A conversion measured without an alignment

A disc galaxy’s outline on the sky is its spin seen in projection, so the alignment of spins that tidal torques produce has to appear in shapes too. The calculation of how much of it survives found a factor of about a sixth for a disc a fifth as thick as it is wide, carried almost entirely by the discs seen close to edge-on. It ended with a test: take one sample of disc galaxies, measure both their spins and their shapes, measure both alignments against the same reconstructed tidal field, and check that the two differ by the factor and nothing else. If they do, the shape alignment term that contaminates every lensing survey can be fixed in advance from spin surveys. If they do not, something besides projection is shaping the discs.

That test sounds as hard as its hardest part. Detecting the alignment at all, in either spins or shapes, takes thousands of galaxies sharing one well-mapped tidal field, and the alignment is weak — a few hundredths in the parameter that describes it today. A test of the ratio of two weak signals ought to be harder still. This essay finds that most of it is not: the conversion factor can be measured without any alignment, because it is a relation each galaxy obeys separately. What remains hard is a narrower question, and it is worth knowing exactly which.

A relation each galaxy obeys #

The alignment of a population of spins is an average. For each galaxy there is a number — call it its spin estimator — that measures how far its spin lies along the directions the tidal field favours: positive for a spin along the field’s intermediate axis, negative for one along either of the others. Averaged over the population, it gives the alignment parameter that the theory of tidal torques predicts and the spin surveys measure. Most of its spread from galaxy to galaxy has nothing to do with the field: an isotropic population has spin estimators scattered from about −0.9 to +0.9 with a mean of zero.

The same galaxy also has an ellipticity on the sky, and that ellipticity has a component along the projected template of the tidal field. Averaged over the population, it gives the shape alignment a lensing survey fits. The previous calculation showed that the second average is the first times a factor fixed by geometry.

The figure above shows something stronger. Sort the galaxies by their own spin estimators, and within each bin average their shape components. A galaxy whose spin happens to lie along the template’s favoured direction shows, on average, an ellipse elongated along the template, by an amount that depends only on the geometry of projection — and that is true of each galaxy whether or not its neighbours are aligned. Three samples with net alignments from 0.36 to zero trace the same curve. The fitted slopes, 0.279 to 0.292, differ by less than the curve’s own departure from a straight line.

That departure is worth naming. The binned points are not exactly on a line: they flatten at both ends. A disc whose spin lies near the plane of the sky is already seen nearly edge-on, and its ellipticity cannot grow further however well its spin is aligned. The slope of a straight-line fit is therefore a weighted average of a gently curving relation, and the weighting differs slightly between a strongly aligned sample, whose galaxies sit further along the curve, and an isotropic one. The two-hundredths spread between the three slopes is that effect, and it sets the precision to which the slope of one sample transfers to another.

The slopes are larger than the factor of a sixth quoted before, and the difference is not a contradiction. A velocity map gives a spin’s position angle on the sky and its tilt to the line of sight, but not which side of the disc is nearer: two spin directions, mirror images in the sky’s plane, fit the same map. The estimator drawn here is averaged over both, which shrinks it to about six-tenths of the true value. The slope per unit of this measured estimator is correspondingly larger, and the product — what actually appears in the shape alignment — is the same.

Why a shape follows a spin at all #

The relation in the figure is not a correlation found in data; it is the geometry of a thin disc, and it is short enough to state. A circular disc whose thickness is $q_0$ of its diameter, tilted so that its spin makes an angle $i$ with the line of sight, projects to an ellipse of axis ratio

$$q^2 = \cos^2 i + q_0^2 \sin^2 i,$$

and the ellipticity a lensing survey records is $(1-q)/(1+q)$. The long axis of that ellipse lies along the line of nodes — where the disc’s plane cuts the plane of the sky — which is perpendicular to the spin’s projection. So two things about a disc’s spin are written into its outline: its position angle, through the orientation of the ellipse, and its tilt, through the ellipse’s elongation. What is not written is which way round it turns, and which side is near.

Everything the conversion does follows. A component of the ellipse along a template is the elongation times the cosine of twice the angle between the long axis and the template’s direction; the spin estimator is a quadratic in the spin’s components along the field’s axes. Both are quadratic forms in the same unit vector, seen from a random direction, and averaging one at fixed values of the other gives the curve the figure draws. Nothing about the field, the torque or the galaxy’s history enters. A population of discs spun by anything at all — torques, mergers, a designer — would trace the same curve.

The field is needed for the alignment, not for the conversion #

The per-galaxy relation has a consequence that removes the hardest part of the original test. A spin estimator and a shape component can be computed against any reference template, true or false. The relation between them is geometric: it says that a disc’s ellipse is perpendicular to its projected spin and elongated in proportion to its tilt, which is true in every frame. A wrong map of the tidal field changes which galaxies are counted as aligned; it cannot change how a disc’s shape follows its spin.

The tidal field around a nearby galaxy is not observed; it is reconstructed from the positions of the surrounding galaxies, smoothed on a few megaparsecs, and its axes are uncertain by tens of degrees. Every alignment measurement is diluted by that error, and the figure shows by how much: an error of thirty degrees costs a quarter of the signal, fifty degrees half. Both alignments fall together, because both are averages against the same wrong template and a wrong template mixes the favoured directions with the others in the same proportion for both.

The slope does not move. The reconstruction that limits the alignment by tens of per cent has no effect on the conversion at all. That splits the original test into two parts with very different costs. The conversion is a property of discs and is measured from discs alone. The alignment is a property of discs in a field and needs the field. Only the second needs the cosmic web to be mapped.

A few hundred galaxies against ten thousand #

Separating the two also separates their costs, and the difference is large.

The slope uses every galaxy’s full range of orientations: a galaxy with a spin estimator of 0.8 and one with −0.8 both pin the line, and the spread of the estimator across an isotropic sample is large. The alignment uses only the shift of the mean, which is a few hundredths of that spread, and pays for it in the square. Fixing the conversion to ten per cent takes about eighty galaxies with good images and velocity maps; detecting a realistic alignment in their shapes takes about ten thousand.

Both numbers grow with the scatter of each ellipticity component, but slowly, because most of the spread of a disc’s shape along a template comes from the direction it happens to be seen from, which no improvement in imaging changes. The integral-field surveys of the past decade hold velocity maps for several thousand disc galaxies at the quality the figure assumes. For the conversion, that is tens of times what is needed. For the alignment, it is about what is needed, and the alignment the field reconstruction allows them to see is diluted further.

What a fixed conversion is worth to a lensing survey #

The reason anyone wants the conversion is the size of what it multiplies. A spin alignment of a few hundredths, times a factor of about a sixth, is a shape alignment of a few thousandths in ellipticity — and the lensing distortion that cosmic shear surveys measure is about one per cent, correlated over the same scales. An intrinsic alignment a tenth the size of the signal, with an angular dependence of its own, is enough to move the inferred amplitude of matter clustering by more than its quoted error.

The surveys handle it by fitting. Their models carry an alignment term for the red, elliptical galaxies, linear in the tidal field and well measured, and a term for the blue discs, quadratic in the field, whose amplitude is left free and comes out consistent with zero. A free parameter that comes out near zero is not a measurement that it is zero; it widens every other error bar by the amount the data could have spent on it. Fixing the disc term from outside — from the spin alignment the integral-field surveys measure, times a conversion measured on the same kind of galaxies — would remove one free parameter from every cosmic shear analysis, at the price of an assumption that can be stated and tested.

The assumption is the transfer. The galaxies in a lensing survey are faint and distant, near a redshift of one, and they are not the bright nearby discs with velocity maps. Their discs are thicker and more turbulent, their bulges smaller, their shapes measured from a few pixels. The conversion depends on thickness, and a disc at a redshift of one is perhaps twice as thick for its diameter as one today. That is a correction a factor of a few smaller than the conversion itself, and it can be measured from the shapes of edge-on discs at each redshift without any velocity map at all.

A twist between the light and the motion #

The geometry assumed so far has one idealisation that real discs break. A disc’s long axis on the sky is drawn exactly perpendicular to its projected spin. In a real galaxy the photometric axis — the direction along which the light is elongated — and the kinematic axis — the direction along which the velocity map is steepest — can differ, by a degree in a quiet spiral and by tens of degrees in a barred, warped or interacting one.

A twist of angle $\psi$ rotates the ellipse against the spin and scales the component that follows the spin by $\cos 2\psi$. Averaged over a Gaussian spread of twists of width $\sigma$, the factor is $e^{-2\sigma^2}$, quadratic at small angles. The points from the simulated discs follow it to within their noise.

For most rotation-dominated discs the integral-field surveys find the two axes within a few degrees of each other, with a tail of larger offsets among barred and disturbed galaxies. A few degrees costs under one per cent. But the figure also says what to look for. A twist that was random would lower the slope everywhere by the same amount. A twist that correlated with the tidal field — bars aligned by the field, warps raised by it — would make the slope depend on environment, and the per-galaxy relation, measured separately in filaments and in sheets, is the place where that would show.

A tilt read from the shape #

There is a second, subtler problem, and it concerns where the spin’s tilt comes from. A velocity map gives the kinematic position angle cleanly. The tilt of the disc to the line of sight is harder: in principle it is in the map’s shape, the way the isovelocity contours close, but in practice most analyses take it from the disc’s own axis ratio — from the very ellipse whose component along the template is the other side of the regression.

The same noise then sits on both sides of the regression, and it does two things that pull in opposite directions. A disc whose ellipse is measured too elongated is also assigned a spin lying closer to the plane of the sky, so its shape and spin errors are correlated and the slope is pushed up. And the noise in the spin estimator itself dilutes the slope, as any noise in the variable a regression is made against does — the attenuation familiar from inverting a noisy parallax into a distance, where a symmetric error on one quantity becomes a bias on another. At small scatter the correlation wins and the slope is about nine per cent high; at large scatter the dilution wins.

A bias of that kind cannot be removed by a rule, because its sign depends on the noise. It can be removed by simulating each sample’s own noise, or avoided by taking the tilt from the kinematics. With an independent tilt the slope is flat across the whole range.

What the test can still find #

With the conversion fixed from the per-galaxy relation, the test reduces to the comparison it was always about: is the shape alignment of a sample equal to its spin alignment times the conversion?

The points scatter about the diagonal, and the scatter about it is only a little smaller than the scatter along it. Measuring both alignments on the same galaxies helps — a survey whose galaxies happen to be more aligned than average is more aligned in both — but only a little, because the shape noise belongs to the shapes alone. The test is about as hard as detecting the shape alignment itself, and no harder. A survey of three thousand bright discs with velocity maps, the scale of the larger integral-field programmes, makes it at about three standard deviations.

What it would find, if it found anything, is narrower than it first appears.

The residual is zero for a disc whose shape is exactly its projected spin, and nothing that moves the spin and the shape together can make it otherwise. A disc tilted against its dark halo is the obvious example: the spin survey measures the disc’s spin, the shape is the disc’s shape, and both inherit whatever alignment the disc has — less than the halo’s, perhaps, but in exactly the projected ratio. The test is blind to it. So is any process that turns the disc as a whole.

What the test sees is a shape that is not a projected spin. The cleanest such effect is a tide stretching the disc’s stars or its outer gas along the field — the mechanism that aligns elliptical galaxies, which have no organised spin at all, linearly in the tidal field rather than quadratically. Lensing models carry such a linear term for ellipticals and assume it vanishes for discs. A residual in this test would be that term, measured in discs, and the figure gives the scale: a stretching that adds about a hundredth to the mean ellipticity along the template is a three-standard-deviation residual in one large survey.

What the simulation assumes #

The discs are thin, axisymmetric and transparent: their ellipticity is fixed by tilt and thickness alone. Real discs have bulges that round them at low tilt, dust lanes that thicken them at high tilt, and spiral arms and bars that add intrinsic ellipticity even when face-on. The scatter per component is meant to stand in for all of that, and it does for the regression’s noise, but not for any dependence of those features on the tidal field.

All discs have the same thickness. A sample with a range of thicknesses has a range of conversions, and the slope measures their average weighted by how strongly each contributes — weighted towards the thin, which have the most ellipticity to give. That is what the lensing term needs, so it is the right average; but a sample selected differently from the lensing galaxies would carry a different one, which is the familiar problem of transferring a calibration between samples and is not solved by anything here.

The spins come from tidal torque theory in a single, fixed field shape, and the frame error is an isotropic random rotation. A real reconstruction errs more in some directions than others, and errs more for galaxies near the boundaries between structures; its effect on the alignments is then not a uniform dilution, though its effect on the slope is still nothing.

And the galaxies are independent. Galaxies in groups share both their tidal field and, sometimes, their histories, and their shapes can be correlated by interactions as well as by the field. In a sample of three thousand, a few hundred such pairs would not move the slope, but they would add correlated scatter to the alignments and make the test’s three standard deviations an overstatement.

Still open: a residual in a narrow window #

The conversion between spin alignment and shape alignment is, it turns out, the easy half. A few hundred discs with velocity maps fix it to a few per cent, with no need for the tidal field and no need for any alignment to exist. The integral-field archives already hold enough for that, and the lensing analyses that fit the torque term for discs as a free parameter could fix it instead, from the spin surveys’ alignment times a conversion measured on nearby discs — provided that the conversion transfers from bright nearby discs to the faint distant ones a lensing survey uses, which is a question about how discs change with time and not about geometry.

What remains hard is whether discs carry any alignment beyond their spins. The test for it needs the alignment itself to be detected, at a scale where the reconstruction of the field and the number of galaxies with velocity maps are both at their limits, and the residual it looks for is a hundredth of an ellipticity. Whether discs are stretched by the field as well as spun by it decides whether the two parameters that lensing measures as one are contaminated by one alignment term for discs or two — and so how much of a lensing measurement of the dark matter is, underneath, a measurement of how galaxies are shaped by where they live.