Why every halo spins equally slowly

Every galaxy turns, and in a smooth expanding universe nothing turns. The rotation was applied before there was anything to spin: while a halo’s material was still a mildly overdense patch spread across megaparsecs, the tidal field of its neighbours pulled unevenly on its irregular shape and set it turning, with an angular momentum that grew steadily until the patch stopped expanding and collapsed. That is tidal torque theory, and it predicts the size of a halo’s spin as well as its direction.

The size is conventionally measured by the spin parameter, $\lambda = J |E|^{1/2}/(G M^{5/2})$ — the halo’s angular momentum compared with what it would need to support itself by rotation alone. Simulated haloes have spin parameters distributed lognormally about 0.035, with a spread of about a factor of 1.6 either way, and a disc’s size follows from its halo’s spin. The distribution is remarkably uniform: dwarf-galaxy haloes and galaxy-cluster haloes, differing in mass by a factor of a million, have nearly the same distribution of spin parameters.

That uniformity is not obvious, and it is not a coincidence. It follows from tidal torque theory by a cancellation that is exact in a universe made only of matter, and it is broken only by dark energy, by a small and calculable amount. This essay follows the cancellation — how much more angular momentum heavier haloes receive, how much more tightly they are bound, why the two are in exact proportion — and the one effect that spoils it.

A spin that grows with the time the torque has had #

In tidal torque theory the angular momentum of a protohalo at time $t$ is

$$J_i(t) = a^2(t)\,\dot D(t)\,\epsilon_{ijk}\,T_{jl}\,I_{lk},$$

the product of three things: the inertia tensor $I$ of the patch’s material, measured in comoving coordinates; the tidal tensor $T$ of the surrounding gravitational field, the second derivatives of the potential at the patch’s position; and a factor $a^2\dot D$ that describes how fast the torque acts as the universe expands and the perturbations grow.

In a universe of matter alone the growth factor $D$ is proportional to the scale factor, which grows as $t^{2/3}$, and $a^2\dot D$ grows linearly with time. Angular momentum accumulates at a steady rate until the patch turns around — stops expanding with the universe and begins to collapse. After that its inertia tensor changes rapidly as the material falls together, and the coherent torque is lost; most of the final spin was acquired before turnaround. The spin a halo has is therefore set by how long the torque acted, which is set by when the halo turned around.

Heavier haloes get far more #

Every factor in the torque depends on the halo’s mass. The inertia tensor of a patch of mass $M$ is its mass times the square of its size, and its comoving size grows as $M^{1/3}$, so the lever arm grows as $M^{5/3}$. The tidal field is set by the fluctuations of the density field on scales near the patch’s size, whose amplitude $\sigma(M)$ falls as the mass rises. And the time the torque has, set by the turnaround, is later for heavier haloes, because a patch of a given rarity collapses when its linear overdensity reaches about one, and smaller fluctuations take longer to get there.

In a matter-only universe with a power-law spectrum of fluctuations, the three factors combine into a single power of the mass: a little steeper than five-thirds, the extra coming from the later turnaround of heavier haloes. A halo ten million times heavier gets about ten trillion times more angular momentum. The exponent depends on the slope of the spectrum, because the spectrum sets how much later heavier haloes collapse; the three spectra drawn span the range relevant to galaxy-mass haloes in the real universe.

and exactly as much more binding energy #

The spin parameter does not compare angular momentum with mass alone. It compares it with what a self-gravitating object of that mass and binding energy would need to be supported by rotation, and the binding energy depends on the mass and on when the halo collapsed. A halo that turns around later does so at a lower density, because the universe has expanded more, and is therefore larger and less tightly bound for its mass. Its binding energy goes as $GM^2/R$, which works out to the five-thirds power of the mass times the fluctuation amplitude $\sigma(M)$.

Put the angular momentum and the binding energy together in the spin parameter and everything cancels. The five-thirds powers of the mass from the lever arm and from the binding energy combine with the $M^{-5/2}$ in the definition to give no dependence on mass at all. The powers of the fluctuation amplitude cancel too: a later turnaround gives more time for torquing, which raises the angular momentum, and a looser halo, which lowers the angular momentum needed for support, in exactly compensating proportions. The spin parameter of a typical halo depends neither on its mass nor on the spectrum of fluctuations that made it.

The reason is dimensional, and it is worth putting plainly. A matter-only universe with a power-law spectrum has no characteristic scale: a halo of one mass at one time looks statistically identical to a halo of another mass at another time, once lengths, times and masses are rescaled. A dimensionless number describing a typical halo — and the spin parameter is dimensionless — cannot depend on which mass or time was chosen. Tidal torque theory respects that symmetry, and the spin parameter inherits the universality that the symmetry guarantees.

How the number was first estimated #

The universality was not obvious to the first people who calculated it. Peebles, in 1969, set out to explain the rotation of galaxies by tidal torques and found that for a spherical protogalaxy the first-order torque vanishes — a sphere has no lever arm for a tidal field to pull on — so he went to second order in the perturbations and found a spin growing as the five-thirds power of time. Doroshkevich showed the next year that a realistic, irregular protogalaxy does have a first-order torque, growing linearly with time as drawn above, and White worked through the full first-order theory in 1984, including the turnaround cut-off that limits the torquing time.

The numbers came from simulations. The first simulations of halo formation with enough particles to measure spins, in the 1980s, found spin parameters of about 0.05 on average, distributed lognormally, with no detectable dependence on mass — the result the scaling argument explains. Later simulations, with a hundred times the resolution and a slightly different definition of the spin parameter that avoids an awkward dependence on the halo’s energy, put the median at about 0.035, and found the dependence on mass to be at most a few per cent across the range they could test. That the theoretical expectation of universality was confirmed before it was clearly understood is common in this subject; the scaling explanation is short enough to have been written down at any time, and it explains what the simulations kept finding.

The universality also survives mergers, which deliver much of a halo’s angular momentum after its first collapse. When two haloes merge, the orbital angular momentum of the pair becomes the spin of the remnant, and spin vectors added in random directions partly cancel. In a scale-free universe the statistics of merger orbits are the same at every mass, for the same dimensional reason, so the spin parameter of merger remnants is also independent of mass. Mergers broaden the distribution — a halo that has just merged spins faster than average, one that has been quiet slower — without introducing a trend.

When haloes of each mass stop being torqued #

The real universe breaks the symmetry in two ways: its spectrum of fluctuations is not a single power law, and it contains dark energy. The first does not spoil the cancellation, because the argument above works mass by mass whatever the local slope. The second does, through the timing.

Small haloes turn around early, at redshifts of five or more, when the universe was overwhelmingly matter. Haloes of the Milky Way’s mass turn around at a redshift of about one, as the universe passes half its present size. Typical haloes of a hundred trillion solar masses have not turned around yet; the galaxy clusters of that mass that exist today are rare peaks, two or three times denser than typical at their birth, which collapsed earlier than their one-sigma counterparts will. Dark energy began to affect the growth of structure when the universe was about half its present size, so the haloes forming since then — the heaviest — acquired part of their spin in a universe that no longer obeys the matter-only scaling.

The one thing that breaks the cancellation #

The cancellation relied on $a^2\dot D$ growing exactly as the three-halves power of $D$, which is true when matter dominates. With dark energy, the growth factor $D$ slows down while the scale factor keeps accelerating, and the ratio of the two drifts.

For haloes that turned around early the factor is exactly its matter-era value. For the late-forming heavy haloes it rises, by about one per cent for haloes slightly heavier than the Milky Way’s and about six per cent for the heaviest haloes collapsing now. That is the leading way the theory allows the spin parameter to depend on mass, and it is small. Changes in how a halo’s binding energy relates to its turnaround in a universe with dark energy enter at a similar level; the detailed sign and size of the net trend depend on definitions — of the halo’s boundary, of when its spin is measured — that the simple scaling does not fix.

The surprising connection: the same moment twice #

The mass at which the cancellation begins to fail is set by the moment dark energy began to matter, when the universe was about half its present size — the moment at which the density of matter and the density of dark energy were within a factor of a few of each other. That moment’s falling so close to the present is a coincidence worth a factor of fourteen that cosmology has never explained, and it appears here in a new guise. Haloes the mass of the Milky Way’s turned around just as the universe was entering that era, and they are the most massive haloes that typically collapsed before it. The galaxies that host most of the stars in the universe sit right on the boundary between the scale-free past, in which their spins are universal, and the dark-energy present, in which spins acquired later carry a small trace of their time.

The coincidence is therefore not only that astronomers happen to live when dark energy has just become important; it is also that the most common large galaxies were assembled just before it did. Both are statements that the characteristic mass of collapsing haloes reached that of a large galaxy at about the time the expansion began to accelerate — which follows from the amplitude of the primordial fluctuations and the value of the cosmological constant, two numbers set independently and nowhere connected by known physics.

A trend buried in the scatter #

The scale of the effect is clearest against the spread that every population of haloes has.

Haloes of any one mass have spin parameters spread over a factor of nearly three within their central two-thirds, because the torque on each depends on the particular arrangement of its neighbours and on how its own inertia tensor happens to be misaligned with the tidal field — a spin that remembers the field and not its sign. The median shifts by a few per cent across the whole range of masses. A galaxy’s spin is decided by which halo it happened to form in, not by how massive that halo was. To detect the dark-energy trend in a simulation at all requires tens of thousands of haloes in each mass bin, so that the median can be measured to better than a per cent.

This is the fact that makes the simplest models of disc galaxies work. If the spin distribution is the same at every mass, then the scatter in disc sizes at fixed mass is the same at every mass, and the size–mass relation of discs has the slope of $R \propto M^{1/3}$ that the haloes’ own sizes give, with a scatter set by the spin distribution alone. Both features are observed. The same uniformity is why the fraction of discs too slowly spinning to avoid forming a bar depends only on the disc’s share of its halo’s mass and not separately on the halo’s mass.

What the derivation leaves out #

The scaling treats a halo as a patch of the linear density field with a definite turnaround time and a single inertia tensor, and assigns it the torque of the field at its centre. Simulations show that tidal torque theory in this form predicts the direction of a halo’s final spin moderately well and overpredicts its magnitude by a factor of a few, because the torque does not switch off cleanly at turnaround and because the halo’s material is not torqued as a rigid body. Those departures affect the normalisation of the spin parameter, not the scaling argument, so long as they act equally at every mass.

The haloes are taken as typical, one-sigma peaks; rarer peaks collapse earlier and sit in different tidal environments, and the spin distribution of a given mass is an average over peaks of all heights. Mergers, which deliver much of a halo’s angular momentum after its first collapse as orbital angular momentum of the merging pieces, are omitted; they obey a similar scaling and do not introduce a mass dependence in a scale-free universe, but they do add scatter. The fluctuation amplitudes $\sigma(M)$ are an approximate tabulation for the present cosmology.

Still open: a spin measured for haloes, not simulated #

The spin distribution of dark-matter haloes is known from simulations, which agree with one another on its median and width to within about ten per cent. It is not measured directly, because a halo’s angular momentum cannot be observed; what is observed is the angular momentum of the galaxy in it, which has been through the gas physics of cooling, feedback and the removal of the slowest gas before it settled. The mass independence predicted here is therefore tested only indirectly, through the uniformity of disc galaxies’ properties across mass.

The indirect test has a weak point that the argument here makes precise. If the spin distribution is universal, then any trend with mass in galaxies’ angular momenta has to come from the gas physics, and the observed relation between a galaxy’s stellar mass and its specific angular momentum, with its slope of about two-thirds, is a statement about how efficiently each mass of galaxy retains the angular momentum its halo gave it. Separating the halo’s contribution from the galaxy’s requires knowing the halo’s mass for each galaxy, which weak lensing and satellite kinematics now provide statistically.

There is one direct handle in prospect. The spins of galaxies are correlated with the tidal field of the large-scale structure around them, and that correlation — the intrinsic alignment that contaminates weak lensing — has been measured for galaxies of different masses. Tidal torque theory predicts that its strength relative to the field should, like the spin parameter, be independent of mass to first order, with departures from dark energy and from the transition between galaxies spinning perpendicular and parallel to their filaments. Whether the measured alignments follow that prediction across a factor of a thousand in mass, from dwarf galaxies to clusters, is a test of whether the cancellation that gives every halo the same spin survives the step from dark matter to light.