Astronomy & Celestial Mechanics
the sky, computed
Cosmology

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.

Assumes Expansion and Luminosity function.

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

Every equation in this field rests on the cosmological principle: that the universe is homogeneous and isotropic. The velocity–distance law follows from it and from nothing else; the Friedmann equations assume it in their derivation; and every distance, age and density quoted so far is computed in a metric that has it built in.

Looked at directly, the universe is nothing of the kind.

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?
Fig. 1 A slice of a galaxy survey a billion light years deep. The galaxies are generated as a Thomas process — Poisson centres with a Gaussian scatter of companions around each — which is not a simulation of structure formation and does not pretend to be; it is the simplest clustered process whose correlation function is known exactly, which is what makes the next figure a comparison rather than a restatement. What it does reproduce is the impression: obvious groups, obvious gaps, and an eye that finds patterns readily, including several that are not there, because an eye finds patterns in a random field too.

The real version of this picture is more striking than the model. Redshift surveys show galaxies strung along filaments, filaments meeting at clusters, and voids tens of megaparsecs across containing almost nothing. Named structures run to hundreds of megaparsecs. The question the cosmological principle poses is not whether the universe looks uniform — it does not — but whether the departures from uniformity get smaller with scale, and how fast.

The statistic that answers it

The two-point correlation function is the fractional excess probability of finding a second galaxy at separation rr from a first, over what a uniform random distribution would give. ξ(r)=0\xi(r) = 0 means no clustering at that separation; ξ=1\xi = 1 means twice as many pairs as random.

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 correlation function measured off the drawing, against the exact prediction for the process that made it. 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 ξ(r)=1/(4πκσ2)exp⁡(−r2/4σ2)\xi(r) = 1/(4\pi\kappa\sigma^2)\exp(-r^2/4\sigma^2), which for this field is 1.37 at zero separation. The agreement is the check that the estimator measures what it claims. The excess falls below one per cent by about 145 Mpc, and that is the useful number: beyond it, knowing where one galaxy is says almost nothing about where the next one is.

In the real universe the same statistic has a different shape and the same character. It is close to a power law, ξ(r)≈(r/r0)−1.8\xi(r) \approx (r/r_0)^{-1.8} with r0≈5 h−1r_0 \approx 5\,h^{-1} Mpc, over three decades of separation — a remarkably simple result for a process as complicated as galaxy formation — and it falls below one per cent somewhere around 50 h−150\,h^{-1} Mpc.

The power law needs one qualification, and it is the same one that attaches to every galaxy statistic. Galaxies are not the matter; they form preferentially at the peaks of the density field, so their correlation function is amplified relative to the matter’s by a bias factor b2b^2, and bb depends on which galaxies were counted. Red, massive galaxies are more strongly clustered than blue star-forming ones at the same redshift, and neither is tracing the underlying field one for one. The correlation length of 5 h−15\,h^{-1} Mpc is a property of a galaxy sample, not of the universe, and comparing two surveys with different selections means comparing two different biases before comparing anything cosmological. It is the same problem a luminosity function has, and it is why weak lensing — which responds to the matter directly and does not care what is shining — has become the preferred way of measuring the clustering amplitude.

A bump at a hundred megaparsecs. The two-point correlation function of galaxies, multiplied by the square of the separation so that the interesting part is not buried under the power law. The smooth dashed curve is the broad-band clustering — the ordinary fact that galaxies are near other galaxies, which carries no cosmological information and is treated as a nuisance term in the real analysis. The bump on top of it is the whole measurement, and its position here is not fitted: it is the comoving sound horizon at the drag epoch, integrated in this file from ∫c_s da/a²H with c_s = c/√(3(1+R)), which comes out at 146.9 Mpc — 99.0 in the h⁻¹ Mpc a survey works in. The excess says that a galaxy is very slightly more likely to have a companion at that separation than at 90 or 115, by about one part in two hundred, and the reason is that a pressure wave in the photon–baryon fluid ran outward from every overdensity for four hundred thousand years and stopped where it was when the photons let go. The points are drawn with the error a survey of a million galaxies achieves; a single pair of galaxies at 100 Mpc means nothing, which is why the measurement waited for the surveys.
Fig. 3 And sitting on top of that power law, at a hundred megaparsecs, is the one feature that is not a smooth decline: the acoustic bump, a one-per-cent excess at the sound horizon. It is drawn here at the position this collection’s own integration of the pre-recombination physics gives, 99.0 h−1h^{-1} Mpc, rather than fitted. Its presence is a reminder that “the clustering dies away smoothly” is only approximately true, and that the exception is the single most useful feature in the whole function.

Counts in spheres, which is the direct question

The correlation function is a statement about pairs. The cosmological principle is a statement about volumes, and the direct test is to count galaxies in spheres and ask how much the counts vary.

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. 4 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 is what a random field gives, 1/Nˉ1/\sqrt{\bar N}; the upper is what a clustered field must give once the circle is much larger than a cluster, (1+μ)/Nˉ\sqrt{(1+\mu)/\bar N}. Both fall as 1/R1/R. Clustering multiplies the scatter by a constant and does not change its power — which is the precise content of the statement that the universe becomes homogeneous on large scales.

That figure is the essay’s central point and it is worth spelling out. Homogeneity is not a property the universe has or lacks. It is a limit: the fractional density fluctuation in a sphere of radius RR falls as RR grows, so for any tolerance there is a radius above which the universe is homogeneous to that tolerance, and below which it is not.

The real numbers put the transition in a definite range. The density fluctuation in spheres of 8 h−18\,h^{-1} Mpc is σ8=0.81\sigma_8 = 0.81 — of order unity, which is why a sphere that size may be a cluster or may be a void. By 100 h−1100\,h^{-1} Mpc it has fallen to a few per cent, and redshift surveys find the counts settling onto the uniform-field scaling somewhere between 70 and 150 h−1150\,h^{-1} Mpc. A range rather than a number, because different surveys and different tolerances give different answers, and pretending otherwise would be the kind of false precision this field is prone to.

The consequence for anything computed in this collection is worth making concrete. The observable universe is 14,000 comoving megaparsecs across and the homogeneity scale is about 150, so a sphere of the observable universe contains some 10610^6 independent homogeneity volumes. That is a large number and it is also a finite one, and it is what limits every cosmological measurement at the largest scales: there is only one sky, and the number of independent samples of any given scale it contains is fixed. The uncertainty on the microwave background’s power spectrum at low multipole is exactly this quantity — cosmic variance — and it cannot be reduced by a better instrument, only by observing from somewhere else.

What is actually measured

A redshift survey delivers two angles and a redshift per galaxy, and turning that into the figures above requires three things that are not in the catalogue.

A cosmology, to convert redshifts into distances. Every analysis states a fiducial model up front. The correlation function is computed in it, and the sensitivity to that choice is a systematic that has to be propagated.

A selection function. A survey is magnitude-limited, so it sees intrinsically faint galaxies only nearby and bright ones everywhere; the number density therefore falls with distance for reasons that have nothing to do with the universe. Correcting for it requires knowing the luminosity function. And a treatment of peculiar velocities. A galaxy’s redshift is not purely cosmological; it contains a Doppler term from the galaxy’s own motion, so the radial coordinate is systematically wrong. Around a cluster the effect is dramatic: infall compresses structures along the line of sight at large radii and the virial motions stretch them into radial spikes at small radii. The distortion is a nuisance for the geometry and a measurement of the growth rate of structure, which is one of the ways the field constrains gravity on large scales.

None of the three is a small correction, and it is worth noting how different that makes this measurement from the ones earlier in this field. A supernova magnitude or an acoustic peak position is a number extracted from a well-controlled measurement; a correlation function is a statistic of a sample whose construction is itself most of the work.

The same generator drawn at two very different depths makes the point the statistic makes, before any statistic is computed.

A slice 300 megaparsecs across. 85 galaxies in a wedge 300 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?
Fig. 5 The same clustering rule in a wedge only three hundred megaparsecs deep. At this depth the structure is the picture: filaments, voids and a strong impression that the distribution has a shape. Nothing in the generating process has changed — the same clustering scale, the same density — and the eye reads it as structured because the box is only a few clustering lengths across.
A slice 2,500 megaparsecs across. 5509 galaxies in a wedge 2500 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?
Fig. 6 The same rule at two and a half gigaparsecs. The filaments are still there and are now too small to see, and the wedge reads as a uniform speckle. Neither picture is more honest than the other; the difference between them is entirely the ratio of the box to the clustering scale, which is the quantity homogeneity is a statement about.

Why the structure is there at all

The agreement between the two is one of the quantitative successes of the model, and it constrains more than the initial amplitude. How fast structure grows depends on how much matter there is to do the collapsing and on how fast the expansion is pulling things apart, so the ratio of the observed clustering today to the observed fluctuations at recombination measures the matter density and the growth history.

There is one place where the two do not agree as well as they might, and it is worth recording rather than skipping. The microwave background, extrapolated forward through ΛCDM, predicts a present-day clustering amplitude a few per cent higher than weak-lensing surveys measure. The discrepancy is two to three sigma — much softer than the disagreement about the expansion rate, and of exactly the same shape: a local measurement against an extrapolation from z=1090z = 1090. It may be a systematic in the lensing calibration, it may be an effect of gas being pushed out of galaxies by their own black holes, and it may be neither. It is called the S8S_8 tension.

What the pictures cannot show

The hero figure’s process is not structure formation. A Thomas process makes blobs; gravity makes filaments and sheets and voids, because collapse happens along one axis first and the third axis last. The real cosmic web has a topology the drawn field does not, and no statistic used in this essay — neither the correlation function nor counts in spheres — can see the difference, because both are insensitive to phase.

That insensitivity is a real limitation, not a drawing artefact. Two fields with identical two-point statistics can look entirely different, and separating them requires higher-order statistics or topological measures. The reason the two-point function dominates the literature is that for Gaussian initial conditions it contains everything, and the initial conditions are Gaussian to the precision available.

And every figure here is a slice. The universe is three-dimensional and the drawings are two-dimensional, which changes the analytic form of everything: the correlation function of a projected catalogue is not the correlation function of the space it was projected from. Real analyses work in three dimensions and pay for it with the selection function and the peculiar velocities.

The principle was an assumption before it was a measurement

Einstein assumed homogeneity in 1917 because the field equations are otherwise unsolvable by hand, and the assumption was frankly acknowledged as a convenience. It survived as one for fifty years.

The first real test was the source counts of radio galaxies in the 1950s and 60s, and they were used to argue against the steady-state theory rather than for homogeneity as such. The first galaxy redshift surveys in the 1980s — the CfA survey in particular — found structure at every scale they could reach, including the “Great Wall” at 200 megaparsecs, and each new survey found something bigger. That pattern, structure on the scale of whatever survey had just been completed, is exactly what a sample too small to reach the homogeneity scale produces, and it was read for a while as evidence that there was no such scale.

What settled it was surveys large enough to contain many independent volumes, from the 2dF and Sloan surveys onwards: a survey whose largest structure is a tenth of its own size is measuring the universe, and one whose largest structure is comparable to its own size is measuring its own boundaries. Modern surveys are in the first regime, and the counts-in-spheres scatter follows the expected 1/R1/R.

The awkward residue is that structures continue to be reported that are larger than the homogeneity scale — quasar groups spanning gigaparsecs, alignments in the microwave background at the largest angles. Whether any of them is a real departure or a selection effect on a low-probability fluctuation is unsettled, and the honest statement is that the principle is verified as a statistical statement at the per-cent level and is not verified as an exact one.

The other half of the principle, and the logic joining them

The cosmological principle asserts two things, and only one of them has been discussed. Homogeneity says the universe is the same at every place; isotropy says it looks the same in every direction from here. They are different claims and they are measured entirely differently.

Isotropy is measured superbly well. The microwave background is the same temperature in every direction to about one part in a hundred thousand, once one thing is removed — a dipole of a few parts in a thousand, which is the solar system’s own motion through the radiation and is a measurement rather than a nuisance. After the dipole, what is left is isotropic at the level the fluctuations themselves sit at.

Homogeneity is measured badly by comparison, at the per-cent level and only in a statistical sense, for the reason this essay has spent its length on: it requires seeing many independent volumes, and there is one sky.

The logical relation between the two is where the interesting move is. Isotropy about a single point does not imply homogeneity — a universe with a spherically symmetric density gradient centred exactly on the observer would look perfectly isotropic and be radically inhomogeneous. What does imply homogeneity is isotropy about every point, and nobody has observed from anywhere else.

The bridge is the Copernican assumption: that this location is not special. Combined with the measured isotropy, it gives homogeneity as a theorem rather than as a measurement.

That is an unusual position for a foundational claim in a quantitative science. The best-measured half of the principle constrains the other half only through an assumption that cannot be tested directly, and the direct test of the second half is the weakest measurement in the field.

There are partial escapes. The scattering of microwave photons off hot gas in distant clusters carries information about how isotropic the radiation looks from that cluster, which is an observation from somewhere else by proxy, and it is consistent. So is the near-uniformity of the radiation’s temperature measured at different redshifts through excited molecular states in absorbing clouds.

The principle is verified directly where it is a statistic, and inferred where it is a statement about places nobody can stand.

Both of the statistics above can be read at a different scale, and the readings do not agree about where the transition sits.

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 60 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 0.075 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 >60 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. 7 The correlation function of the same slice, measured only out to sixty megaparsecs. Over this range the power law is an excellent fit and nothing suggests it ever ends — which is what the measurement looked like for most of the twentieth century, when the surveys were this deep and no deeper. A power law with no break in it is exactly what a fractal universe would produce.
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 25 Mpc, roughly 3 cluster radii, and sit 3.3 times above pure Poisson at the largest circle. That factor is the practical content of clustering: a survey counting 245 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 Counts in spheres for a distribution whose clustering scale is eight megaparsecs rather than twenty-two. The scatter falls to any given tolerance at a proportionally smaller radius, so the homogeneity scale that comes out is not a property of the universe alone: it is the clustering scale multiplied by whatever tolerance the definition demands.

The void that would be convenient

There is one specific inhomogeneity that has been argued about for two decades, and it deserves stating because it sits at the junction of this essay and the field’s largest open disagreement.

If the solar system sat near the centre of a large underdense region — a void several hundred megaparsecs across, with a density a few per cent below the mean — then the local expansion would be slightly faster than the global one, because an underdense region expands faster than its surroundings. A local measurement of the expansion rate would then come out high, and an inference from the microwave background, which averages over the whole sky at great distance, would come out at the true global value.

That is exactly the shape of the observed disagreement, so a local void has been proposed as its resolution repeatedly.

The evidence for such a void is mixed and has not converged. Galaxy counts in some directions and to some depths suggest an underdensity of several per cent out to a couple of hundred megaparsecs; other analyses of other tracers find nothing. The measurements are hard for the reason described above: converting counts into a density needs a selection function, and a systematic error in that function looks exactly like a density gradient.

Two constraints bear on it independently of the counts. A void deep enough to explain the whole discrepancy would distort the microwave background’s dipole and the observed supernova magnitudes in ways that have been searched for and not found. And a void of that size is itself improbable in the standard model — not impossible, but a fluctuation large enough that invoking it trades one problem for another.

A local inhomogeneity is the one explanation of the tension that requires no new physics and does require this place to be unusual, which is precisely the assumption the previous section showed the whole framework rests on.

The generalisation

The move that makes the cosmological principle testable is one worth naming: turn a qualitative claim into a statistic with a scale, and the claim becomes a measurement.

“The universe is homogeneous” is not falsifiable as stated, because any distribution is inhomogeneous somewhere. “The fractional density fluctuation in spheres of radius RR falls as R−1R^{-1} above 100 h−1100\,h^{-1} Mpc” is falsifiable, and it has been tested.

The same conversion appears throughout this collection. “A galaxy has a mass” becomes a mass profile, and the profile is what is measured. “Planets are common” becomes an occurrence rate per star per logarithmic interval of period, and the rate is what is measured. “The main sequence is a track” becomes a place stars sit with a computable dwell time.

In each case the general claim was in circulation for decades and the quantitative version is what allowed it to be wrong. A statement that cannot fail is not a foundation, however often it is used as one, and the cosmological principle spent fifty years as exactly that.

Where the ladder goes next

Every figure in this essay traced structure back to fluctuations at recombination, and every essay in this field so far has treated the surface of last scattering as a given. The last essay asks what that surface is, and why it is at three thousand kelvin rather than at a hundred and fifty thousand.

Later rungs on this anchor: the power spectrum as the Fourier counterpart of the correlation function, and why it is the more natural object; the turnover at the horizon scale at equality; redshift-space distortions as a measurement of the growth rate and a test of gravity; the S8S_8 tension between the clustering measured by weak lensing and the clustering the microwave background predicts; voids as objects in their own right; and the cosmic web’s topology, which the two-point function cannot see.

Where this goes next

A bump at a hundred megaparsecs. The two-point correlation function of galaxies, multiplied by the square of the separation so that the interesting part is not buried under the power law. The smooth dashed curve is the broad-band clustering — the ordinary fact that galaxies are near other galaxies, which carries no cosmological information and is treated as a nuisance term in the real analysis. The bump on top of it is the whole measurement, and its position here is not fitted: it is the comoving sound horizon at the drag epoch, integrated in this file from ∫c_s da/a²H with c_s = c/√(3(1+R)), which comes out at 146.9 Mpc — 99.0 in the h⁻¹ Mpc a survey works in. The excess says that a galaxy is very slightly more likely to have a companion at that separation than at 90 or 115, by about one part in two hundred, and the reason is that a pressure wave in the photon–baryon fluid ran outward from every overdensity for four hundred thousand years and stopped where it was when the photons let go. The points are drawn with the error a survey of a million galaxies achieves; a single pair of galaxies at 100 Mpc means nothing, which is why the measurement waited for the surveys. The same ruler measured twice, ten billion years apart The sound wave that produced the acoustic peaks in the microwave background also left a faint excess in how galaxies are spaced, at a separation of about a hundred megaparsecs. It is the only cosmological distance indicator whose length is set by physics rather than by a chain of calibrations. The luminosity function has a knee, and the knee is the point. A Schechter function with a faint-end slope of -1.25, a characteristic magnitude of -20.9 and a normalisation of 0.0093 per cubic megaparsec per magnitude, plotted logarithmically. Fainter than the knee the curve is a straight line — a power law — and brighter than it the count falls off exponentially, which is why there is no such thing as a galaxy ten times brighter than the brightest. Integrated across the range drawn, galaxies brighter than the knee are 1.3 per cent of the number and 26 per cent of the light: almost every galaxy is a dwarf, and almost all the light is not in one. A count with a knee in it Count galaxies by luminosity and the answer is a power law at the faint end and an exponential cut-off at the bright one. Almost every galaxy is a dwarf; almost none of the light is in one; and the bend between those two statements is where galaxy formation stops being efficient. 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.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 14 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Correlation functionCosmic webCosmological principleCounts in spheresGalaxy biasHomogeneity scaleLarge-scale structurePower spectrumRedshift surveySample variance