An alignment a disc shows only when it is tipped

Two kinds of survey have been looking at the same mechanism and reporting different things about it.

The surveys that measure galaxy spins — from images whose spiral arms reveal which way a disc turns, and more recently from velocity maps that record the rotation directly — find that disc spins line up with the tidal field around them at a few standard deviations, in the direction tidal torque theory says they should: preferentially along the field’s intermediate axis, and so across the filaments of the cosmic web for massive galaxies. The surveys that measure weak gravitational lensing need to model the same thing, because an intrinsic alignment of galaxy shapes contaminates the shear they are trying to measure. Their model for disc galaxies includes a term quadratic in the tidal field, put there precisely because torque theory says discs should carry it. When the term is fitted, it comes out consistent with zero.

The obvious reading is a tension, and it would be a serious one: the same physical alignment detected by one method and absent in the other. The less obvious reading is that the two surveys are measuring different projections of one quantity, and that the projection a lensing survey sees is a small fraction of the one a spin survey sees. The second reading is a calculation rather than an opinion, and it can be done.

A shape is a spin with one number thrown away #

A thin circular disc has one direction that matters: the axis it spins about, perpendicular to its plane. Seen from any direction, it projects to an ellipse. The ellipse’s short axis is the projection of the spin axis onto the sky, and its long axis is perpendicular to it. How elongated the ellipse is depends on the angle between the spin and the line of sight — the inclination. A disc seen face-on, spin pointing straight at the observer, is a circle; a disc seen edge-on is a line, or for a real disc of finite thickness a narrow ellipse whose axis ratio is that thickness.

So the image of a disc encodes its spin in two pieces. The orientation of the ellipse on the sky gives the direction of the spin’s projection, up to a sign that the image cannot settle — the near side and the far side of a tilted disc look identical, and only a dust lane or the sense of the spiral arms supplies that bit. The elongation gives the inclination, again up to a sign. What the image does not give is any weighting of the orientation by how informative it is. A face-on disc’s ellipse has no orientation at all, because a circle has none; a disc tipped by thirty degrees has an orientation that is barely defined, because its ellipse is nearly round.

The ellipticity used here is the one lensing surveys use, $(1-q)/(1+q)$ with $q$ the axis ratio, because it adds to a weak shear in the simplest way. With that definition a disc tipped by sixty degrees from face-on shows about a third of the elongation it would show edge-on, and that sixty degrees is not an extreme: inclinations are distributed as $\sin i$, so the cosine of the inclination is uniform between zero and one and half of all discs lie between face-on and sixty degrees. The median disc in a random sample is an ellipse with an axis ratio of about a half, and its orientation on the sky records its spin direction with a weight of about 0.3 against the weight of 1 that a measured spin vector would carry.

What survives of the spin alignment #

The tidal-torque alignment is a weak preference among spin directions, inherited from the torque that set each halo turning. In the frame of the tidal field, the expected value of the product of two spin components is

$$\langle \hat L_i \hat L_j\rangle = \frac{1+a}{3}\,\delta_{ij} - a\,(\hat T^2)_{ij},$$

where $\hat T$ is the unit traceless tidal tensor and $a$ is the alignment parameter Lee and Pen introduced: about 0.6 for spins exactly as linear theory produces them, and a few tenths or less after the evolution that follows. The template a lensing survey fits for discs is the same tensor, $\hat T^2$, projected onto the sky. The question is how much of $a$ survives the journey from a three-dimensional spin direction to a two-dimensional ellipse.

The calculation in the opening figure answers it by brute force. Spins are drawn with a known alignment in a known field. The field is then turned to a random line of sight, each disc is projected into an ellipse of the right elongation with its long axis perpendicular to the projected spin, and the mean ellipticity along the projected template is measured. No parameter is fitted. The ratio of that mean to the alignment parameter is a property of the geometry alone.

It comes out at 0.17 for a disc a fifth as thick as it is wide, which is a fair description of a spiral galaxy’s stellar disc, rising to about 0.2 for the thinnest discs and falling linearly to zero for a sphere. The linearity matters, and it is checked in the next figure rather than assumed.

Because the relation is a straight line through the origin, it can be read in either direction. A spin survey that measures an alignment of 0.1 predicts a shape alignment for the same discs of about 0.017 in mean ellipticity along the template. A lensing survey that bounds the shape alignment of its discs to below some value has, by the same factor, bounded their spin alignment to below six times that value. The two kinds of measurement are not independent tests of different effects. They are one effect, seen at two different magnifications.

The discs that carry it #

The factor of a sixth is an average over a population, and it is worth asking which members of the population are doing the carrying. The answer is lopsided enough to change how the measurement should be thought about.

A shape survey of disc galaxies is, in effect, a survey of the discs seen close to edge-on. The other two thirds of the sample contribute their shape noise in full and their signal hardly at all. This is not a flaw in any estimator. It follows from the projection: a disc’s shape can only point somewhere to the extent that it is elongated, and a disc is only strongly elongated when its spin lies close to the plane of the sky.

The consequence for a lensing survey is uncomfortable. Edge-on discs are exactly the galaxies a shape-measurement pipeline handles worst. They are often reddened by their own dust, which moves them out of the colour selections used to define disc samples; their surface brightness is high along a thin line and their photometric redshifts are poorer; and some pipelines down-weight or reject them because their measured ellipticities are large and noisy. Every one of those choices removes the galaxies that carry the quadratic alignment, and leaves the term to be fitted from the population that cannot show it.

Noise that does not scale away #

The size of a signal is only half of whether it can be detected. The other half is what it is detected against, and here the two methods differ in a way that a factor of a sixth does not capture.

A measured spin direction is a unit vector, and its scatter about the expected alignment is the scatter of a random direction on the sphere — fixed, and of order one. A measured shape carries two kinds of scatter. The first is the spread of inclinations: a population of perfectly aligned spins still produces ellipses of every elongation from round to needle-thin, depending on where each one happens to be seen from. The second is the shape noise that every lensing survey contends with — the intrinsic non-circularity of real discs, their spiral arms and bars, and, for the faint galaxies at redshift one that lensing surveys are built on, the measurement error on an image a few pixels across. Together these give a scatter of about a quarter in each component of the ellipticity.

The ratio between the two lines is the square of the projection factor, partly compensated by the larger spread of the spin estimator, and multiplied by the shape noise. For the faint galaxies a lensing survey uses, the shapes need five to six times as many galaxies as the spins to reach the same significance. Reducing the shape scatter to what bright, nearby, well-resolved galaxies allow does not remove the gap.

That residual factor of 2.7 is worth dwelling on, because it says something about what kind of measurement each method is. A spin survey with integral-field velocity maps measures the thing the theory predicts — a direction. A shape survey measures a function of that direction which throws away most of the information for most of the galaxies, and no quality of imaging gets it back. The shape method is cheaper per galaxy by orders of magnitude, which is why lensing surveys have hundreds of millions of shapes and spin surveys have thousands of velocity maps. Whether the cheap method wins depends on whether the factor in sample size is larger than the factor in cost, and for this particular signal it is not close.

Reading the lensing result again #

With the conversion in hand, the apparent tension can be examined directly.

Simulations that follow haloes past the linear regime find that the spin alignment falls from its linear value of about 0.6 to a few tenths, and continues to fall as mergers and accretion deliver angular momentum from directions the original field did not set. For the stellar discs observed today, alignments near 0.05 to 0.1 are the expectation. The shape alignment that implies, from the second figure, is a mean ellipticity of one to two hundredths along the projected template — for a population composed entirely of discs, measured with perfect knowledge of the field.

A lensing survey’s sample is not all discs. It is a mixture in which the pressure-supported galaxies carry a different alignment — linear in the field rather than quadratic, and stretched towards the mass rather than spun across it. The quadratic term is fitted alongside the linear one, the two templates are correlated on the scales where both are measured, and the disc fraction varies with redshift and colour. The signal from discs is then diluted by their fraction of the sample, smeared by photometric redshift errors that mix galaxies from different tidal environments along the line of sight, and estimated from a sample that under-represents the edge-on discs that carry it.

Put those together and the expected amplitude of the quadratic term is small enough that a measurement consistent with zero is what should be found. That is not a detection of the term’s absence. It is a statement that the lensing measurement has, for this term, a sensitivity well above the size of the effect, and the conversion factor says by how much.

The spin surveys, meanwhile, are measuring the quantity directly in galaxies chosen to make it visible, from velocity maps that turn a Doppler width into a rotation point by point across each galaxy: bright, nearby, resolved, with a measured rotation sense. Their detections at a few standard deviations correspond to alignments of the size simulations predict. The two results are compatible because one of them is measuring something the other can barely see.

What the conversion assumes #

The calculation is clean because the model is simple, and each simplification has a direction.

The discs are perfect circles of uniform thickness. Real discs are not round even when seen face-on; their intrinsic ellipticity of a few hundredths, from bars, lopsidedness and spiral structure, adds noise and contributes no signal, which lowers the effective transfer further. Real discs also have bulges, and a bulge-dominated galaxy seen edge-on is rounder than a thin disc, so the thickness parameter for a real population is an effective one somewhere between the stellar disc’s and the whole galaxy’s.

The disc’s orientation follows its halo’s spin. That is the assumption, the same one that sets a disc’s size from its halo’s spin, that connects tidal torque theory, which is a theory of haloes, to galaxy shapes at all, and it is known to hold only loosely. Simulations find the angle between a disc’s spin and its halo’s spin to have a median of tens of degrees, larger for haloes that have recently merged. Any such misalignment is an additional dilution, acting on the spin surveys and the shape surveys alike, and it does not change the ratio between them. What it changes is the absolute size of both signals, and it is part of why the observed alignment of discs is so much weaker than the linear theory’s 0.6.

The field is known. In the calculation the tidal template is exact; in a survey it is estimated from the galaxy distribution, smoothed on a chosen scale, and distorted along the line of sight by the same flows the field drives. The lensing analyses do not estimate it at all for this purpose; they model the alignment’s correlation function rather than its value galaxy by galaxy. Neither changes the projection factor, which is purely geometric, but both lower the significance either kind of survey can reach with a given sample.

And the line of sight is random with respect to the field. Over a whole sky it is. For a single filament seen end-on, where the projection of its tidal field onto the sky nearly vanishes, it is not — and selection by the visibility of a structure on the sky could in principle select orientations. That effect is small for the large-area surveys considered here and could matter for targeted studies of individual filaments.

Why the edge-on discs are worth more than their number #

One practical conclusion follows from the third figure and is not usually drawn. If the shape signal comes overwhelmingly from discs inclined by more than about seventy degrees, then a sample restricted to those discs loses two thirds of its galaxies and about a third of its signal, and gains almost all of the noise reduction that dropping the rest provides. For a measurement of the quadratic alignment specifically — as opposed to a lensing measurement, which wants every galaxy — the edge-on subsample is the efficient one.

The same reasoning applies in reverse to the contamination problem. A lensing survey concerned about the quadratic term can test for it by splitting its disc sample by apparent axis ratio. Rounder discs should show a smaller quadratic signal than flatter ones, in a ratio fixed by the projection, and a contaminant that does not follow that ratio is not a torque alignment. The test uses information every shape catalogue already has, and it separates the mechanism from anything that merely correlates with the field.

There is a nice symmetry with the problem of measuring the spin of a single galaxy. A disc seen face-on has a perfectly measured inclination — zero — and an undefined position angle. A disc seen edge-on has a perfectly defined position angle and a spin sense that can only be read from which side is approaching. The kinematic method needs inclination to turn a velocity field into a rotation, and loses precision face-on; the shape method needs elongation to turn an image into an orientation, and loses everything face-on. Each fails where the disc’s geometry hides the quantity it uses, and each is best where the other is worst.

A disc is a lens on its own spin #

The general point is about what an observable does to a signal before it is measured. A shape is not a noisy measurement of a spin. It is an exact measurement of a different quantity — the spin projected onto the sky and weighted by the square of the sine of its inclination — and the statistics of that quantity are fixed by geometry before any instrument is involved. The same structure appears whenever a three-dimensional orientation is inferred from a two-dimensional image: the inclination problem of a spectroscopic binary, the projected obliquity of a transiting planet’s orbit, the apparent axis ratios from which the true shapes of elliptical galaxies have to be deconvolved. In each case the projected quantity is distributed in a way that concentrates the information in a minority of orientations, and an estimator that weights all orientations equally pays for it.

For the torque alignment the weighting is unusually severe, because the quantity being measured is a quadratic preference among directions and the projection discards the part of the direction that carries most of it. The factor of a sixth is not a small correction to be applied at the end. It is the dominant fact about whether the effect can be seen in shapes at all.

Still open: the same galaxies measured both ways #

The conversion here is a prediction, and it has not been tested on one sample. What would test it is a set of disc galaxies with both a kinematic spin measured from a velocity map and a shape measured from an image of lensing quality, all in a volume where the tidal field can be reconstructed from the surrounding galaxies. The spin alignment and the shape alignment of those galaxies, measured against the same field, should differ by the projection factor and nothing else. If they do, the quadratic term in a lensing survey can be fixed in advance from the spin surveys, rather than fitted and found consistent with zero. If they do not, something other than the projection — a dependence of disc shape on environment, a misalignment of disc and halo that differs between the two samples — is at work, and it would be a new piece of the connection between haloes and the galaxies inside them. The integral-field surveys of the last decade hold velocity maps for thousands of galaxies with deep imaging; the sample sizes the noise figure asks for are within a factor of a few of what they contain.