Moore, Benjamin; Diemand, Jurg; et al. (20 December 2006). "Empirical Models for Dark Matter Halos". The Astronomical Journal 132 (6): 2685–2700. arXiv:astro-ph/0509417. Bibcode:2006AJ....132.2685M. doi:10.1086/508988.
Merritt, David; et al. (May 2005). "A Universal Density Profile for Dark and Luminous Matter?". The Astrophysical Journal 624 (2): L85–L88. arXiv:astro-ph/0502515. Bibcode:2005ApJ...624L..85M. doi:10.1086/430636.
Klypin, Anatoly; Zhao, HongSheng; Somerville, Rachel S. (10 July 2002). "ΛCDM-based Models for the Milky Way and M31. I. Dynamical Models". The Astrophysical Journal 573 (2): 597–613. arXiv:astro-ph/0110390. Bibcode:2002ApJ...573..597K. doi:10.1086/340656.

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