The NFW profile

Cold-dark-matter halos of every mass share one density shape — the Navarro–Frenk–White profile. It is the benchmark against which fuzzy dark matter's cored halos are measured, so it pays to know its cusp, its scale radius, and the concentration that summarizes it.

A universal shape

Navarro, Frenk & White found in simulations that CDM halos of all masses follow one two-parameter density law,

$$\rho(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},$$

with a scale radius $r_s$ and normalization $\rho_s$. Every cold halo, from dwarf to cluster, is described by this form (figure).

Cusp and outskirts

Two limits define it. Toward the centre $\rho\propto r^{-1}$ — a cusp that formally diverges. Far out $\rho\propto r^{-3}$. The scale radius $r_s$ marks the transition ($\rho\propto r^{-2}$ there). The diverging central density is the crucial feature: cold dark matter packs ever more mass into the centre, with nothing to stop it.

The NFW profile: an $r^{-1}$ cusp toward the centre steepening to $r^{-3}$ outside, with the transition at the scale radius $r_s$. The universal shape of cold-dark-matter halos.

Why halos look like this

The cusp is a generic outcome of hierarchical, collisionless collapse: small halos merge into large ones, violently relaxing into this attractor. Nothing in CDM physics halts the density climb inward — which is exactly what fuzzy dark matter's quantum pressure will change.

Worked example — mass in an NFW halo

Integrating gives the enclosed mass $M(

The FDM contrast

FDM halos share the NFW envelope in their outskirts — gravity is gravity — but replace the central cusp with a flat solitonic core (Topic 8.4). NFW is thus the yardstick for Task 2: we ask precisely where and how the FDM profile departs from it.

The NFW circular-velocity curve: rising inside $r_s$, gently falling beyond.

In our research

NFW is the $\Lambda$CDM baseline in Task 2 (our halo-structure figure). We build the FDM profile as a soliton core embedded in an NFW envelope, and our GADGET-4 halos follow the NFW concentration–mass relation — validating the control against which FDM's core is compared.

Key references

Navarro, Frenk & White (1996), The structure of CDM halos, ApJ 462, 563 (arXiv:astro-ph/9508025).
Navarro, Frenk & White (1997), ApJ 490, 493 (arXiv:astro-ph/9611107).
Wang et al. (2020), Universal structure of DM halos, Nature 585, 39.

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