How does Dark Matter form a halo?

Question

Here are the velocity components of the Milky Way (according to Sofue et al. 2013) out to 20 kpc. The vertical axis is $k$ $s^{-1}$. The horizontal axis is $kpc$.

The purple line is the rotation curve of all the components. The green line is the contribution from the bulge, the red is the contribution from the disk and the blue is the contribution from Dark Matter. Here is the mass of each of those components (in $10^9$ solar masses):

Notice that the Dark Matter basically forms a hollow sphere. There's no Dark Matter in the core where you'd expect it to be and it increases steadily to 20 kpc and beyond. Every other gas we know clumps. Gravitating matter obeys the Seric profile. What physical explanation does Dark Matter have for not collecting in the core of the sphere?

Answer 1

You have not put any axis labels on your plots.

I am guessing that the top plot show the rotation speed (in km/s) at a given radius (in kpc). Very roughly, $$v(R) \propto M(<R)^{1/2} R^{-1/2},$$ where $M(<R)$ is the mass inside radius $R$.

The second plot then actually shows $M(<R)$ vs $R$.

Any smooth mass distribution will not contain much mass within a small radius. Specifically, the mass contained within a radius $r$ will not increase towards smaller radii unless the density is increasing towards the centre more rapidly than $\rho \propto r^{-3}$.

i.e. $$M(<R) = \int^{R}_{0} 4\pi r^2 \rho(r)\ dr$$ if $\rho = Ar^{-n}$, then $$M(<R) = 4\pi A\int^{R}_{0} r^{2-n}\ dr = \frac{4\pi A}{3-n}[r^{3-n}]^{R}_{0}$$ (for $n \neq 3$). $M(<R)$ only increases as $R$ decreases, if $n>3$.

A typical model for dark matter is the Navarro, Frenk & White profile, which has a density that depends on $r^{-1}$ (so $n=1$) near the centre of the galaxy; this leads to $M(<R) \propto R^2$ (which you can just about see in your plot).

Normal matter will have a steeper profile because it loses kinetic energy due to interactions, and those processes are usually more effective in the denser regions. As a result, normal matter will relax deeper into the potential well. i.e. The density of dark matter does increase towards the centre; there is no hole; it just increases less steeply than for normal matter, which is why normal matter ends up dominating near the Galactic centre.