Astronomy & Celestial Mechanics
the sky, computed
Concept

Nfw profile — where it appears

The density profile Navarro, Frenk and White found fits dark-matter halos in simulations: a slope of −1 near the centre steepening to −3 far out. Its two parameters are a mass and a concentration, and its central cusp is what the cusp–core problem is about.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

A cusp and a core in the same halo: 39 times apart in density, 2.34 in rotation. Two dark-matter density profiles for a halo of 10¹⁰ solar masses, drawn on logarithmic axes so a power law is a straight line and its exponent is a slope. The upper curve is the profile cold dark matter simulations produce: density rising inward without limit, with a logarithmic slope tending to -1.01 — a cusp. It is not a fit to observations; it emerges from simulations run by many groups with different codes, and its robustness is what makes it a prediction worth testing. The lower curve is a cored profile with the same virial mass, flat inside a kiloparsec, slope -0.01. What the rotation curves of gas-rich dwarf galaxies prefer is the second. The disagreement is stark where it is drawn — a factor of 39 in density at 0.10 kiloparsecs — and much less stark in what is actually measured, because a rotation speed depends on the mass enclosed rather than on the local density, and integrating a cusp over a small radius does not accumulate very much. At 1 kiloparsecs the two halos differ by a factor of 2.34 in circular speed — a real difference, and a far smaller one than the 39 in density that produced it. That compression is the whole difficulty of the problem. The observable is an integral of the quantity in dispute; integrating a cusp over a small radius does not accumulate much mass, so the sharpest disagreement lives where the instrument is bluntest. And the innermost points of a rotation curve are also the ones most affected by non-circular motions, by beam smearing, and by the inclination assumed — so the measurement is hardest exactly where it matters most. Whether the resolution is astrophysical — supernova feedback moving gas repeatedly and dragging the dark matter outward with it — or a statement about what dark matter is remains open, and the figure deliberately shows only the alternatives rather than choosing.

An argument about the innermost kiloparsec

Simulations of cold dark matter have produced the same halo for thirty years — a density rising inward without limit. The rotation curves of the smallest galaxies say the density flattens off. The disagreement is confined to a region a thousandth of the halo's size, and it has not been settled.

galaxies · Halo profiles A cusp pulled deeper by the disc inside it. The dark-matter density of a halo of 10¹² solar masses and concentration 10 before and after an exponential disc of 5·10¹⁰ solar masses and scale length 3 kpc settles slowly into its centre, computed shell by shell from the adiabatic invariant r·M(r). Far outside the disc the two profiles coincide — 4.8% apart at 83 kpc — because contraction moves mass inward without changing how much there is. Inside, the dark matter is pulled in after the baryons: at half a disc scale length the density is 2.9 times higher, and at 1 kpc the logarithmic slope goes from −1.09 to −1.25. The cusp that dark-matter-only simulations predict is therefore the shallowest the inner halo of a disc galaxy can be before feedback acts, not the profile feedback starts from. The calculation assumes circular orbits and slow settling, and simulations that include gas find the true response weaker than this.

The halo a galaxy drags inward

The cusp that dark-matter simulations predict is a halo with no galaxy in it. When gas cools and settles into a disc at the centre, the dark matter is pulled in after it, and a quantity conserved on slowly changing orbits says by how much. In a large disc galaxy the pull triples the dark matter at the centre and multiplies tenfold the energy needed to dig a core; in a dwarf it barely registers.

galaxies · Halo profiles Fitted to one curve, three halo histories agree at the Sun and not at the centre. The dark-matter density from 1 to 15 kpc from the Galactic centre, on logarithmic axes in GeV per cubic centimetre, for three halos fitted to the same flat rotation curve with the same disc of 5·10¹⁰ solar masses: one as dark-matter-only simulations produce it, and two contracted by the disc under different assumptions about the dark matter's orbits. At the Sun's 8.2 kpc they give 0.45, 0.41, 0.40 GeV/cm³, a spread of 12 per cent. At 2 kpc they give 3.2, 4.6, 4.6, a spread of 44 per cent. Inside the region the curve constrains, a fit has no freedom to move dark matter about, and the history drops out; inward of it, where the disc and bulge dominate the curve and the dark matter is a small part of the total, the history decides the answer.

A history the rotation curve absorbs

Whether the Milky Way's dark halo was pulled inward by its disc changes the dark matter at the Sun by half again — if the halo is taken from a simulation. Fit it to the Galaxy's own rotation instead, and the history drops out at the Sun's radius and survives only near the centre. The local density that detection experiments depend on is set by the disc's mass, not by the halo's past.

galaxies · Halo profiles A cusp steepened into a spike by the black hole growing in it. The dark-matter density near the Galaxy's central black hole of 4.3·10⁶ solar masses, in solar masses per cubic parsec on logarithmic axes. The dashed line is the halo's NFW cusp, falling inward as one over the radius. If the black hole grew slowly at the centre of that cusp, the dark matter would have followed it inward on adiabatically shrinking orbits, steepening the cusp inside about a fifth of the black hole's sphere of influence — here 0.4 parsecs, with the sphere of influence at 1.8 — into a spike falling as r to the minus 7/3, measured off the drawn curve. Close in, the dark matter's own annihilation would cap the density at the particle mass over the annihilation rate times the Galaxy's age — for a 100 GeV particle with the thermal cross-section, 2.8·10⁸ solar masses per cubic parsec — the flat plateau. Between the spike's edge and its plateau the density rises by 6 orders of magnitude.

A cusp a black hole would sharpen

A black hole that grows slowly at the centre of a dark-matter cusp drags the dark matter in after it, turning a density that rises as one over the radius into a spike rising as the radius to the minus seven-thirds. Whatever the cusp was, the spike comes out nearly the same — and the annihilation signal it would produce is set less by the particle than by whether the spike survived the stars around it.

galaxies · Halo profiles A cluster's mass written as a dipole on the microwave background. The change in microwave-background temperature caused by lensing behind a cluster of 3·10¹⁴ solar masses at redshift 0.5, where the unlensed background has a uniform gradient of 13 microkelvin per arcminute running left to right. Contours at ±0.5, ±1, ±1.5, ±2, ±2.5 μK: blue where the sky is warmer than it would have been, orange where it is cooler. The cluster bends light towards itself, so each point on the sky shows the temperature of a point a little closer to the cluster's centre: on the side towards which the gradient rises that point is colder, and the sky there is cooler than it would have been; on the downhill side it is warmer. The pattern is a dipole aligned with the gradient, peaking at 2.9 μK, and it carries the cluster's mass profile in its radial shape. Its orientation is set entirely by the background, not by the cluster, which is what lets it be told apart from the cluster's own emission.

A cluster weighed against the background behind it

Every cluster of galaxies has the same source behind it — the last-scattering surface, at a distance known to a fraction of a per cent — and bends that source's smooth gradient into a dipole of a few microkelvin. The dipole knows nothing about gas or galaxy shapes, so stacked over hundreds of clusters it is a mass scale that no hydrostatic assumption can bias.

cosmology · Sunyaev zeldovich

Named alongside it

The objects these essays reach for when they reach for this one.

Dark matter haloAdiabatic contractionAdiabatic invariantCusp core problemGalactic centreInner slopeRotation curveStellar feedbackBaryon fractionBeam smearingCluster mass functionDamping tail

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