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Cosmology

A map stretched by the thing it measures

A redshift survey plots galaxies at distances derived from their redshifts, and a galaxy's redshift contains its own motion as well as the expansion. So the map is systematically distorted — squashed on large scales, drawn out into radial spikes on small ones — and both distortions are caused by the gravity the survey exists to measure.

Assumes Large-scale structure, Expansion and Virial theorem.

16 min read 8 figures Measuring the unreachableScales that defeat intuitionSmall perturbations, large consequences

A redshift survey does not measure positions. It measures two angles, which are exact, and a redshift, which is converted into a distance by assuming the galaxy is moving with the expansion and nothing else.

The redshift itself is not the problem. It is not a Doppler shift but a ratio of scale factors, and as a measurement it is superb: a spectrum gives it to a part in ten thousand in a few minutes, for objects a hundred times too faint to have a distance measured any other way. Everything about a redshift survey’s reach depends on that.

The assumption underneath the conversion is what is false, and it is false for every galaxy in the survey. Galaxies fall towards mass, at hundreds of kilometres a second in the field and at over a thousand inside a cluster, and that motion adds to the cosmological redshift indistinguishably. The resulting map is therefore not a map of where things are. It is a map of where things are, displaced along the line of sight by their own velocities — and since the velocities are caused by the very structure being mapped, the displacement is correlated with the structure, which turns it from noise into signal.

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.
Fig. 1 The same simulated volume plotted twice: once at true positions, once at the positions a redshift survey would infer. Two distortions appear and they act in opposite directions. Dense clusters are drawn out into elongated features pointing directly at the observer — fingers of God, produced by the random internal motions of a virialised system. Large-scale overdensities are squashed along the line of sight instead, because material on the far side is falling towards them and appears blueshifted, while material on the near side is falling away and appears redshifted.

Why the two distortions point opposite ways

The distinction is between coherent and random motion, and it maps exactly onto the distinction between a structure that has collapsed and one that has not.

Inside a cluster the galaxies have long since virialised. Their motions are random with respect to the cluster’s centre, with a dispersion of order a thousand kilometres a second, and a thousand kilometres a second corresponds to about ten megaparsecs of apparent radial displacement. A cluster three megaparsecs across therefore appears as a filament thirty megaparsecs long, aimed at the observer. It is the most conspicuous artefact in any redshift map and it is entirely an artefact — the elongation is a factor of ten and it points at the observer from every direction at once, which is how it announces itself.

The effect is large because of an accident of units. A peculiar velocity of a thousand kilometres a second is trivial dynamically, and yet at a Hubble constant near seventy kilometres per second per megaparsec it is fourteen megaparsecs of apparent displacement, which is enormous compared with a cluster. The conversion factor between a velocity and a distance in this subject is a small number, and that is what magnifies a modest motion into a gross distortion. On large scales nothing has collapsed. An overdense region is still growing, and the material around it is falling in coherently: the whole far side moves towards the observer and the whole near side moves away. Converted to distance, that pulls the far side nearer and pushes the near side further, and the structure is compressed along the line of sight.

The infall is the same phenomenon that shows up locally as galaxies departing from a smooth velocity–distance law: a peculiar velocity is a record of the gravitational field integrated over the age of the universe, so a map of infall is a map of where the mass is.

The compression is small — of order ten per cent — and it is not random, so it survives averaging over the whole survey. That is what makes it measurable, and what makes it the point of the exercise.

What the squashing is a measurement of

The amplitude of the large-scale flattening depends on how fast the infall is happening, which depends on how fast structure is growing, which depends on gravity.

Write the growth rate as f=dln⁡D/dln⁡af = \mathrm{d}\ln D/\mathrm{d}\ln a, the logarithmic rate at which the amplitude of density fluctuations grows with the scale factor. In general relativity with ordinary matter, f≈Ωm0.55f \approx \Omega_m^{0.55} — a relation that holds across a wide range of expansion histories and that theories of modified gravity generically violate.

The flattening measures ff multiplied by the amplitude of the fluctuations, and it does so through a very clean geometrical signature: the correlation function acquires a quadrupole. Structure that would be isotropic if positions were true becomes measurably anisotropic in a way that depends only on the angle to the line of sight.

The excess, measured and predicted. The two-point correlation function of the 1499 points in the slice, measured off the drawing rather than assumed. ξ(r) is the fractional excess probability of finding a second galaxy at separation r over what a uniform random field would give: ξ = 0 means no clustering at all. The points are counted in annuli around every galaxy at least 150 Mpc from an edge, so no annulus crosses the boundary and there is no edge correction to get wrong; the curve is the exact correlation function of the process the points were drawn from, ξ(r) = 1/(4πκσ²)·exp(−r²/4σ²), which for this field is 1.37 at zero separation. The two agree to 0.166 in ξ over the clustered range, which is the check that the estimator is measuring what it claims. The excess falls below one per cent by 145 Mpc, and that is the useful number: beyond it, knowing where one galaxy is tells almost nothing about where the next one is. In the real universe the same statistic falls below a per cent at about 100 Mpc, by a rather different route — a power law rather than a Gaussian — and with the acoustic bump sitting on top of it.
Fig. 2 The statistic being distorted. The correlation function measures the excess probability of finding two galaxies at a given separation over what a random distribution would give, and in true space it depends only on separation. In redshift space it depends on separation and on the angle between the pair and the line of sight — and decomposing that dependence into multipoles separates the growth rate from everything else.

The line of sight is a special direction, and it is the survey’s own. That is the structural fact that makes the whole method work: the universe has no preferred direction, so any anisotropy in the measured clustering must have been introduced by the observation. There is nothing to disentangle it from, because there is nothing else that could have produced it.

Homogeneity is a scale with a tolerance attached. The fractional scatter in the number of galaxies inside a circle, against the circle's radius, measured by dropping four hundred circles of each size wholly inside the slice. Two analytic curves are drawn against it and neither is fitted. The lower one is what a completely random field gives, 1/√N̄. The upper one is what this clustered field must give once the circle is much larger than a cluster, √((1+μ)/N̄) with μ = 13 galaxies per group — clustering multiplies the scatter by a constant and does not change its power: both fall as 1/R. The measured points join the upper curve above about 118 Mpc, roughly 5 cluster radii, and sit 3.1 times above pure Poisson at the largest circle. That factor is the practical content of clustering: a survey counting 244 galaxies in a circle has the statistical power of about 17 independent ones, which is why counting more galaxies in the same volume stops helping. Nothing here says the universe is homogeneous, and the figure is not able to. What it shows is the shape of the question: the scatter falls as 1/R, so "homogeneous" is not a property but a radius chosen against a tolerance. In the real universe the corresponding measurement is that the density scatter in spheres of 8 h⁻¹ Mpc is about 0.81, and redshift surveys find the counts settling onto the uniform-field scaling somewhere between 70 and 150 h⁻¹ Mpc — a range rather than a number, because different surveys and different tolerances give different answers, which is the honest state of it.
Fig. 3 And the property that guarantees it. Above about a hundred megaparsecs the distribution is homogeneous to a stated tolerance, which is what makes “isotropic in true space” a testable claim rather than an assumption. Homogeneity is a scale with a tolerance attached, and the redshift-space measurement lives at separations where that tolerance is comfortably met.

What the quadrupole actually looks like

It helps to be concrete about the shape. Plot the correlation function on a plane whose axes are the separation along the line of sight and the separation across it, and contours of constant correlation would be circles if positions were true.

They are not. At small transverse separations the contours are stretched enormously along the radial axis — the fingers. At large separations they are flattened, pushed in along the radial axis by a few per cent. Between the two the contours pass through nearly circular, and the crossover scale is roughly where structures have just stopped collapsing.

A single figure therefore carries both signals, at different radii, produced by the same physics in different regimes. The analysis extracts them by expanding the angular dependence in Legendre polynomials: the monopole is the ordinary isotropic clustering, the quadrupole carries most of the growth signal, and the hexadecapole helps break the remaining degeneracies.

Fingers 5.9 times long, a large scale squashed to 0.91, 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, 1400 kilometres a second of them, which at H₀ = 67.36 is 21 megaparsecs of smearing on an object a few across: the clusters become fingers 5.9 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 0.91 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.
Fig. 4 The same construction with the virial dispersion inside a cluster nearly doubled, to 1,400 kilometres a second. The fingers stretch from 3.5 times their real length to 5.9, and the large-scale squashing does not move at all — 0.91 against the same figure before. The two distortions are set by two different velocities and the figure separates them by construction: one is the random motion inside a collapsed object and the other is the coherent flow onto one that has not collapsed yet. A survey measures both at once and has to model the first to get at the second.

Growth is a different measurement from expansion

This is the part worth sitting with, because it is why anybody goes to the trouble.

Nearly every cosmological measurement in this collection is a measurement of geometry: the expansion history, through distances and angles. Supernova distances, the acoustic scale as a ruler, the angular size of the microwave background’s features — all of them constrain how the scale factor has behaved, and all of them are silent about how structure grew inside it.

Growth is a separate function, and in general relativity it is determined by the expansion history. Given how fast the universe expanded, the rate at which perturbations grew follows with no freedom. So measuring both is a consistency test of the theory rather than two measurements of the same thing.

A concrete instance makes the point. Two models can be constructed that agree exactly on every distance ever measured — the same supernova Hubble diagram, the same acoustic scale, the same microwave background peak positions — and that predict growth rates differing by ten per cent. Nothing geometric distinguishes them. A redshift-space measurement does.

That test is the main reason dark energy is not simply accepted as a modification of gravity. A modified theory can be tuned to reproduce any expansion history — that is only one function — but it will generically predict a different growth rate. Comparing the two therefore separates “the expansion is accelerating because of a new component” from “the expansion is accelerating because gravity is not what is assumed”, which no distance measurement can do.

The bias problem, which does not go away

There is a difficulty sitting underneath all of this and it deserves its own section.

Galaxies are not matter. They form at the peaks of the density field, and peaks are more clustered than the field they are peaks of — so the clustering measured from galaxies is amplified by a bias factor relative to the clustering of the mass. Worse, the bias depends on what kind of galaxy is being counted: luminous red galaxies are more biased than blue star-forming ones, and both are more biased than the mass.

Bias is degenerate with the fluctuation amplitude in every measurement of clustering alone. What redshift-space distortions supply is a way out, because the velocities respond to the total mass rather than to the galaxies. So the ratio of the anisotropic part of the clustering to the isotropic part cancels the bias to leading order, and what survives is the growth rate multiplied by the mass fluctuation amplitude — a combination that is genuinely about the matter.

The excess, measured and predicted. The two-point correlation function of the 1499 points in the slice, measured off the drawing rather than assumed. ξ(r) is the fractional excess probability of finding a second galaxy at separation r over what a uniform random field would give: ξ = 0 means no clustering at all. The points are counted in annuli around every galaxy at least 300 Mpc from an edge, so no annulus crosses the boundary and there is no edge correction to get wrong; the curve is the exact correlation function of the process the points were drawn from, ξ(r) = 1/(4πκσ²)·exp(−r²/4σ²), which for this field is 1.37 at zero separation. The two agree to 0.282 in ξ over the clustered range, which is the check that the estimator is measuring what it claims. The excess falls below one per cent by 90 Mpc, and that is the useful number: beyond it, knowing where one galaxy is tells almost nothing about where the next one is. In the real universe the same statistic falls below a per cent at about 100 Mpc, by a rather different route — a power law rather than a Gaussian — and with the acoustic bump sitting on top of it.
Fig. 5 The correlation function of the same slice measured out to three hundred megaparsecs rather than a hundred and fifty. The excess falls towards zero and stays there, which is the real-space statement this essay’s whole distortion is measured against: past a scale, one galaxy tells you nothing about where the next one is. The distortion is a distortion of a function that is asymptotically flat, and that is what makes an anisotropy in it legible at all — there is nothing else at those separations to confuse it with.

The other anisotropy, which is geometric

A second distortion afflicts the same map for a completely different reason, and separating the two is a large part of the analysis.

Converting redshifts into distances requires a cosmology, and if the assumed cosmology is wrong the conversion stretches the map by different factors along and across the line of sight. Radial separations are converted using the Hubble rate at that redshift; transverse ones using the angular-diameter distance. Get either wrong and a sphere becomes a spheroid.

A sphere reconstructed as a spheroid, and two distortions 0.15 apart. Left: the acoustic scale in the plane of separation across the line of sight against separation along it, one quadrant of it. In the cosmology that actually holds, the sound horizon is a sphere of 99.0 h⁻¹ Mpc and its locus here is a quarter circle. That is the whole content of the Alcock–Paczyński test: nothing about the early universe distinguishes the radial direction from the transverse one, so any departure from a circle is a statement about the observer's arithmetic rather than about the ruler. Converting angles into transverse separations needs the transverse comoving distance and converting redshift intervals into radial ones needs H(z), so assuming distances 1.1 times too large and rates 0.94 times too small returns an ellipse of axis ratio 0.855 — and the ellipticity measures that distance times the expansion rate over c, in which the sound horizon has cancelled. A ruler of unknown length still measures a shape. The third curve is the difficulty: peculiar velocities also distort the same correlation function along the same axis, squashing it by 1/(1+β) = 0.704 for β = 0.42, and a squashing is a squashing. Separating a geometric distortion from a dynamical one is the entire art of the measurement, and it is done by using the fact that they have different dependences on scale — the velocities act on the broad-band shape and the ruler is a feature. Right and below: the two numbers the same feature gives at each redshift. Across the line of sight, the transverse distance divided by the sound horizon; along it, c divided by the expansion rate times the sound horizon. Two functions of the expansion history, from one bump in one correlation function, and their agreement with a single model is one of the sharper consistency tests in the subject.
Fig. 6 The Alcock–Paczynski test, which turns that nuisance into a measurement. Any population known to be statistically spherical — and clustering is, in true space — can be used: the assumed cosmology is varied until the reconstructed shapes come out round. It measures the product of the Hubble rate and the angular-diameter distance, which is a purely geometric quantity, and it is completely independent of the growth measurement even though both are read off the same anisotropy.

Disentangling the two anisotropies is the central technical problem of a modern redshift survey. They have different dependences on scale and on angle, which is what makes it possible, and the fits are done jointly rather than in sequence.

Reconstruction, and undoing the distortion

There is a third use for the velocity field, and it turns the distortion into a repair rather than a measurement.

The infall is caused by the density field, and the density field is what the survey has measured. So the velocities can be estimated from the map itself, and each galaxy moved back along the line of sight by its estimated displacement — a procedure that partly undoes both the distortion and the smearing that gravitational evolution has done to the acoustic scale over ten billion years.

This is not circular, because it does not need to be accurate to be useful. Displacing galaxies by an estimate that is right on average sharpens the acoustic peak in the correlation function substantially, and a sharper peak is a more precise ruler. The improvement is close to a factor of two in the distance precision from the same data, which is why every modern survey does it.

The reconstruction also gives, as a by-product, a map of the velocity field, and that map can then be compared against velocities measured directly for the subset of galaxies with independent distances. Where the two agree, the mass model is right; where they do not, something is missing.

Fingers 3.1 times long, a large scale squashed to 0.92, and a test of gravity. Left: a 200-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.
Fig. 7 The same effect in a box a little over half the size. The fingers are 3.1 times their true length rather than 3.5 and the squashing is 0.92: both numbers move, because a smaller box holds fewer superclusters and the statistics of a small sample are what they are. That sensitivity is the practical difficulty of the measurement rather than a defect in it — the quantity being measured is a property of a distribution, so a survey volume is a sample size, and the errors on fσ8f\sigma_8 in the next section are dominated by exactly this.

What the fingers are good for

The small-scale distortion is usually treated as a contaminant to be modelled and marginalised over. It is not only that.

The length of the fingers is a direct statistical measurement of the pairwise velocity dispersion of galaxies — a map that is not of positions but of motions — which is a measurement of the mass in the systems they inhabit — a redshift survey therefore contains a mass census that requires no cluster to be identified and no virial equilibrium to be assumed about any individual object. It is a much blunter instrument than weighing a cluster three ways, and it applies to the whole survey at once.

Fitting a simulation instead of a formula

The modelling difficulty described above — that most of the information sits at scales where linear theory has failed and the phenomenological damping is a guess — has produced a change in how the analysis is done, and the change is worth describing because it trades one kind of uncertainty for another.

The older approach writes down an analytic model with a handful of nuisance parameters, restricts the fit to scales where that model has been shown to be unbiased against simulations, and discards everything smaller. The restriction is severe: it throws away the majority of the measured pairs, because the number of pairs at a given separation grows as the cube of the separation only up to the survey’s size and the small separations are where the counts are.

The newer approach fits the simulations directly. A large suite of N-body runs is produced across a grid of cosmological parameters; galaxies are placed into the resulting haloes according to a prescription with its own parameters; the clustering statistics are computed for every run; and a fast interpolator — an emulator — is trained to predict those statistics anywhere in the grid. The survey’s measurements are then compared against the emulator rather than against a formula, and the fit runs down to separations of a megaparsec or less.

What that buys is information. What it costs is that the result now depends on the prescription for how galaxies occupy haloes, which is a model of galaxy formation rather than of gravity, and on the simulations having converged at the scales being used.

The two approaches are not in competition so much as in tension, and the honest position is that they answer slightly different questions. The analytic fit measures a growth rate with a well-understood systematic and a large error bar; the emulator measures a growth rate with a smaller error bar and a systematic that is a modelling choice. Where the two are run on the same data they agree within their errors, which is reassuring and is not the same as either being validated.

There is a third position that is gaining ground and that sidesteps the choice. Rather than fitting a model to a compressed statistic, a field-level analysis attempts to infer the initial conditions of the observed volume directly — the density field that, evolved forward through gravity and through a galaxy-formation prescription, reproduces the catalogue galaxy by galaxy. The velocities then come out as a by-product rather than being inferred from an anisotropy at all, and the distinction between the two distortion regimes disappears, since a forward model produces both automatically.

Whether that is progress depends on something the method cannot check about itself. Inferring a field with millions of degrees of freedom from a catalogue requires a prior, and the prior is that the initial conditions are a Gaussian random field with a spectrum the analysis is also fitting. Where that is right the method is close to optimal; where it is wrong the failure is not localised to any one statistic and is correspondingly hard to notice.

The scales that carry the most information are the scales where the theory is weakest, and every methodological development in this subject for two decades has been an attempt to move the boundary between them.

Homogeneity is a scale with a tolerance attached. The fractional scatter in the number of galaxies inside a circle, against the circle's radius, measured by dropping four hundred circles of each size wholly inside the slice. Two analytic curves are drawn against it and neither is fitted. The lower one is what a completely random field gives, 1/√N̄. The upper one is what this clustered field must give once the circle is much larger than a cluster, √((1+μ)/N̄) with μ = 13 galaxies per group — clustering multiplies the scatter by a constant and does not change its power: both fall as 1/R. The measured points join the upper curve above about 20 Mpc, roughly — cluster radii, and sit 2.8 times above pure Poisson at the largest circle. That factor is the practical content of clustering: a survey counting 244 galaxies in a circle has the statistical power of about 17 independent ones, which is why counting more galaxies in the same volume stops helping. Nothing here says the universe is homogeneous, and the figure is not able to. What it shows is the shape of the question: the scatter falls as 1/R, so "homogeneous" is not a property but a radius chosen against a tolerance. In the real universe the corresponding measurement is that the density scatter in spheres of 8 h⁻¹ Mpc is about 0.81, and redshift surveys find the counts settling onto the uniform-field scaling somewhere between 70 and 150 h⁻¹ Mpc — a range rather than a number, because different surveys and different tolerances give different answers, which is the honest state of it.
Fig. 8 The counts-in-cells scatter for a slice whose clusters are nearly twice as extended — a Gaussian of forty megaparsecs about each centre rather than twenty-two. The scatter approaches the random 1/N1/\sqrt{N} curve more slowly and from further out, because there is correlated structure on larger scales to average over. The homogeneity scale is not a constant of nature; it is the scale at which whatever clustering the universe has stops mattering, and it therefore moves with the clustering — which is why quoting it requires quoting a tolerance as well.

Where the picture stops

The small-scale modelling is not first-principles. The transition between the coherent infall regime and the virialised one happens over a range of scales where neither limit applies, and it is handled with a phenomenological damping function whose form is chosen for convenience. Results are quoted only above a scale where that choice demonstrably does not matter, and establishing where that scale is consumes a large fraction of the analysis effort.

Bias is only cancelled to leading order. Beyond the linear approximation the relationship between galaxies and mass involves further parameters, and the cancellation is incomplete. Whether the residual is small enough for the next generation of surveys, which will be an order of magnitude more precise, is an open question rather than a settled one.

Survey geometry leaks into the anisotropy. A survey covering a limited patch of sky has a window function that is itself anisotropic with respect to the line of sight, and its effect on the measured multipoles has to be computed and divided out. On a small or oddly shaped field the correction is not small, which is one reason the field has moved towards surveys covering enormous contiguous areas.

And the observer’s own motion has to be removed first. The Sun moves at some 370 kilometres a second with respect to the microwave background, which is a dipole in every redshift in the survey. It is subtracted using the background’s own dipole — which assumes that dipole is entirely kinematic, an assumption that is itself under examination.

Where this ladder goes next

Later rungs on this anchor: the multipole decomposition in detail, and which combinations of growth, bias and amplitude each moment constrains; direct peculiar-velocity surveys using distance indicators, which measure the velocity field rather than inferring it; the cross-correlation of galaxy positions with lensing maps, which breaks bias by a second route; the void statistics, where the distortions have the opposite sign and different systematics; and the joint fit with the acoustic scale, which is how one survey delivers a growth rate and two distances at once.

Where this goes next

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. 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. The same recession law from two different galaxies. Twenty galaxies at fixed comoving positions after the whole picture has been multiplied by 1.34, with each galaxy's displacement drawn from where it was to where it is. Every arrow's tail sits at the old separation and its tip at the new one, so the arrow is exactly 0.34 times its tail's distance from the highlighted galaxy — proportional to separation for one reason and no other: a uniform scaling moves everything in proportion to its distance from whatever point the scaling is measured about. The right-hand panel measures from a different galaxy and gets the identical law with the identical constant. That is the content of a linear velocity–distance relation. It is the signature of an expansion with no centre, and the observation that every galaxy recedes is therefore not evidence that this one is the centre — it is evidence that none of them is. The shift that is not a Doppler shift Every galaxy beyond the Local Group has its lines shifted to the red, by an amount proportional to its distance. Read as a velocity that looks like a confession that everything is fleeing from here; read as a change of scale it says the opposite, because a uniform expansion produces exactly the same law measured from any galaxy in it.

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Alcock paczynski testAnisotropic clusteringCorrelation functionFingers of GodGrowth rateKaiser effectLinear biasModified gravityPeculiar velocityRedshift space distortionStructure growthVelocity dispersion