Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal tensor — where it appears

The matrix of second derivatives of the gravitational potential, which says how the field stretches along one axis and squeezes along others. In cosmology its principal axes fix the directions along which a protogalaxy is torqued and along which its shape is aligned.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

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 Stretched towards the mass, sheared across it. Two rings of galaxies around the same concentration of mass. Left, galaxies physically beside it: its tidal field stretches each one along the line to the centre, so their long axes point radially and their mean tangential ellipticity is −0.30. Right, galaxies far behind it: their light is deflected past the mass and each image is sheared tangentially, with mean tangential ellipticity +0.30. The two patterns are perpendicular. A weak-lensing survey correlates the shapes of foreground and background galaxies to measure the shear, and when a foreground galaxy was aligned by the very structure that lenses the background one, their product is negative — −0.090 here — and subtracts from the signal rather than adding noise to it. What the picture exaggerates is the size: real intrinsic alignments are a per cent or two in the mean, buried under a random shape scatter of about 0.3, and are found only statistically.

The shape a galaxy was given before it was lensed

Weak lensing assumes that galaxies point in random directions, so that a coherent stretch can only have been put there by the light's journey. The tidal field that set every galaxy's spin also stretched its shape — towards the very mass that lenses the galaxies behind it — and the resulting error is not noise but a signal of the opposite sign.

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 fraction of a spin alignment a disc can show. The ratio of a disc population's shape alignment — its mean ellipticity along the projected tidal template — to the Lee–Pen alignment of its spins, against the discs' thickness as a fraction of their diameter, each point from 30,000 discs in a field of shape 0.2. The thinnest disc drawn, 0.05 of its diameter, passes 0.204 per unit of spin alignment; a disc a fifth as thick, typical of a spiral's stellar disc, 0.171; a body as thick as it is wide passes nothing, because a sphere has no shape to align. The ratio is the conversion between what a spin survey measures and what a lensing survey fits, and it is small for two reasons that multiply: a spin alignment is a weak preference among directions, and a disc shows its direction only in proportion to how far it is tipped towards edge-on.

An alignment a disc shows only when it is tipped

A disc galaxy's outline on the sky is its spin seen in projection, so the torque alignment of spins has to appear in shapes too. It does, multiplied by a factor near a sixth and carried almost entirely by the discs seen close to edge-on — which is how a lensing survey can fit the term and find nothing while a survey of spins finds it.

cosmology · Tidal torque theory A conversion read galaxy by galaxy, with or without an alignment. The mean ellipticity of disc galaxies along a tidal template, in bins of each galaxy's own spin estimator — the quantity whose population average is the Lee–Pen alignment — for 60,000 discs a fifth as thick as they are wide, in three samples: spins set by the torque (net alignment 0.36), 85% of them randomised (0.054), and all of them random (0.002). The points trace the same curve through the origin in every sample — not quite straight, flattening at both ends, because a disc's shape saturates when its spin lies in the sky's plane — and a straight line fitted through each sample (dashed) has slope 0.279, 0.290, 0.292. That slope is the projection factor, what a disc's shape shows of its spin, and it does not care whether the population is aligned. A sample with no alignment at all measures the conversion as well as one with a strong one; only the population's mean, where all its galaxies pile up, moves.

A conversion measured without an alignment

The factor that turns a disc galaxy's spin alignment into the shape alignment a lensing survey fits looked like something only a sample with a detected alignment could test. It is not. It is a relation each galaxy obeys on its own, so it can be read off a few hundred discs with velocity maps whether or not their spins point anywhere in particular — and what is left for the hard measurement to test is only whether a disc's shape is anything more than its projected spin.

cosmology · Tidal torque theory and exactly as much more binding energy as cancels it. The spin parameter — angular momentum times the square root of the binding energy, divided by G and the five-halves power of the mass — relative to its value at 10¹² solar masses, for the same three spectra. The binding energy grows as the five-thirds power of the mass times σ, and when it is combined with the angular momentum every power of the mass and every power of σ cancels. The spin parameter is the same at every mass and for every spectrum: 1.000, 1.000, 1.000 at 10⁸ solar masses.

Why every halo spins equally slowly

A halo of a hundred trillion suns receives a trillion times more angular momentum from the tidal field than a halo of a hundred million. It is also bound a trillion-fold more tightly, and the two cancel so exactly that the spin parameter — how fast a halo turns compared with how fast it would need to turn to hold itself up — comes out the same at every mass and for any spectrum of fluctuations. Only dark energy, by changing how fast structure grows after the universe is half its present size, leaves a trace of mass in it, and the trace is a few per cent.

cosmology · Tidal torque theory

Named alongside it

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

Tidal torque theorySpin alignmentIntrinsic alignmentAngular momentumInertia tensorLarge-scale structureLinear growthShape noiseSpin parameterTurnaroundCorrelation functionCosmic shear

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