Dark Matter Halo Cores and Cusps

Why might a dwarf galaxy’s dark matter have a flat core instead of a cusp? Cold dark matter keeps getting denser toward a halo’s centre. Collisions between dark matter particles, the wave nature of extremely light particles, or gas blown out again and again by supernovae can each spread that centre into a flatter core. Warm dark matter mainly removes the smallest halos.

Switch between cold, warm, fuzzy and self-interacting dark matter and watch the centre of a dwarf-galaxy halo flatten or stay sharp. Compare how much small-scale structure each model leaves in the early density field.

Dark matter density at 150 pc $0.0398\,\mathrm{M_\odot\,pc^{-3}}$

Density slope at 300 pc $-0.123$

Circular speed at the star $22.6\,\mathrm{km\,s^{-1}}$

Halos of this mass, relative to cold $1.00$

Choose the dark matter

Halo mass

M_{\rm vir}

10^{10}\,\mathrm{M_\odot}

Scattering cross section per mass

\sigma/m

2.00\,\mathrm{cm^2\,g^{-1}}

Drag the gold star to read the circular speed at any radius.

Halo details, probe radius and random phases

Radius where the slope reaches minus one half $r_{-1/2}$: $0.644\,\mathrm{kpc}$
Concentration of the outer halo $c_{\rm vir}$: $17.2$
Scale radius $r_s$: $3.30\,\mathrm{kpc}$
Virial radius $r_{\rm vir}$: $56.8\,\mathrm{kpc}$
Half-mode wavenumber $k_{\rm hm}$
Wavenumber where the power halves $k_{1/2}$
Half-mode mass $M_{\rm hm}$
Soliton core radius $r_c$
Radius of one scattering per particle $r_1$: $2.57\,\mathrm{kpc}$
Core velocity dispersion $\sigma_0$: $22.7\,\mathrm{km\,s^{-1}}$
Core strength from star formation $n$

Probe radius

1.00\,\mathrm{kpc}

Both density slices use the same random phases. New phases redraw the field; the halo does not change.

Model assumptions

Each halo is a single sphere at the present day. Baryons are left out except in the feedback model. The cold halo follows the Navarro–Frenk–White profile with the Dutton and Macciò concentration relation, which is extrapolated below about fifteen billion solar masses. Warm dark matter uses the Viel fitting function for the linear cutoff and the Schneider fits for halo numbers and concentrations; any core from its thermal motion is left out, and simulations find no large constant-density core. Fuzzy dark matter places the Schive soliton inside the cold halo out to the radius where the two densities cross. When the soliton is less dense than the cusp at every radius, the page says so and joins them where they come closest. The fuzzy halo-number fit was calibrated at redshift four and above. Self-interacting dark matter joins an isothermal core to the cold halo where each particle has scattered about once in ten billion years, following Kaplinghat, Tulin and Yu; scattering is independent of speed and core collapse is not modelled. The feedback model is the coreNFW fit of Read and collaborators with ten billion years of star formation. The density slices are linear Gaussian fields filtered from a frozen CLASS spectrum; they are neither simulations nor observations. Dots trace the enclosed mass, and their circular paths are a drawing choice.

ColdWarmFuzzySelf-interactingCold + gas feedbackCLASS calculationGold star

Small-scale linear power

T^2(k)=P_X/P_{\rm CDM}

Lines: published fits. Circles: CLASS at 2, 3 or 6.5 keV.

Density profile

\rho(r)

Rotation curve

V_c=\sqrt{GM(<r)/r}

Halo numbers

n_X/n_{\rm CDM}

Focus question: If two models give the same flat core, what else could tell them apart?
One-sentence intuition: A core can come from collisions, from waves or from gas feedback, but only warm and fuzzy dark matter also remove small-scale structure from the start.

Core mathematical model

The cold halo and its concentration

\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},\qquad \log_{10}c_{\rm vir}=1.025-0.097\log_{10}\frac{M_{\rm vir}}{10^{12}\,h^{-1}\,\mathrm{M_\odot}}

Near the centre the density rises as one over radius. The concentration, the virial radius divided by the scale radius, follows the Dutton and Macciò fit to cold simulations.

Where warm dark matter cuts the power

T(k)=\left[1+(\alpha k)^{2\nu}\right]^{-5/\nu},\quad \nu=1.12,\qquad \frac{n_{\rm WDM}}{n_{\rm CDM}}=\left(1+\frac{M_{\rm hm}}{M}\right)^{-1.16}

Fast-moving warm particles smooth out the smallest ripples. The half-mode mass is the mass inside a sphere whose radius is half the wavelength where the amplitude has fallen to one half; halos near and below it become rare.

The soliton at the centre of a fuzzy halo

\rho_{\rm sol}(r)=\frac{1.9\,(m/10^{-23}\,\mathrm{eV})^{-2}\,(r_c/\mathrm{kpc})^{-4}}{\left[1+0.091\,(r/r_c)^2\right]^{8}}\,\mathrm{M_\odot\,pc^{-3}},\qquad r_c=1.6\,\frac{10^{-22}\,\mathrm{eV}}{m}\left(\frac{M_{\rm vir}}{10^{9}\,\mathrm{M_\odot}}\right)^{-1/3}\mathrm{kpc}

Schive and collaborators fitted these to their wave simulations at the present day. A lighter particle has a longer de Broglie wavelength and a larger, less dense core.

Where self-interactions have had time to act

\frac{\sigma}{m}\,\langle v\rangle\,\rho(r_1)\,t_{\rm age}=1,\qquad \sigma_0^2\,\nabla^2\ln\rho=-4\pi G\rho

Inside the radius where each particle has scattered about once, heat flows outward and the dark matter settles into an isothermal core. Density and enclosed mass match the cold halo at that radius.

A core made by gas outflows

M(<r)=M_{\rm NFW}(<r)\,\tanh^{n}\!\left(\frac{r}{r_c}\right),\qquad n=\tanh\!\left(\kappa\,\frac{t_{\rm SF}}{t_{\rm dyn}}\right),\quad r_c=\eta R_{1/2}

Read and collaborators fitted this to dwarf-galaxy simulations with repeated supernova outflows. Longer star formation gives a larger exponent and a flatter centre.

Common difficulties

A core and the nature of dark matter

Typical misconceptionA flat core proves that dark matter is not cold.

Better mental modelGas blown out by supernovae can also lower the central density of a cold halo. A core points to new physics only after feedback has been ruled out for that galaxy.

What warm dark matter changes

Typical misconceptionWarm dark matter is mainly a way to make cores.

Better mental modelIts main effect is earlier: it removes small-scale power, so fewer small halos form and the ones that do are less concentrated. Simulations find no large constant-density core from it.

A denser fuzzy core

Typical misconceptionA fuzzy dark matter core is always less dense than a cold cusp.

Better mental modelIn this Lab, heavier particles in more massive halos give a soliton denser than the cold cusp at 150 parsecs, once the published core–halo relation is applied.

Run the experiment

01
Compare a core with a cusp
Start from Collisions flatten the centre, then choose A cold cusp.
What to observe: The dots pile into a bright centre and the density at 150 parsecs rises by about six times. The slope at 300 parsecs goes from almost flat to steeper than minus one.
02
Turn up the collisions
Select Self-interacting and move the cross-section slider from its lowest to its highest value.
What to observe: The dashed core circle grows from less than a tenth of a kiloparsec to about one kiloparsec. Outside the radius where the core joins the cold halo, the rotation curve returns to the cold curve.
03
Watch small structure disappear
Choose Warm: fewer small halos and lower the warm particle mass.
What to observe: The fine grain disappears from the upper density slice, the warm curve in the halo-number chart drops at larger masses, and the halo-number readout falls.
04
Find the look-alike
Choose Gas feedback mimics a core and compare its rotation curve with the self-interacting preset.
What to observe: Both lower the inner circular speed below the cold curve. The density slice shows full cold structure in both, so the cores alone do not tell you which physics acted.

Sources and checks

The linear power uses CLASS 3.4.2 with Planck 2018 parameters. Separate Python scripts check the warm cutoff against CLASS, the fuzzy cutoff against a wave-pressure growth equation and the soliton against a Schrödinger–Poisson ground state.

Read the course on dark matter at small scales

Dark Matter Halo Cores and Cusps

What sets the density at the centre of a small halo

Core mathematical model

The cold halo and its concentration

Where warm dark matter cuts the power

The soliton at the centre of a fuzzy halo

Where self-interactions have had time to act

A core made by gas outflows

Common difficulties

A core and the nature of dark matter

What warm dark matter changes

A denser fuzzy core

Run the experiment

Compare a core with a cusp

Turn up the collisions

Watch small structure disappear

Find the look-alike

Sources and checks

Related experiments and articles

Articles

From a cusp to counting satellites