Concept

Spin parameter — where it appears

The dimensionless measure of how fast a dark matter halo turns, comparing its angular momentum against what rotational support would require. Its median is about 0.035 across four decades of mass, which is why it points at a scale-free origin.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.

A disc the size its halo was born with

A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe.

galaxies · Galaxy spin

A torque that stops when the patch lets go. Left, the mechanism: a protogalactic patch drawn as an ellipsoid of axis ratios 1:0.72:0.5, with the principal axes of the surrounding tidal field drawn across it at 30 degrees to its own. The torque is proportional to the difference of the patch's principal moments times the sine of twice that angle, so it is exactly zero when the two sets of axes agree — checked at both alignments — and largest at forty-five degrees. A spherical patch takes no torque whatever the field around it does, which is why the spin of every galaxy begins as a statement about its shape. Right, the angular momentum against time in units of the turnaround time: in linear theory the torque acts on a patch still expanding with the universe and the angular momentum grows as the first power of time, measured off the drawn curve as t^1.000. At turnaround the patch detaches from the expansion, its quadrupole shrinks, and the torque switches off — so a galaxy's spin is fixed before it has collapsed at all, by neighbours it will never interact with again. What the figure cannot show is the sign: the same mechanism gives no preferred direction, and the observed near-absence of alignment between neighbouring galaxies' spins is the check on that.

Spin acquired before there was anything to spin

Every galaxy turns, and nothing in a smooth expanding universe turns. The rotation was applied while the material was still a mildly overdense patch spread across megaparsecs — torqued by the tidal field of its neighbours, growing steadily with time, and switching off the moment the patch stopped expanding.

cosmology · Tidal torque theory

A spin that prefers the axis in the middle. Where the spin of a tidally torqued patch points, measured against the three principal axes of the tidal field acting on it: the distribution of the absolute cosine between the spin and each axis, from 20,000 patches with random shapes in one field of principal values −0.75, 0.09, 0.66. Principal values are those of the second derivative of the potential, so a positive one compresses. An isotropic spin would give three flat lines at one. The spin avoids the most compressive and the most stretching axes and prefers the intermediate one, because the torque about each axis is proportional to the difference of the field's principal values about the other two, and the intermediate axis is the one whose two neighbours differ most: squared, those weights are 0.11, 0.66, 0.24 of the total. Fitted as Lee and Pen's alignment parameter, the directions give a = 0.617. The preference is real and it is weak — the densest bin is 3.37 times the isotropic value — which is the whole difficulty of using spins to map a field.

A spin that remembers the field and not its sign

Tidal torque theory says where a galaxy's spin should point, not only how large it should be. It points along the middle axis of the tidal field, it is exactly perpendicular to an ideal filament, and it is an even function of the field — so a map of spins can find the directions of the cosmic web and can never say which of them is collapsing.

cosmology · Tidal torque theory $The angular momentum a halo has and the angular momentum a disc needs. The fraction of the mass with specific angular momentum below j, against j in units of the mean, for a halo's material — the universal profile fitted to simulated haloes, with shape parameter μ of 1.1, 1.25, 1.5 — and for an exponential disc with a flat rotation curve (black). Both have the same mean, as a disc that kept its halo's angular momentum would. They do not have the same shape. Below a tenth of the mean a disc holds 2% of its mass; a halo with μ = 1.25 holds 11%, and one with μ = 1.1, 15%. The halo also stops at a largest j, 4.0 times the mean for μ = 1.25, beyond which the disc still has a thin tail. A disc built from a halo's gas without any exchange of angular momentum would be too concentrated at its centre and cut off at its edge.$

A halo with too much slow gas for a disc

A disc's size follows from its halo's average spin, and the average is all the simplest account of disc formation uses. The whole distribution does not match. A halo holds five times as much of its material at low angular momentum as an exponential disc can accommodate, so gas that kept its angular momentum exactly would build a dense centre, a thin body and a hard edge — and the bulgeless discs that are common in the nearby universe say that something removed a fifth to a quarter of the slowest gas before it settled.

galaxies · Galaxy spin

Named alongside it

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

Angular momentum Tidal torque theory Dark matter halo Exponential disc Gaussian random field Inertia tensor Large-scale structure Specific angular momentum Spin alignment Tidal tensor Turnaround Adiabatic contraction

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