A spin that remembers the field and not its sign

The size of a galaxy’s spin, as tidal torque theory computes it, is set by three things multiplied together: the strength of the tidal field around the protogalaxy, how far the protogalaxy was from round, and how badly the two were misaligned. The theory predicts the distribution of that size and gets it roughly right. It also predicts something that is easier to state and harder to test: the direction of the spin, relative to the field that produced it.

The torque on a patch of matter is a vector, and its components are not symmetric. In the frame of the tidal field’s own principal axes the angular momentum about each axis is proportional to one off-diagonal component of the patch’s inertia tensor, multiplied by the difference of the field’s principal values about the other two axes:

$$L_1 \propto (\lambda_2-\lambda_3)\,I_{23},\qquad L_2 \propto (\lambda_3-\lambda_1)\,I_{31},\qquad L_3 \propto (\lambda_1-\lambda_2)\,I_{12}.$$

If the patch’s shape is random, the three off-diagonal components are statistically equal, and the direction of the spin is decided entirely by the three differences. Those are not equal. Order the principal values from the most stretching to the most compressive, and the largest difference is always between the two extremes — which is the coefficient of the torque about the middle axis.

Why the middle axis wins #

The result has a flavour of paradox, because the intermediate axis is the one that the field distinguishes least. It is neither the direction along which the patch is being stretched nor the one along which it is being squeezed.

The resolution is that a torque about an axis is produced by the forces in the plane perpendicular to it. A torque about the intermediate axis is produced by the difference between the most stretching and the most compressive directions — the two that differ most — acting on whatever part of the patch lies diagonally between them. A torque about either extreme axis is produced by the difference between the intermediate direction and the other extreme, which is smaller. Spin accumulates about the axis whose perpendicular plane contains the strongest contrast.

There is a curious echo of this elsewhere. A rigid body spun freely about its intermediate principal axis is unstable and turns over on its own, while here the intermediate axis is the preferred direction of spin. The two statements are about different tensors — the free-rotation result concerns the body’s own moments of inertia, the torque result concerns the external field — and nothing connects them except that in both cases the middle axis is special because it sits between the other two.

The preference is strong in a single patch’s expected torque and weaker in the directions actually drawn. Normalising each spin to a unit vector gives a weakly torqued patch the same vote as a strongly torqued one, and it flattens the distribution. The standard measure of the resulting alignment is a single number, introduced by Lee and Pen, that describes how much the spin–spin correlation tensor departs from isotropy in the direction of the squared tidal tensor. For the field drawn it comes out at about 0.62, and it varies between roughly 0.6 and 0.7 depending on the field’s shape. That is about as strong as the linear theory makes the alignment of spin directions, and every subsequent effect reduces it.

What a number like 0.62 means on the sky #

An alignment parameter is an abstraction, and it is worth translating into something that could be looked for.

The parameter is defined so that it would be one if every spin pointed exactly along the direction the squared tidal tensor favours most and zero if spins were isotropic. For the unnormalised angular momentum the torque formula gives exactly one, since the expected squared components are the squared differences of principal values and nothing else. Normalising each spin to a direction pulls it down to about 0.6, and the difference between the two numbers is itself informative: the alignment is carried disproportionately by the strongly torqued patches, which are the ones that will end up as the most rapidly rotating galaxies. A sample selected for large spin should therefore show a stronger alignment than a random sample of the same galaxies — a prediction that follows from the formula and that is worth making before anyone has checked it.

On the sky, the opening figure’s histogram corresponds to a median angle between the spin and the intermediate axis of 41 degrees, against sixty for a random direction. Twenty degrees of preference, in a quantity that cannot be measured for any single galaxy to better than ten or twenty degrees, and in a field whose intermediate axis is itself uncertain by a similar amount. The numbers explain why the alignment was predicted explicitly in 2000 and why convincing detections against the filaments of the galaxy distribution came more than a decade later.

An ideal filament spins its haloes across itself #

The shape of the tidal field changes which axis loses. A field whose two most compressive principal values are equal is one that is squeezing a region from two sides and stretching it along the third — the tidal field of a filament. A field whose two most stretching values are equal is squeezing along one direction only — the field of a sheet.

That is a clean and unambiguous prediction. In a perfectly cylindrical filament linear theory gives no spin whatever along the filament: every halo spins about an axis perpendicular to it. In a perfectly flat sheet every spin lies in the sheet.

Simulations and surveys agree for massive haloes. Their spins are preferentially perpendicular to the filaments they sit in, as the right-hand side of the figure requires. For low-mass haloes they disagree with the figure outright. Haloes below a transition mass of order $10^{12}$ solar masses at the present day spin preferentially along their filaments, which the right-hand edge says is impossible.

The flip was described in the first account of this mechanism as a history: early, smooth accretion from the surrounding sheet delivers angular momentum aligned with the filament, and later accretion along the filament delivers it across. The figure says what that history implies about the theory. Linear torque theory evaluated at a random point, with the patch’s shape independent of its environment, can only ever give the perpendicular answer. The parallel alignment of the low-mass haloes requires information that the random-point calculation throws away — namely where in the web the patch sits. A version of the theory constrained to patches near the saddle point of a filament, where the flow converges from the sheets on either side, reproduces the low-mass alignment and the transition mass. So the flip is not a failure of tidal torquing; it is a failure of assuming that a protogalaxy is at a typical point of a Gaussian field rather than at a special one.

That distinction has a general form worth keeping. A statistical theory evaluated at random points predicts what a random point does, and galaxies do not form at random points. They form at peaks of the density field, and near a peak the field’s derivatives are correlated with the fact that it is a peak: the curvature is larger than average, the neighbouring saddles are arranged in a particular way, and the tidal field has a predictable relation to the local shape. Every prediction of torque theory that has been compared in detail with simulations has needed that conditioning, and the ones that did not need it — the scale-free distribution of spin magnitudes, the growth linear in time — are the ones that depend only on the field’s two-point statistics.

A sheet and a filament that spin identically #

The most consequential property of the torque formula is not in any one component but in its symmetry. Every component is a difference of principal values multiplied by a component of the inertia tensor, and the direction statistics depend only on the squares of those differences. Reversing the sign of every principal value leaves every squared difference unchanged.

A field and its negative are physically very different. One is a region of excess density pulling matter in along two axes; the other is a region of deficit letting matter escape along two axes. One is a filament forming and the other is the empty space between filaments. A census of spin directions cannot tell which is which.

This was noticed as soon as the idea of reconstructing the density field from galaxy spins was proposed. The spin–spin correlation gives the square of the normalised tidal tensor, and taking a matrix square root returns the tensor only up to the signs of its eigenvalues. The principal directions come back, and the principal values come back in absolute value, and the half of the information that says whether the region is overdense or underdense is gone. It has to be supplied from somewhere else — from the galaxy counts, from peculiar velocities, or from a prior that most of the volume is underdense.

The contrast with shapes is exact, and it is the reason the two kinds of alignment behave so differently in a lensing survey. A pressure-supported galaxy stretched by the tidal field carries an alignment linear in the field: beside a cluster its long axis points at the cluster, beside a void it points along the void’s edge, and the two are perpendicular. Its shape knows the sign. A disc’s orientation follows its spin, which is quadratic, and a disc forming beside a cluster and one forming beside a void can be spun identically. What is a signal of the field’s sign for one population is blind to it in the other, and a survey that mixes the two has a contaminant whose sign depends on the mixture.

There is a second, more mundane loss of sign that stacks on top. A disc galaxy’s spin axis is inferred from its image as a direction along the minor axis of the projected ellipse, and the image alone does not say which side of the disc is nearer. That single bit is supplied by dust lanes or by the sense in which the spiral arms trail, when it can be supplied at all. The spin–spin statistic is quadratic and is indifferent to it, which is convenient; but it means that the only statistic available from shapes alone is exactly the one that cannot recover the sign of the field.

A shape that has already lined up spins less and points the same way #

The torque theory’s cleanest calculation assumes that the patch’s shape is random with respect to the field acting on it. It is not. The same gravitational potential that generates the tidal field generates the density perturbation that defines the patch, so the patch’s principal axes tend to line up with the field’s, and simulations find the two strongly correlated.

That correlation has an effect on the torque that is easy to state: if the patch’s axes coincide exactly with the field’s, the torque is zero, since two tensors with the same principal axes commute. Partial alignment reduces the torque in proportion to the misalignment that remains. That is one reason halo spins are as small as they are.

What does not follow is that the correlation erodes the spin’s memory of the field. It does not, for a reason visible in the formula. The components of the inertia tensor that line up with the field are its diagonal components in the field’s frame, and they do not appear in the torque at all. Only the off-diagonal components drive the spin, and shrinking all three of them by the same factor shrinks the spin without changing its direction.

The observed weakness of spin alignments therefore needs a different cause. The candidates are the ones that act after turnaround, when the linear theory has stopped applying: the collapse along the shortest axis first, which continues to torque the patch after the moment the calculation freezes it; the mergers that deliver orbital angular momentum from directions set by the local geometry rather than by the linear field, and whose outcome depends on how the encounter’s two timescales compare; and, for the baryons, the redistribution inside the disc that makes the disc’s angular momentum a fraction of the halo’s rather than a copy of it. N-body simulations that measure the alignment parameter directly find it considerably smaller than the linear value, a few tenths at most, and falling as the haloes are followed to later times.

How many spins it takes to find one axis #

The practical question that follows is whether spins can ever be used as a map. The tidal field is correlated over several megaparsecs, so all the galaxies in a region of that size share approximately one field, and the principal axis of their spin directions estimates the field’s intermediate axis.

The error falls as one over the alignment strength times the square root of the number of galaxies, so a signal diluted by a factor of ten costs a factor of a hundred in sample size. And the sample is not a count of galaxies but of galaxies with a measured spin direction inside one coherent patch of the field — a sphere a few megaparsecs across, which at the density of disc galaxies bright enough for kinematics contains tens rather than thousands.

The spin measurement itself is the other bottleneck. The oldest method infers the spin axis from the shape alone: a thin disc seen at an angle projects to an ellipse, and the spin lies along the ellipse’s minor axis, tipped towards or away from the observer by the inclination. That leaves the near-side ambiguity already described, and it fails outright for thick discs and for any galaxy whose shape is not a disc. The modern method measures rotation directly. Integral-field spectrographs record a spectrum at every point of the galaxy’s image, and the Doppler shift across the image is a velocity map — the same measurement that turns a line width into a rotation speed for a single spectrum, spread out over the face of the galaxy. The kinematic axis of that map is the projected spin, with its sense determined. Surveys of this kind now hold velocity maps for thousands of galaxies, which is the first sample in which the spin can be measured for elliptical and lenticular galaxies as well as for thin discs — the population for which the mass-dependent flip was predicted and had never been observable.

Detections exist. Surveys of spiral galaxies whose spin sense has been determined from their images, and more recently integral-field surveys that measure the rotation of each galaxy’s stars and gas directly, have found spin–filament alignments at the level of a few standard deviations, with the sign depending on mass as the transition requires. What has not been done is to invert them: no map of the tidal field has been made from galaxy spins that competes with one made from galaxy positions. The first paragraph of this section is the reason.

What the picture leaves out #

Every figure here is drawn in the tidal field’s own frame, with the field known exactly. A survey knows neither. The field has to be estimated from the distribution of galaxies, smoothed on a scale that has to be chosen — too small and the estimate is dominated by shot noise, too large and it averages over the very filaments whose axes are wanted, on a web that only becomes homogeneous above a hundred megaparsecs. That estimate introduces its own noise, and its own distortions from peculiar velocities along the line of sight — and the spin has to be estimated from an image or a velocity map, which is a projection of a three-dimensional vector onto the sky with one component missing. Both errors enter the alignment statistic multiplicatively and both reduce it.

The figures also use a Gaussian random inertia tensor. A real protogalaxy is not a smooth ellipsoid with random axes; it is a region defined by where the density first exceeded a threshold, and its shape statistics depend on the threshold and on the smoothing scale. The qualitative results — the preference for the middle axis, the exact zeros for ideal filaments and sheets, the insensitivity to the sign, the independence of direction from shape alignment — follow from the algebraic form of the torque and do not depend on those details. The numbers do.

And the whole calculation stops at turnaround. What is drawn is the spin a patch is given, not the spin a galaxy ends up with. For the dark-matter halo the two are related through a history of mergers; for the visible disc they are related through that history and through everything the gas did on its way in.

Still open: whether shapes and spins see the same field #

A disc galaxy’s shape on the sky is its spin direction seen in projection, so the alignments measured from spins and the alignments that contaminate a lensing survey are, for discs, the same quantity. The lensing surveys model it as a torque term quadratic in the field, and they find its amplitude consistent with zero. The spin surveys find the torque-driven alignment at a few standard deviations. Those two statements are not in tension only because the two kinds of survey look at different galaxies, on different scales and at different redshifts. A measurement of both in one sample — shapes and kinematic spins for the same galaxies — would say whether the quadratic term is small because the mechanism is weak or because the lensing surveys’ galaxies are too faint and too far away to show it.