Astronomy & Celestial Mechanics
the sky, computed
Cosmology

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.

Assumes Large-scale structure, Expansion and Galaxy spin.

16 min read 8 figures Scales that defeat intuitionSmall perturbations, large consequences

Start with a universe that is expanding uniformly and is very nearly smooth. Nothing in it rotates. The velocity field of a pure Hubble expansion has zero curl everywhere, and a curl-free velocity field carries no angular momentum about any point. Whatever spin galaxies have, they were not born with it.

They also cannot have acquired it after they collapsed. A collapsed, isolated object has no external torque acting on it, and there is nothing for it to exchange angular momentum with. So the spin had to be applied in between: while the future galaxy was still an extended, expanding, slightly overdense patch — big enough to feel the gravity of its neighbours in a way that a compact object would not.

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.
Fig. 1 Left, the mechanism: a protogalactic patch drawn as an ellipsoid with the principal axes of the surrounding tidal field crossing it at thirty degrees. The torque is proportional to the difference of the patch’s own principal moments times the sine of twice that angle, so it vanishes exactly when the two sets of axes agree — checked at both alignments — and is largest at forty-five degrees. Right, the angular momentum against time: linear growth while the patch is still expanding, and nothing after turnaround.

Why a sphere gets nothing

The first thing the mechanism explains is why the spin of a galaxy is a statement about its shape.

A torque about the centre of mass requires a force that varies across the body in a way that is not radially symmetric. The external gravitational field of everything around the patch, expanded about the patch’s centre, has a constant part — which accelerates the whole patch and produces no torque — and a linear part, the tidal tensor TijT_{ij}, which is what remains. Contracting that against the patch’s own inertia tensor IjkI_{jk} gives

Li  ∝  ϵijk Ijl Tlk,L_i \;\propto\; \epsilon_{ijk}\,I_{jl}\,T_{lk},

and the antisymmetrisation is decisive. If the patch is spherical, Ijl∝δjlI_{jl} \propto \delta_{jl} and the contraction with a symmetric tensor gives exactly zero. If the patch is aspherical but its principal axes coincide with the tidal field’s, the two tensors commute and the result is zero again.

So the torque is a measure of misalignment between two shapes: the shape of the patch and the shape of the field it sits in. It is largest when they are at forty-five degrees, and a universe in which perturbations were spherical would contain no rotating galaxies at all.

A torque that stops when the patch lets go. Left, the mechanism: a protogalactic patch drawn as an ellipsoid of axis ratios 1:0.95:0.9, 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.
Fig. 2 The near-spherical case, with the patch’s three axes within ten per cent of each other. The torque collapses, because the expression carries the difference of the principal moments and a nearly round patch has almost none — and the growth curve on the right flattens accordingly. That is the first prediction the mechanism makes and the one it is easiest to check against the wrong intuition: spin is not delivered to a patch in proportion to how much matter is pulling on it, but in proportion to how far the patch is from being round.

Why the growth is linear, and why it stops

The timing argument is the part that makes the theory predictive rather than merely plausible.

While a perturbation is small, its shape and the external tidal field both grow in a known way. In an expanding matter-dominated universe the density contrast grows in proportion to the scale factor, and the tidal field and the patch’s quadrupole grow with it. Working through the scalings, the torque acts on a patch that is still being carried apart by the expansion, and the angular momentum accumulates as the first power of the cosmic time:

L∝t.L \propto t.

Then the patch stops expanding. At turnaround its own gravity has overcome the expansion, it begins to collapse, and two things change at once. Its physical size shrinks, so its quadrupole moment against the external field falls; and it decouples from the surrounding flow, so the coherence between its own orientation and the field’s is lost. The torque switches off, and the angular momentum a galaxy will have for the rest of its existence is fixed.

Everything after that — collapse, virialisation, cooling, the settling of gas into a disc — is a redistribution of an amount that was decided before any of it began.

The calculation, done in the right coordinates

There is a technical move that makes the whole argument tractable, and it is worth naming because it is what turns a hopeless integral into two lines.

The obvious way to compute the torque is to follow the patch as it moves — to work out where its matter is at each moment, evaluate the external field there, and integrate. That is intractable, because the patch’s shape is changing while it is being torqued and the field is changing too.

The trick is to label each parcel of matter by where it started rather than by where it is. In those coordinates, while the perturbations are small, every displacement is proportional to one growing function of time multiplied by a gradient of the initial potential. The patch’s inertia tensor and the external tidal tensor are then both evaluated once, on the initial conditions, and the entire time dependence factors out into a single scalar. The torque integral becomes an integral over a fixed region of space with a fixed integrand, multiplied by a known function of time.

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 8 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.
Fig. 3 The other way the torque can vanish: an aspherical patch whose principal axes nearly agree with the field’s. Eight degrees of misalignment gives a quarter of the torque that thirty does, because the dependence is on the sine of twice the angle. Two independent conditions therefore have to be met for a patch to be spun at all — it must be aspherical, and it must be misaligned — and neither is guaranteed by anything. That both are satisfied for essentially every patch is a statistical statement about a Gaussian random field rather than a property of gravity.

That is why the growth law is clean. It is not that the physics is simple; it is that in the right variables the geometry stops moving. The approximation fails exactly when the displacements become comparable to the region’s own size, which is another way of saying it fails at turnaround — which is where the theory says it stops applying anyway. The two limits coinciding is not luck; it is why the theory has a well-defined domain rather than a fudge factor.

What it predicts

Three things follow, and all three are checkable.

A dimensionless spin of a few hundredths. Combining the linear growth with the statistics of a Gaussian random density field gives a distribution of the spin parameter with a median of around 0.03 to 0.05 and a spread of about a factor of e in the logarithm. That is what the sizes of disc galaxies require, and it was a prediction rather than a fit: the value was computed from perturbation theory before it was measured from galaxy sizes.

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.9, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0156, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.
Fig. 4 The same distribution nearly twice as wide in the logarithm, which is what the misalignment angle alone would produce if it were the only source of scatter. The disc sizes it implies spread correspondingly, and the observed spread of disc scale lengths at fixed rotation speed is narrower than this — which is a constraint the theory has to satisfy and does, but only because the other factors in the product partly cancel the angle’s contribution. A lognormal’s width is as much a prediction as its median, and it is the half less often checked.
A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.
Fig. 5 The distribution the theory delivers, and the disc sizes it implies. A lognormal is what one expects when a quantity is the product of several independent factors, which is what the torque expression is — an amplitude, a shape difference, and the sine of a misalignment angle. The width comes almost entirely from the misalignment, which is uniformly distributed and therefore contributes a broad factor.

No dependence on mass. The torque expression contains no scale. In a universe whose initial fluctuations have no preferred scale either, the resulting distribution of the dimensionless spin should be the same for a dwarf galaxy and for a cluster. That is a strong prediction, since almost nothing else about galaxy formation is scale-free, and it is confirmed: the measured λ distribution is the same across four decades in mass.

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹⁴-solar-mass halo of radius 958 kiloparsecs gives 23.7 kiloparsecs at the median, against the 8 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.
Fig. 6 The claim drawn: the same distribution at a halo mass a hundred times larger. The λ axis has not moved and could not, because λ is dimensionless and the torque expression has no scale in it; what has moved is the disc size the distribution implies, and it has moved by the cube root of a hundred through the virial radius. So a two-order-of-magnitude change in mass produces no change at all in the predicted spin and a factor of five in the predicted size, and both halves of that are testable against a survey.

Weak alignment between neighbouring spins, and a specific kind of it. Because the torque is set by the local tidal field, and the field is correlated over megaparsecs, nearby galaxies should have partially correlated spins. But the correlation is second-order — the spin depends on the tidal field quadratically, since both the patch’s shape and the field derive from the same underlying density — so the alignment is weak, and it is an alignment with the filament rather than with each other. The observed pattern is that low-mass disc galaxies tend to spin with their axes along the filament they inhabit, and massive ellipticals tend to spin perpendicular to it. The transition mass is the mass at which a galaxy’s history becomes dominated by mergers, which is the process that overwrites the tidal-torque signal.

That flip is worth dwelling on, because it is the one place where the theory makes a qualitative prediction rather than a statistical one. A patch that collapses along its shortest axis first becomes a sheet, then drains along the sheet into a filament; the material arriving late comes in along the filament, and its orbital angular momentum is perpendicular to that direction. Early, smooth accretion therefore aligns a spin with the filament and late, merger-dominated accretion turns it across. Measuring the transition mass measures where in the galaxy population one mode of growth gives way to the other, and the answer — a few times ten to the tenth solar masses — is close to the mass at which galaxies stop being predominantly discs.

The measurement is difficult in a specific way: a galaxy’s spin axis is not directly observable, and what a survey records is the orientation of its projected image, which is the spin axis only if the galaxy is a thin disc. For an elliptical it is not, and the alignment signal for exactly the population where the flip is predicted has to be extracted from stellar kinematics rather than from shapes.

Where the theory stops being right

Linear theory gets the order of magnitude and the scalings, and it gets the details wrong by a factor of two or three.

The reason is that turnaround is not a clean switch. Real patches are not ellipsoids, their collapse is not simultaneous along all three axes, and the collapse along the shortest axis happens first — producing a sheet, then a filament, then a knot. During that sequence the object is still extended in some directions and still being torqued, and the linear calculation’s assumption that the accumulation stops at a single moment is an approximation.

More importantly, the mass that ends up in a galaxy does not all arrive as one patch. It arrives as a series of mergers, and each merger delivers the orbital angular momentum of the pair as well as the internal spin of the incoming object. For a massive galaxy assembled through many mergers, the tidal-torque contribution is a minority of the final answer.

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.02 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0156, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.02 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 2.9 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.
Fig. 7 The distribution shifted down to a median of 0.02, which is roughly what would be needed for the naive disc-size prediction to match the observed sizes without any loss of angular momentum by the baryons. Nothing in the theory permits that value — the torque calculation gives 0.035 to 0.05 and has no free parameter to move it with — so the discrepancy has to be absorbed somewhere else. The section above says where: the gas does not keep what it was given, and this figure is the size of what it has to have lost.
One slope, three heights. Specific angular momentum against stellar mass, both logarithmic, for disc galaxies, for spheroids, and for the dark haloes they formed in. All three loci are drawn with slope 0.6667 — measured off the drawn line rather than assumed — because a spin parameter that does not depend on mass makes j scale as the two-thirds power of mass, and the observed relations do have that slope over three decades of mass. What separates the three lines is not their shape but their height: a disc keeps roughly 53 per cent of the specific angular momentum of its halo, and a spheroid of the same stellar mass has about 20 per cent of a disc's. That is the whole morphological sequence written as one number. A galaxy is a disc because it kept its spin and a spheroid because it lost it, and the losing happens in mergers, where the orbital angular momentum of the pair goes into the outer halo and the remnant keeps almost none of it. The figure cannot show the scatter, which is about a factor of two at fixed mass and is itself the spread in λ from the previous panel.
Fig. 8 Where the two histories separate. Discs and spheroids of the same stellar mass sit on parallel lines with the same two-thirds slope and different heights, and the height is what fraction of the halo’s specific angular momentum the visible galaxy kept. A galaxy that grew smoothly kept most of it; a galaxy assembled by major mergers threw most of it into the outer halo along with the stars it scattered there. Tidal torque theory sets the top line; what happens afterwards decides which of the lower two an object lands on.

What makes it worth trusting anyway

The theory’s strongest evidence is not numerical agreement, which is only ever approximate. It is that the mechanism explains a fact that is otherwise very hard to account for: that galaxy spins are small.

A halo with λ=0.035\lambda = 0.035 has a few per cent of the angular momentum it would need to hold itself up by rotation. If galaxies had acquired their spin by some direct means — primordial vorticity, a rotational mode in the initial conditions — there would be no reason for the answer to be small, and no reason for it to be the same small number everywhere. Tidal torque theory produces a small number because it is a second-order effect operating for a limited time, and it produces the same small number everywhere because it is scale-free.

There is a stronger version of that argument, and it is about what an expanding universe does to vorticity rather than about what it does to spin. In a smooth expansion, any rotational velocity field decays: the circulation is conserved while the physical scale grows, so the vorticity falls as the inverse square of the scale factor. Anything rotational present in the initial conditions is therefore erased by the time structure forms, by a factor of a thousand or more. That is why the spin has to be generated late and locally rather than inherited — the theory is not competing with a primordial alternative so much as filling a gap the expansion has already cleared.

It also explains why the answer arrives as a distribution with no preferred direction. The torque on any one patch is set by its own neighbours, the neighbours are a random realisation of a statistically isotropic field, and nothing in the construction picks out an axis. A measured alignment between galaxy spins and the large-scale structure is therefore not a violation of that isotropy but a correlation between two quantities drawn from the same field — which is exactly the signal the flip in the previous section is about.

The quantity nobody measures

Every prediction above is about the spin parameter, and the spin parameter has never been measured for any galaxy.

What it is defined from is the halo’s angular momentum, its mass and its energy — three quantities of a dark component whose extent nobody knows. None of them is observable.

What is observable is a disc: its size, its rotation speed, and its surface brightness profile. The inference from those to a spin parameter runs through a model in which the disc formed by gas cooling inside the halo while conserving its angular momentum, so that the disc’s scale length is a fixed fraction of the halo’s virial radius times the spin parameter.

That chain has three assumptions in it, and each is known to be imperfect. The gas is assumed to have started with the same specific angular momentum as the dark matter — which is reasonable, since both were torqued by the same field, and which simulations find to be true to a few tens of per cent rather than exactly. The gas is assumed to have retained it during collapse — which it does not, since some is transported outward and some is carried away by outflows. And the halo is assumed to respond to the disc forming inside it in a calculable way, which requires a model of how the dark matter contracts.

So a published distribution of spin parameters for observed galaxies is a distribution of a derived quantity, and comparing it with the theory’s prediction is comparing two model outputs rather than a prediction with a measurement.

What can be compared honestly is the shape of the distribution rather than its position. The theory predicts a lognormal of a particular width, and the width is a property of the torque mechanism rather than of the baryonic physics — so a measured distribution with the right shape and the wrong median is evidence that the mechanism is right and the retention fraction is not one.

That is the state of the comparison, and the median is discrepant by about a factor of two in the direction that says the baryons lost angular momentum on the way in.

The comparison is therefore between a theory that predicts a distribution and an observation that reports a modelled quantity, and the honest reading of the discrepancy is that it measures the modelling rather than the mechanism.

Initial conditions that are a realisation rather than a fact

The theory is statistical by construction, and it is worth being explicit about what that means for what it can predict.

The tidal field acting on a protogalaxy is a realisation of a Gaussian random field. The theory computes the distribution of torques over all possible realisations, and therefore predicts the distribution of spins over a population. It cannot predict the spin of any individual galaxy, because that depends on which realisation happened.

There is a technique that gets round this for one particular case. Given a map of where the matter actually is now, a simulation can be initialised so that its initial conditions are consistent with producing that observed structure — a constrained realisation, in which the random field is drawn subject to the constraint that it evolves into the observed local universe.

Such simulations can then be asked what the Milky Way’s own spin should be, and what the tidal field acting on the Local Group was. The answers are not predictions in the strong sense, since the constraints were derived from the same universe being predicted, and they are a genuine test of consistency: a mechanism that produced the right statistics but the wrong answer for the one system whose environment is known in detail would be in trouble.

The limitation is the quality of the map. Constrained simulations are built from redshift surveys and from peculiar-velocity measurements, both of which are incomplete and both of which are worst in the region hidden by the Galaxy’s own disc — which is a substantial fraction of the sky and is not a random fraction.

Where the ladder goes

The immediate consequence is the one this collection takes up under what a halo’s spin does to the disc inside it: the theory delivers a distribution of λ, and λ delivers a disc size, and the observed sizes come out about half of the naive prediction — which is a measurement of the angular momentum the baryons lost on the way in.

The second thread runs into weak lensing. A galaxy’s intrinsic shape is set by the same tidal field that set its spin, and a redshift-space map is stretched by the very flows that field produced, so galaxy shapes are intrinsically correlated with the surrounding structure — which is a contaminant of exactly the signal a weak lensing survey is trying to measure and which the strong lensing of individual systems does not suffer from. Modelling that contamination requires the same theory, applied to the shape rather than to the spin, and it is currently one of the leading systematic uncertainties in cosmology from lensing.

Where this goes next

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in. A disc the size its halo was born with A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe. A slice 1,000 megaparsecs across. 872 galaxies in a wedge 1000 comoving megaparsecs deep, generated as a Thomas process — Poisson centres at 0.00012 per square megaparsec, 13 galaxies each on average, scattered about their centre with a Gaussian of 22 Mpc. That is not a simulation of how structure formed and does not pretend to be; it is the simplest clustered process whose correlation function is known exactly, which is what lets the next figure check a measurement against something rather than against itself. What it does reproduce is the impression: at this scale the distribution is obviously not uniform, there are groups and there are gaps, and the eye finds patterns in it readily — including several that are not there, because the eye finds patterns in a Poisson field too. The question the next two figures ask is the only one that settles it: at what separation does the clustering stop, and is there a scale above which a box of this universe looks like any other box? Homogeneous above a hundred megaparsecs Every cosmological calculation in this field assumes the universe is the same everywhere. Looked at on any scale a person can picture, it plainly is not — it is stars in galaxies in groups in clusters in filaments around voids. The assumption is not a claim about appearance; it is a claim about a statistic, and the statistic has a scale attached. Fingers 3.1 times long, a large scale squashed to 0.92, and a test of gravity. Left: a 360-megaparsec slice of a clustered universe, as the galaxies actually sit. Right: the same galaxies as a redshift survey records them, with the line of sight up the page. Nothing has moved sideways, because an angle is an angle; every displacement is along the line of sight, because that coordinate came from a redshift and a redshift is the expansion plus whatever the galaxy is doing on its own. Two effects, opposite in sign and separated by scale. Inside a cluster the motions are virial and random, 720 kilometres a second of them, which at H₀ = 67.36 is 11 megaparsecs of smearing on an object a few across: the clusters become fingers 3.1 times longer than they are wide, all pointing at the observer, which is the one structure in cosmology that is definitely not real. On the scale of a supercluster the motions are coherent — everything is falling in, so the far side is approaching and the near side receding — and the structure is compressed rather than stretched, to 0.92 of its true extent here, measured on cluster centroids so the fingers have already averaged away. Below: why that compression is worth having. Its amplitude is the rate at which structure is currently growing, and the growth rate is Ωₘ(z) raised to a power that general relativity fixes at about 0.55. A theory of gravity that differs from general relativity on cosmological scales while matching every solar-system test changes that exponent and nothing else, and the two curves drawn — γ = 0.55 and γ = 0.68 — differ by only 4 per cent at redshift a half, against error bars of 12 per cent on the points beside them. The worst systematic in a redshift survey is the measurement — and it is a hard one, because a quarter of a change in the exponent that governs how gravity assembles structure moves the observable by less than the width of the curve it is drawn on. What the picture cannot show is the degeneracy that limits it: what is measured is fσ₈, a product, and separating the growth rate from the amplitude of clustering needs something else entirely. A map that is not of positions One axis of every redshift survey is not a distance but a velocity, and the difference is not noise. Inside a cluster it smears the galaxies into a finger pointing at the observer; on the scale of a supercluster it compresses the structure — and the amount of that compression is a test of gravity.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Angular momentumCorrelation functionDensity perturbationGaussian random fieldInertia tensorLagrangian perturbation theoryLarge-scale structureLinear growthProtogalaxySpin alignmentSpin parameterTidal tensorTidal torque theoryTurnaround