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Carlos S. Frenk used gravitational N-body simulations with Julio F. Navarro and Simon D. M. White to investigate dark-halo structure. Their 1997 study found a common fitted radial shape over a broad range of simulated halo masses, now represented by the Navarro--Frenk--White profile.^[ref]

Lower-mass haloes were typically denser, which the authors related to earlier assembly at a higher cosmic background mass density. The profile is an empirical fit over resolved radii, rather than a proof of a universal central singularity.

Julio F. Navarro, with Carlos S. Frenk and Simon D. M. White, quantified a common density profile for simulated dark-matter haloes. Writing $x=r/r_s$, their Navarro--Frenk--White profile isIt approaches $r^{-1}$ at small radii and $r^{-3}$ at large radii. Their 1997 work also connected characteristic halo mass density with assembly epoch.^[ref]

A halo model needs a specified outer boundary when assigning total mass: the untruncated formula has logarithmically divergent integrated mass.

The mass-concentration relation of dark-matter haloes describes a statistical trend of halo concentration with mass and epoch. The original Navarro--Frenk--White profile simulations connected denser inner structure with earlier assembly, helping explain why smaller halos were typically more concentrated in those hierarchical models. The relation has scatter and depends on the cosmology and halo definition; it is not a unique concentration fixed by mass alone.^[ref]

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.