Astronomy & Celestial Mechanics
the sky, computed
Concept

Linear growth — where it appears

The regime in which density fluctuations in the expanding universe are small and every one grows by the same factor, the growth function, independent of scale. Tidal-torque theory and the linear alignment model are both calculated in it.

Named by 4 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 How much each change moves the growth rate. The percentage difference in fσ₈ from growth in relativity inside a cosmological constant's expansion, against redshift, for three alternatives started from the same early amplitude. Changing the expansion to the evolving fit with DES-SN5YR and keeping relativity moves it by −1.2 per cent at redshift 0.5, and by at most 2.1 per cent anywhere below redshift 2. Keeping the constant's expansion and changing gravity to γ = 0.68 moves it by −8.6 per cent; doing both, −10.3. The shaded band is ±8 per cent, about the error of one of the better redshift-space measurements so far. The difference between a constant and the evolving fit is several times smaller than any single measurement; the difference between two laws of gravity is comparable to one. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is gravity.

The growth that tells a substance from a law

In relativity an expansion history leaves no freedom in how fast structure grows inside it, so every fitted history predicts its own growth. The evolving dark energy the surveys prefer changes that growth by one or two per cent; a change in the law of gravity, with the expansion held fixed, changes it by nine. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is there at all.

cosmology · Dark energy 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 tensorTidal torque theoryAngular momentumCorrelation functionInertia tensorLarge-scale structureSpin alignmentSpin parameterTurnaroundCosmic shearDark energyDensity perturbation

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