Astronomy & Celestial Mechanics
the sky, computed
Cosmology

A test that can only fail one way

Single-field inflation predicts a local non-Gaussianity of 0.015, a skewness in the primordial potential of a few parts in a million. The measurement is −0.9 ± 5.1. A detection at the level of one would eliminate every model with a single clock at once; a null result at any reachable precision confirms nothing, because the prediction lies below anything the sky has enough independent modes to measure.

Assumes Inflation and Microwave background.

17 min read 6 figures Small perturbations, large consequencesMeasuring the unreachable

Most tests in physics are symmetric. A prediction is made, a measurement is taken, and the two either agree or they do not, so a theory stands or falls with the result. A few tests are not like that, and the one this essay is about is the cleanest example in cosmology: a measurement that can falsify a whole class of theories in one stroke and cannot, at any precision anyone can reach, confirm it.

The class is every model of inflation in which a single field sets the clock — every potential drawn in the essays on the tilt and the tensor ratio. The measurement is whether the primordial fluctuations are Gaussian. The prediction is that they are, to a degree so extreme that the residual departure is smaller than the statistical noise of any sky, and the question is what it means that nobody has yet seen a departure at all.

One sky with and without a local non-Gaussianity of fNL·σ = 0.3. The same scale-invariant random field drawn twice, from one seed: on the left as it is, Gaussian, and on the right after the local transformation Φ → Φ + fNL(Φ² − ⟨Φ²⟩) with fNL·σ = 0.3. Solid contours are one and two standard deviations above the mean, dashed ones below, and each map is measured against its own mean and spread. The transformation adds to every value in proportion to its square, so peaks are pushed up and troughs are pulled back towards the mean: the area above +2σ goes from 1.8 to 4.7 per cent of the map and the area below −2σ from 2.4 to 0.0, and the skewness of the values rises from −0.073 to 1.484. This is exaggerated by a factor of about 2200. The primordial potential varies by about 3 × 10⁻⁵, so even fNL = 5 — the size of the current uncertainty — makes fNL·σ about 10⁻⁴.
Fig. 1 One random field drawn twice from the same seed, with the same scale-invariant spectrum: on the left Gaussian, on the right after the local transformation Φ→Φ+fNL(Φ2−⟨Φ2⟩)\Phi \to \Phi + f_{\rm NL}(\Phi^2 - \langle\Phi^2\rangle) at fNLσ=0.3f_{\rm NL}\sigma = 0.3. Solid contours are one and two standard deviations above the mean and dashed ones below. The area above +2σ rises from 1.8 to 4.7 per cent of the map, the area below −2σ falls from 2.4 per cent to nothing, and the skewness of the values goes from −0.07 to 1.48. The distortion is about two thousand times larger than the current uncertainty allows.

A field made lopsided by its own square

A Gaussian random field is fixed entirely by its power spectrum. Every higher statistic — the three-point correlation, the skewness of the distribution of values, the relative number of hot and cold peaks — follows from the two-point one or vanishes. The fluctuations measured in the microwave background’s standing wave are Gaussian to every precision reached so far, and inflation predicts it: the fluctuations are quantum zero-point motions of a nearly free field, and the ground state of a free field is exactly Gaussian.

Departures come from interactions and from non-linear evolution, and the standard way of parameterising the simplest kind is to write the primordial gravitational potential as a Gaussian field plus a multiple of its own square,

Φ=ΦG+fNL(ΦG2−⟨ΦG2⟩).\Phi = \Phi_{\rm G} + f_{\rm NL}\left(\Phi_{\rm G}^2 - \langle\Phi_{\rm G}^2\rangle\right).

The subtraction keeps the mean at zero. The square is positive wherever the field is large in either direction, so for positive fNLf_{\rm NL} the peaks are pushed further up and the troughs are pulled back towards the mean, and the distribution of values acquires a tail on one side. It is called local because the correction at each point depends only on the field at that point, and that locality is the signature of a non-linearity that acted after each scale had left the Hubble radius, when no process could couple distant points — after the crossing drawn in the diagram of the comoving Hubble radius.

The size of the effect is not fNLf_{\rm NL} but fNLf_{\rm NL} multiplied by the amplitude of ΦG\Phi_{\rm G}, and the amplitude is tiny: about 3×10−53\times10^{-5} for fluctuations within a factor of ee in scale. So a value of fNL=5f_{\rm NL} = 5, the size of the present uncertainty, distorts the field by about a part in seven thousand of its own spread, and the opening figure had to be exaggerated by three orders of magnitude to show anything.

Two maps that differ at the level the sky allows. The same scale-invariant random field drawn twice, from one seed: on the left as it is, Gaussian, and on the right after the local transformation Φ → Φ + fNL(Φ² − ⟨Φ²⟩) with fNL·σ = 0.00014. Solid contours are one and two standard deviations above the mean, dashed ones below, and each map is measured against its own mean and spread. At this value, which is fNL = 5 on a potential that varies by 3 × 10⁻⁵, the two maps are the same drawing: the area above +2σ is 1.75 and 1.75 per cent, and no contour has moved by the width of its own line. That is the signal the microwave background has been searched for, and why the search is a statistical one over millions of pixels rather than anything a map can show.
Fig. 2 The same two maps at fNLσf_{\rm NL}\sigma = 0.00014, which is fNLf_{\rm NL} = 5 on a potential that varies by 3 × 10⁻⁵ — the largest distortion the present measurement permits. The area above +2σ is 1.75 per cent in both, and no contour has moved by the width of its own line. This is the signal the microwave background has been searched for, drawn at its real size.

The second map is the honest one, and it says what kind of measurement is involved. Nothing about a non-Gaussianity at the permitted level can be seen in a picture. It lives in the correlation of three points averaged over millions of triangles, and the whole difficulty is statistical before it is instrumental.

Why one clock fixes the number

The reason a single field leaves so little is an argument due to Juan Maldacena, made in 2003 and generalised by Paolo Creminelli and Matias Zaldarriaga the following year, and it is one of the few results about inflation that holds without choosing a potential.

Consider three waves whose wave vectors close into a triangle, with one side much shorter than the other two — one long wave and two short ones, the configuration called squeezed. By the time the short waves leave the Hubble radius the long one has been outside it for many e-folds and is frozen. From the short waves’ point of view a frozen long wave is not a perturbation at all; it is a constant rescaling of the local coordinates, as if that patch of the universe had been slightly larger or smaller. In a single-field model the only thing such a rescaling can do is shift the moment at which the short waves left, and the only effect of leaving slightly earlier or later is to sample the power spectrum at a slightly different point along its tilt.

The picture has a name, the separate-universe picture, and it is more literal than it sounds. A region sitting inside a long crest is, to anything local within it, simply a universe with a slightly different density budget, expanding on its own slightly shifted schedule. Everything that happens in it is the same physics run against a clock that started a fraction of an e-fold early. If there is only one clock, that shift is all the long wave can do.

So the correlation between one long wave and two short ones is proportional to how much the power changes between one moment of exit and the next, which is the tilt. The result is

fNLsqueezed=512 (1−ns)=0.0146,f_{\rm NL}^{\rm squeezed} = \frac{5}{12}\,(1 - n_s) = 0.0146,

with the measured index. Every single-clock model gives it, whatever its potential, because the argument used nothing but the existence of one clock. It is called a consistency relation, and it converts a measurement of the tilt into a prediction about three-point correlations.

The subtlety that makes the test asymmetric goes one step further. The same argument says the squeezed-limit signal is a coordinate effect, and a later analysis by Enrico Pajer, Fabian Schmidt and Zaldarriaga showed that an observer who describes the short waves in their own local coordinates sees even that 0.015 vanish. What remains in a real measurement are the non-linearities of gravity itself and of the way light and galaxies trace the potential, which contribute amounts of order one and have to be modelled and removed. Single-clock inflation predicts no physical local non-Gaussianity at all. A detection of a local signal clearly above one — above what gravity and projection add by themselves — rules out every model with a single clock simultaneously, and there is no measurement that could confirm the prediction, because the prediction is zero.

Counting the modes

How far from zero can a measurement get? The crudest statistic gives an answer that turns out to be close to the best one.

How lopsided the sky would be, against fNL. The skewness a local non-Gaussianity gives the primordial potential, 6 fNL σᵩ to first order, against fNL, both logarithmic, for a potential varying by σᵩ = 2.75e-5 per logarithmic interval of scale — the value fixed by the measured amplitude. The horizontal line is how precisely a skewness can be measured by counting: √(6/N) for N independent values, taken here as the 3.2 million modes of a map resolved to multipole 2000 over 80 per cent of the sky. It crosses the line at fNL ≈ 8, which is within a factor of two of the ±5.1 the optimal estimators actually reached — so the whole difficulty is already present in the crudest possible statistic. The squeezed-limit value single-field inflation leaves, (5/12)(1 − nₛ) = 0.0146, is 2.8 decades below that line and corresponds to a skewness of 2.4e-6. Nothing about a better map closes that gap; only more independent modes do, which is why the next measurements are planned in the three-dimensional distribution of galaxies rather than on the two-dimensional sky.
Fig. 3 The skewness a local non-Gaussianity gives the primordial potential, 6 fNLf_{\rm NL} σ, against fNLf_{\rm NL} on logarithmic axes, for σ = 2.75 × 10⁻⁵. The horizontal line is the precision with which a skewness can be measured by counting, 6/N\sqrt{6/N}, for the 3.2 million independent modes of a map resolved to multipole 2000 over 80 per cent of the sky; the curve crosses it at fNLf_{\rm NL} ≈ 8, against the ±5.1 the optimal analysis achieved. The single-clock squeezed value, 0.0146, is a skewness of 2.4 × 10⁻⁶, nearly three decades below the line.

The skewness of a set of NN independent Gaussian numbers has a statistical scatter of 6/N\sqrt{6/N}, whatever instrument measured them. A map of the microwave background resolved to multipole ℓ\ell has about ℓ2\ell^2 independent modes, and the resolution cannot usefully go past ℓ≈2000\ell \approx 2000, because beyond that the thickness of the last-scattering shell has erased the primordial pattern and what remains is foreground and noise. That gives a few million modes, a skewness uncertainty of about 10−310^{-3}, and a sensitivity to fNLf_{\rm NL} of about eight. The analyses Planck actually performed use estimators, descended from one Eiichiro Komatsu, David Spergel and Benjamin Wandelt built in 2005, that weight each triangle of multipoles optimally, and they reach ±5.1 — a factor of 1.6 better than counting pixels, and nowhere near a factor of a hundred.

The limit is not the detector. It is the number of independent modes the last-scattering surface contains, and Planck is already close to it. The microwave background will never measure fNLf_{\rm NL} much better than it has, which is why the next measurements are planned elsewhere.

Where in triangle space the signal lives

The local form is one shape among several, and the shape is diagnostic. The three-point correlation of the potential in Fourier space, the bispectrum, is a function of three wave vectors that close into a triangle, and different physics puts its signal in different triangles.

Where each kind of non-Gaussianity puts its signal among triangles. The bispectrum — the correlation among three waves whose wave vectors close into a triangle — as a function of the triangle's shape, for local and equilateral non-Gaussianity. Each point in a panel is one shape, k₂/k₁ across and k₃/k₁ up, with the longest side k₁ fixed; contours are in units of each template's value on the equilateral triangle, at 1.2, 1.5, 2, 3, 5, 10, 20 for the local shape, which never falls below one, and at 0.1, 0.25, 0.5, 0.75, 0.9 for the equilateral shape, which never rises above it, each labelled where it meets the right-hand edge. The local template grows without limit towards the squeezed corner, where one wave is much longer than the other two: it describes a long wave modulating the amplitude of short ones, which is what a field that became non-Gaussian after leaving the Hubble radius does. The equilateral template vanishes there and peaks where the three waves are the same size, which is what interactions of the inflaton at the moment of crossing do. The two are nearly orthogonal, so they are measured separately, and a single-clock inflaton can produce the second but not — beyond the (5/12)(1 − nₛ) = 0.0146 its own tilt fixes — the first.
Fig. 4 The shape of the bispectrum for local and equilateral non-Gaussianity, each normalised to one on the equilateral triangle. Across each panel is the ratio of the second-longest side of the triangle to the longest, and up it the ratio of the shortest; contours run from 1.2 to 20 for the local shape, which never falls below its equilateral value, and from 0.1 to 0.9 for the equilateral shape, which never rises above it. The local template climbs without limit towards the squeezed corner, where one wave is much longer than the other two, and the equilateral template falls to zero there and peaks where the three waves are the same size.

The two templates are nearly orthogonal, so they are measured separately and constrain different physics. Local non-Gaussianity is what a field that became non-linear outside the Hubble radius produces, and it concentrates in the squeezed corner — the long wave modulating the amplitude of the short ones. Equilateral non-Gaussianity comes from interactions at the moment of horizon crossing, when all three waves are about the same size, and it is what an inflaton whose fluctuations propagate slower than light produces: the lower the sound speed, the larger the signal. A third, orthogonal template picks up what lies between.

Planck’s 2018 values are fNLlocal=−0.9±5.1f_{\rm NL}^{\rm local} = -0.9 \pm 5.1, fNLequil=−26±47f_{\rm NL}^{\rm equil} = -26 \pm 47 and fNLortho=−38±24f_{\rm NL}^{\rm ortho} = -38 \pm 24. None is a detection. The equilateral and orthogonal uncertainties are an order of magnitude larger than the local one because their signal is spread over the whole triangle rather than piled into one corner, but they are not uninformative: a single field with a sound speed much below a tenth of light’s would have produced an equilateral signal of hundreds, and it did not.

A bias that remembers the long wave

The squeezed configuration has a physical reading that opens a second route to the measurement, one with far more modes than the sky.

If a long wave modulates the amplitude of the short fluctuations in its region, it modulates how many of the short peaks rise high enough to collapse into galaxies. A region sitting in a long-wavelength crest then has more galaxies than its matter density alone would predict, and a trough fewer. That is an extra contribution to galaxy bias — the factor by which galaxies over-trace the matter — and because it is driven by the potential rather than by the density, it grows towards large scales as the inverse square of the wavenumber.

A bias that grows on the largest scales, if fNL is not zero. The correction a local non-Gaussianity makes to how strongly galaxies trace the matter, Δb = 3 fNL (b − 1) δc Ωₘ H₀² / (c² k² T(k) D), against wavenumber, for a population with bias b = 2 at redshift 1, both axes logarithmic. The correction arises because a long-wavelength fluctuation in a non-Gaussian field changes the local amplitude of the short ones, and so changes how many of the short peaks cross the threshold for collapse. It rises as k⁻² towards large scales, where nothing in Gaussian clustering does — so it is a signal with a shape as well as a size, which is what makes it measurable. The horizontal line is a tenth of the bias itself; fNL = 1 reaches it at k = 0.0014 h/Mpc, a wavelength of 4574 Mpc/h, and fNL = 5 reaches it at k = 0.0031 h/Mpc, a wavelength of 2018 Mpc/h. Scales that large contain few independent modes in any survey, which is the limit on this method, and they are also where every systematic in a galaxy survey's selection is largest.
Fig. 5 The change in galaxy bias a local non-Gaussianity produces, Δb=3fNL(b−1) δc ΩmH02/(c2k2T(k)D)\Delta b = 3f_{\rm NL}(b - 1)\,\delta_c\,\Omega_m H_0^2/(c^2k^2T(k)D), against wavenumber on logarithmic axes, for a population with b = 2 at redshift 1. The correction rises as k−2k^{-2} on large scales, where Gaussian clustering has no such feature. The dashed line is a tenth of the bias itself: fNLf_{\rm NL} = 1 reaches it at k = 0.0014 h/Mpc, a wavelength of 4,600 Mpc/h, and fNLf_{\rm NL} = 5 at k = 0.0031 h/Mpc, 2,000 Mpc/h.

The signature was pointed out by Neal Dalal and collaborators in 2008, and its virtue is that it has a shape: nothing in ordinary gravitational clustering rises as k−2k^{-2} on scales of gigaparsecs. Above the scale at which the matter distribution becomes homogeneous, ordinary clustering is weak and nearly featureless, which is what gives a rise of that kind room to be seen.

Its difficulty is visible in the wavelengths in the caption. A survey measures a wavenumber only if its volume contains several wavelengths, and the scales where the effect is large are close to the size of the largest surveys ever made. Those are also the scales where every error in how the survey chose its targets — a patch of sky with more stars contaminating the sample, a band of dust dimming the galaxies behind it, a region observed in worse conditions — produces spurious power, and all of those errors grow towards large scales too. The measurement is a contest between a k−2k^{-2} signal and systematics with a similar trend, won by mapping the systematics rather than by observing longer.

A galaxy survey has one great advantage over the sky: it is three-dimensional. The number of independent modes grows with the volume, not with the area, and the redshifts that place galaxies along the line of sight — the same redshifts that stretch the map along the line of sight — are what supply the third dimension. Surveys completed so far give uncertainties on fNLf_{\rm NL} of order ten, larger than Planck’s, and the SPHEREx satellite, launched in 2025 to map the whole sky in low-resolution spectra, was designed with an uncertainty of about one as its primary cosmological goal.

What a second field would have left

The threshold of one is not an arbitrary round number, and a specific alternative shows why.

Suppose the fluctuations were not generated by the inflaton at all but by a second light field — a curvaton — that was present during inflation, carried its own quantum fluctuations through it without driving the expansion, and decayed into radiation long afterwards. Its fluctuations become curvature only when its energy density becomes a significant fraction of the total, and a small component has to fluctuate by a large fraction of itself to supply the whole observed amplitude. A large fractional fluctuation of an energy density is non-linear, so the curvaton leaves local non-Gaussianity behind.

A second field that would leave the sky lopsided. The local non-Gaussianity produced by a curvaton — a second, light field that plays no part in driving inflation, carries its own fluctuations through it, and later decays into radiation — against the fraction of the energy density it carried when it decayed, r, on a logarithmic axis. In the sudden-decay approximation fNL = 5/(4r) − 5/3 − 5r/6, which is −5/4 if the curvaton dominated completely and grows without limit as its share shrinks, because a small component has to fluctuate by a large fraction of itself to supply the whole observed amplitude, and a large fractional fluctuation of an energy density is non-linear. The shaded region is excluded at two sigma by the measured fNL = −0.9 ± 5.1; it requires r above 0.11, so a curvaton would have had to make up at least 11 per cent of the universe at decay. The dotted line is single-field inflation's squeezed-limit value, 0.0146, indistinguishable from zero on this axis. The comparison is the argument: an alternative mechanism for the same fluctuations predicts a skewness the data can reach, and the single-field one predicts a skewness it cannot.
Fig. 6 The local non-Gaussianity a curvaton produces, fNLf_{\rm NL} = 5/(4r) − 5/3 − 5r/6 in the sudden-decay approximation, against r, the fraction of the energy density it carried when it decayed. A curvaton that dominated completely gives −5/4; one that carried a small share gives a large positive signal. The shaded region is excluded at two standard deviations by the measured −0.9 ± 5.1, which requires r above 0.11. The dotted line is the single-clock value, 0.0146.

That is a model the data already constrain: if the fluctuations came from a curvaton, it made up at least eleven per cent of the universe when it decayed. It also marks where a precision of about one matters. A curvaton that dominated completely leaves fNL=−5/4f_{\rm NL} = -5/4, and a single-clock model leaves zero plus the effects of gravity; separating the two requires an uncertainty well below one, and that is the precision the three-dimensional surveys are aiming for.

The curve carries a warning as well. It passes through zero, at rdec=0.58r_{\rm dec} = 0.58, so a curvaton that happened to carry a little over half the energy when it decayed would leave no local signal and be indistinguishable from a single clock by this measurement alone. That is the asymmetry of the test in a second form: a large signal points to a specific kind of physics, and a null result is compatible with more than one.

A curvaton also has a way to fail that has nothing to do with skewness. If it decayed after the dark matter or the excess of matter over antimatter had been produced, its fluctuations would vary the ratio of those components to the radiation from place to place — an isocurvature perturbation, as distinct from the curvature perturbation that changes all of them together. The microwave background limits any such component to a few per cent of the total, and the uniformity of the ratio four light-element abundances measure points the same way. So a surviving curvaton must have decayed before those relics formed, which is a constraint on its lifetime that the three-point function never sees.

What is actually measured

The quantity fitted to the microwave background is not the potential but temperature and polarisation, and between them lies the acoustic physics of the plasma, which is itself slightly non-linear. The largest contaminant of the local estimator is the correlation between the gravitational lensing of the maps and the integrated Sachs–Wolfe effect of the potentials the light crossed, which produces a squeezed-limit signal of its own of about seven in the units used here; it is computed and subtracted, and the quoted −0.9±5.1-0.9 \pm 5.1 is after that subtraction. The Galaxy’s own emission, point sources and the scan pattern of the satellite are handled the same way, and consistency across four independent component-separation methods is what makes the null result credible.

The scale-dependent bias measurement is younger and less mature. Its published values so far come from quasars and luminous galaxies in the largest spectroscopic surveys, and each analysis spends most of its length on the large-scale systematics described above, with results whose uncertainties are still dominated by those systematics rather than by the number of galaxies.

What the maps leave out

The maps are two-dimensional toys. The field drawn has power falling as k−2k^{-2} in two dimensions, which gives equal variance per logarithmic interval of scale, like the real potential; it is not a simulation of the sky, and the contour fractions depend on its particular seed.

The sky is temperature, not potential. On the largest scales the temperature is proportional to the potential, and on the scales that carry most of the information the acoustic transfer mixes them, which spreads the local signal across multipoles and is the reason optimal estimators work in harmonic space rather than on maps.

The templates are not the only shapes. Specific models predict bispectra that oscillate, or peak at flattened triangles, and are searched for separately. A null result for three templates is not a null result for every shape.

And the curvaton curve assumes a sudden decay. A gradual decay changes the coefficients, including where the curve crosses zero, without changing the conclusion that a subdominant component leaves a large signal.

Still open: whether the universe had one clock

The measurement will tighten by perhaps a factor of five over the next decade, from the microwave background’s ±5 to the ±1 the three-dimensional surveys are designed to reach, and possibly beyond with the redshifted twenty-one centimetre line of the neutral hydrogen that filled the universe before reionisation ended, whose volume is larger still. At that precision a local signal would rule out every single-clock model and point to something like a curvaton; a null result would rule out the curvaton that dominated, and nothing more. Whether the primordial fluctuations were generated by the same field that drove the expansion is the question that result would bear on — and a zero, however precise, is the one answer that leaves it open.

Where this goes next

m²φ²: where the observed scales left, and where inflation ends. The m²φ² potential, drawn as a shape with its height divided out, against the field in reduced Planck masses. The field rolls downhill towards zero and inflation ends where the slow-roll parameter ε reaches one, at φ = 1.41. The shaded band is the stretch of field the scales now seen on the sky left the Hubble radius from: 60 e-folds before the end at φ = 15.56 and 50 before it at φ = 14.21. Nothing about the sky depends on the rest of the curve. Across that band the two numbers the tilt is made of are ε = 9.01e-3 and η = 0.0090, which give a spectral index of 0.9640 and a tensor-to-scalar ratio of 0.1441 at 55 e-folds. The height is not in either: it is fixed separately by the amplitude of the fluctuations, which puts the potential at (2.0 × 10¹⁶ GeV)⁴ there — and multiplying the whole curve by any constant leaves the band, the tilt and the ratio exactly where they are, because every slow-roll quantity is a ratio of the potential to its own derivatives. The field travels 13.49 Planck masses from the middle of the band to the end. The tilt knows the slope and not the height The measured spectral index, 0.965, is quoted as the strongest evidence for inflation, and it is a statement about two dimensionless numbers — how steeply the potential fell and how sharply that slope was changing, over the few e-folds the sky can see. The height of the potential is not in it at all, which is why potentials that look nothing alike reproduce it. The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has. A standing wave frozen at one instant The temperature of the microwave background varies across the sky by one part in a hundred thousand, and the sizes of the patches are not random. There is a preferred angular scale near one degree, and it is a sound wave that stopped ringing four hundred thousand years after the beginning. 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 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.

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BispectrumConsistency relationCurvatonGalaxy biasInflatonNon gaussianityPrimordial fluctuationsScalar spectral indexScale dependent biasSqueezed limit