Navarro--Frenk--White profile (NFW profile)

The Navarro--Frenk--White profile is an empirical model for the spherically averaged mass density of collisionless dark matter halos. Julio F. Navarro, Carlos S. Frenk and Simon D. M. White found that the same two-parameter shape fitted halos in their hierarchical-clustering N-body simulations over a broad resolved radial interval.^[ref]

Writing $x=r/r_s$, the profile iswith positive scale radius $r_s$ and characteristic density ${\rho }_s$. Its inner density cusp scales as $r^{-1}$ and its outer tail as $r^{-3}$. Integrating $4\pi r^2\rho$ gives the enclosed massAlthough the central density diverges, the enclosed mass tends to zero as $r^2$. At large radius the total mass diverges logarithmically, so a quoted halo mass requires a finite boundary convention, such as a specified mean enclosed overdensity. The halo concentration compares that boundary radius with $r_s$. The simulation fit is useful for consistent halo descriptions and gravitational lensing calculations; its empirical success does not require every halo or baryon-modified inner region to follow it exactly.

For an NFW profile, $x=r/r_s$ and $v_c^2=GM(r)/r$. The circular speed rises as $r^{1/2}$ at small radius and eventually decreases as $[\log (r/r_s)/r{]}^{1/2}$. Its maximum occurs at $x\simeq 2.16258$, with $v_{c,\max }^2\simeq 2.71706G{\rho }_s{r}_s^2$.

The NFW scale radius is where the logarithmic density slope equals $-2$.

The characteristic density of a Navarro--Frenk--White profile is its normalization ${\rho }_s$ at fixed scale radius of a Navarro--Frenk--White profile $r_s$. It is not the density at that radius: substituting $r=r_s$ in the NFW profile gives $\rho (r_s)={\rho }_s/4$. Once $r_s$ is fixed, doubling ${\rho }_s$ doubles the enclosed mass at every radius. This separates a profile's density normalization from its radial scale.

The cusp--core problem is the tension between the central cusps of collisionless cold-dark-matter halo profiles and the approximately constant-density cores inferred for many dwarf and low-surface-brightness galaxies.