Astronomy & Celestial Mechanics
the sky, computed
Cosmology

A ruler measured along and across

The sound horizon is a sphere, and a sphere in a redshift survey is measured twice over — across the line of sight it gives an angle, along it a redshift interval. Two different functions of the cosmology out of one feature, and their ratio is a measurement with no ruler in it at all.

Assumes Baryon acoustic oscillations, Large-scale structure and Expansion.

17 min read 8 figures Everything is a conicScales that defeat intuitionMeasuring the unreachable

The rung below this one established the sound horizon as a standard ruler: a fixed comoving length, imprinted on the matter distribution at recombination, visible as a one-in-two-hundred excess in the galaxy correlation function at about 150 megaparsecs, and calibrated from the microwave background.

A ruler measures a distance by subtending an angle. But this particular ruler is not a rod — it is a shell, a preferred separation in every direction at once, and a shell in a redshift survey is measured twice over.

Across the line of sight, the separation between two galaxies is an angle times a distance, so the feature’s angular size gives DM(z)/rdD_M(z)/r_d. Along the line of sight, the separation is a redshift interval, and a redshift interval is Δz=H(z) rd/c\Delta z = H(z)\,r_d/c — so the same feature gives the expansion rate at that redshift rather than a distance integrated over one.

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. 1 The two measurements, and the test that needs neither. Left: the acoustic scale in the plane of transverse against radial separation. In the cosmology that holds, it is a sphere and its locus is a quarter circle — nothing about the early universe distinguishes the radial direction from the transverse one. Assume distances ten per cent too large and rates six per cent too small and it comes back as an ellipse; the ellipticity measures DMH/cD_M H/c, in which the sound horizon has cancelled. The third curve is the difficulty: peculiar velocities squash the same correlation function along the same axis.

Two numbers from one bump

The distinction is worth stating precisely because it is the whole content of this rung.

A transverse separation is inferred from an angular separation on the sky. Converting the two requires the comoving angular-diameter distance DM(z)D_M(z), which is an integral of 1/H1/H from zero to zz. So the transverse acoustic scale measures DM/rdD_M/r_d, a quantity that has the whole expansion history between here and there inside it.

A radial separation is inferred from a difference in redshift. Converting the two requires only H(z)H(z) — the expansion rate at that redshift, not integrated. So the radial acoustic scale measures c/(H(z) rd)c/(H(z)\,r_d).

No other observation in cosmology gives an expansion rate at a redshift. Every distance-based measurement — supernovae, lensing, the microwave background — constrains an integral, and integrals are insensitive to what happens over any short interval. A direct H(z)H(z) at half a dozen redshifts is a differential measurement of the same history, and it constrains a time-varying dark-energy equation of state in a way distances do not.

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. 2 The feature itself, in one dimension. The correlation function of galaxies has a broad smooth part carrying no cosmological information and a bump at the acoustic scale, of amplitude about half a per cent above the smooth part. The bump’s position is the measurement and everything else is a nuisance, which is why the analysis fits a broad-band polynomial alongside it and reports only the position.

The test that needs no ruler

The ratio of the two is the useful part, and it is a different kind of measurement.

Write FAP=DM(z) H(z)/cF_{\rm AP} = D_M(z)\,H(z)/c. It is dimensionless. Both observables carry a factor of rdr_d and the ratio does not, so the ratio is a measurement of the geometry that does not require the ruler’s length to be known.

That is the Alcock–Paczyński test, proposed in 1979 for a completely different purpose. Its logic is not about the acoustic scale at all: it is the observation that any population of objects known to be statistically isotropic — spherical on average, with no preferred direction — must appear isotropic when the coordinates are computed correctly, and appears squashed or stretched when they are not.

The acoustic feature is the ideal application because its isotropy is not an assumption but a consequence: it was imprinted by a sound wave propagating through a medium with no preferred direction, at a time when nothing in the universe had a preferred direction, and it has expanded with the background since.

A ruler of unknown length can still measure a shape.

One length, two distances, two angles. The same comoving ruler — the sound horizon, 147 Mpc at the drag epoch and 144 Mpc at last scattering — seen from here at the two distances it has been measured across. At z = 0.57 it sits 2183 Mpc away in comoving distance and subtends 3.86 degrees; in the microwave background it sits 13866 Mpc away and subtends 0.60, which is the first acoustic peak at ℓ = 302. The distances along the page are to scale with each other. The two angles are both magnified by 8, because drawn true the near one is the width of a thumbnail at arm's length and the far one is a fifth of that; magnifying both by one number leaves their ratio, 6.47, exactly as computed. That ratio is the measurement. A ruler of known length seen at two distances gives the ratio of the distances with no standard candle, no ladder, and no calibration carried up from a parallax — which is what makes the acoustic scale a different kind of distance from every other one in this collection.
Fig. 3 What is being calibrated when the length is used. The comoving sound horizon at the drag epoch is an integral of the sound speed over the age of the universe up to recombination, and the sound speed is fixed by the ratio of baryons to photons. So rdr_d is computed from parameters the microwave background measures directly — which is why the BAO scale is a standard ruler rather than a measured one, and why every BAO distance carries the microwave background’s calibration inside it.
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. 4 The same bump in a much nearer sample, at z=0.15z = 0.15 rather than 0.57. The feature is at the same comoving separation — it is a ruler and a ruler does not change length — and what changes is the volume available to measure it in and therefore the scatter on the points. A low-redshift survey measures the same number less well, not a different number, and the whole design of a survey is the trade between the volume at high redshift and the density of tracers at low.

What contaminates it

A geometric distortion squashes the correlation function along the line of sight. So does a dynamical one, and separating them is the whole difficulty of the measurement.

Every galaxy’s radial coordinate comes from a redshift, and a redshift contains the galaxy’s own motion as well as the expansion. On large scales those motions are coherent — everything is falling towards the overdensities — so structures are compressed along the line of sight by a factor that depends on the growth rate of structure divided by the galaxy bias.

The compression is not small. For a typical survey it is of order twenty per cent in the quadrupole, which is far larger than the few per cent of Alcock–Paczyński distortion being looked for. Three things make the separation possible.

They have different scale dependences. The dynamical squashing acts on the broad-band shape of the correlation function at all separations; the geometric one acts on the position of the feature. Fitting the bump’s position while marginalising over a flexible broad-band model removes most of the sensitivity to the first.

They have different angular signatures at higher order. Both produce a quadrupole, but the velocity distortion also produces a hexadecapole with a fixed relation to it in linear theory, and that relation is a handle.

And the reconstruction step helps twice. Modern analyses estimate the displacement field from the observed density, move each galaxy back along it, and re-measure the correlation function. That sharpens the acoustic peak — undoing the smearing from bulk flows, which had broadened it by several per cent — and it removes much of the linear redshift-space distortion at the same time.

The ruler being laid down. The comoving sound horizon against the scale factor, integrated from the big bang forwards. A pressure wave in the photon–baryon fluid travels at c/√(3(1+R)), which begins at 0.577c while the photons dominate and falls to 0.451c by the time the baryons have caught up; the curve is the distance it has covered, in comoving megaparsecs. It stops when the photons stop pushing, which is the drag epoch at z = 1059.94, and the value it has reached there is 146.9 Mpc. That is the length of the ruler, and it is set entirely by physics before recombination — by the radiation density, the matter density and the baryon density, all three of which the same sky map measures. Nothing about the later universe enters it. The dashed mark at z = 1089.92 is last scattering, forty thousand years earlier in redshift and 2.7 Mpc shorter, which is why the microwave background's ruler and the galaxy survey's ruler are two slightly different numbers rather than one.
Fig. 5 The growth of structure that supplies the velocities. Perturbations grow while matter dominates and stop growing once the expansion accelerates, so the rate at which they are growing now is a function of the matter density — and the peculiar velocities are proportional to that rate. The nuisance and the signal come from the same physics, which is why the same data set that measures the geometry also measures the growth, and why the two are fitted together.

What was actually measured

The observation is a catalogue of positions and redshifts — a million or two galaxies over several thousand square degrees, from a spectroscopic survey — and the statistic is the two-point correlation function or its Fourier counterpart, measured in bins of separation and of angle to the line of sight.

Because the coordinates cannot be computed without a cosmology, the analysis assumes one: a fiducial cosmology, used only to turn angles and redshifts into separations. The fitted parameters are then the ratios of the true distances to the fiducial ones,

α⊥=DM/rd(DM/rd)fid,α∥=(Hrd)fidHrd,\alpha_\perp = \frac{D_M/r_d}{\left(D_M/r_d\right)_{\rm fid}}, \qquad \alpha_\parallel = \frac{\left(H r_d\right)_{\rm fid}}{H r_d},

and the fit returns those two. The Alcock–Paczyński combination is their ratio, and it is the better-measured direction in the likelihood for surveys with modest volume.

For the eBOSS and DESI samples the individual precisions are one to two per cent on DM/rdD_M/r_d and two to three per cent on HrdH r_d, at each of several effective redshifts from 0.3 to 2.3. The radial measurement is always the noisier one, because a survey has fewer independent modes along a narrow redshift shell than across the sky.

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. 6 And the other end: a sample at z=1.5z = 1.5, where quasars and Lyman-α forests do the measuring rather than galaxies. The bump is again at the same separation, and the point of drawing it three times is that this is a claim rather than a construction — the same comoving length recovered from three epochs is what makes the ruler standard. Anything that moved the feature between these three drawings would be a systematic, and the flatness of the comparison is the measurement.

Why gravity does not move the feature

The whole method rests on a claim that deserves to be examined rather than asserted: that the acoustic scale is a fixed comoving length, unaffected by thirteen billion years of gravitational evolution.

It is not obviously true. Structure grows, matter flows towards overdensities, and everything in the density field moves. Why does a feature at 150 megaparsecs stay at 150 megaparsecs while everything around it changes?

The answer is that the displacements are small compared with the scale. Galaxies move by of order ten megaparsecs over the age of the universe, which is a few per cent of the acoustic scale — so the feature is blurred by that amount rather than shifted by it. A blur is symmetric to first order and does not move a peak’s centre.

To second order it does, and by a computable amount. The displacement field is itself correlated with the density, so a pair separated by the acoustic scale is slightly more likely to be pulled together than apart, and the peak shifts inwards by something like a third of a per cent at redshift zero. That is comparable with the statistical error of a modern survey, so it is not ignorable — it is calculated in perturbation theory, checked against simulations, and applied as a correction.

The reconstruction step reduces both effects at once. Estimating the displacement field from the observed density and moving the galaxies back undoes most of the blurring, sharpening the peak, and undoes most of the systematic shift with it. What is left after reconstruction is a shift of well under a tenth of a per cent, which is where the method’s robustness claim actually sits.

A standard ruler that is only approximately standard is still a ruler if the departure is calculable, and the reason this one is trusted at the per-cent level is that the departure has been computed three ways — analytically, in simulations, and by the improvement reconstruction produces — and the three agree.

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. 7 The same measurement over a wider range of separation — forty to two hundred and fifty megaparsecs rather than twenty-five to a hundred and seventy-five. The bump sits in the same place and the broad-band power law now dominates more of the plot, which is the practical difficulty: the feature is a per cent-level wiggle on a falling curve, and everything about extracting it is about modelling the curve well enough to see the wiggle. The signal is a bump on a background that is itself the thing being fitted.

Where the model stops

The fiducial cosmology leaves a residue. The analysis is designed so that the fitted α\alpha parameters absorb the difference between the fiducial and the true cosmology, and to first order they do. If the fiducial is badly wrong the reconstruction step and the broad-band model are computed in the wrong coordinates, and the recovered position acquires a small bias. Tests on simulations bound this at a fraction of a per cent, which is currently below the statistical error and will not always be.

The bias of the tracers is not known. Galaxies are not matter; they are peaks in the matter field, and the relation between the two is a function that has to be marginalised over. It does not shift the acoustic position — that is the method’s central robustness claim, and it survives every simulation test — but it enters the velocity distortion and therefore the separation of the two effects.

The peak is broadened by non-linear evolution. Bulk flows displace galaxies by several megaparsecs, smearing a feature that started as a sharp shell. Reconstruction recovers most of it and not all, and how much is recovered depends on the density of the tracer sample.

And the picture cannot show the sphere. No shell is visible around any galaxy. What is measured is a half-per-cent excess in a statistic computed over millions of pairs, and the “sphere” is a property of the ensemble rather than of any object. The Alcock–Paczyński test is applied to something that is spherical only on average and only in a statistical sense, which is why it needs a survey volume of cubic gigaparsecs rather than a good image.

Why the radial measurement is the harder one

The asymmetry between the two observables is not an accident of any one survey and it is worth understanding, because it decides what the next generation of instruments is built for.

A survey covering a solid angle Ω\Omega over a redshift shell of thickness Δz\Delta z contains a certain number of independent Fourier modes. Modes across the line of sight are limited by the survey’s angular extent, which for a wide survey is enormous; modes along it are limited by the shell’s depth, which is not. At a given redshift the radial extent of a useful shell is a few hundred megaparsecs and the transverse extent is several thousand, so there are simply fewer independent radial modes to measure the feature with.

There is a second, subtler cost. The radial coordinate is contaminated by peculiar velocities and the transverse one is not, so every systematic in the velocity model degrades the radial measurement alone.

The consequence is that H(z)rdH(z) r_d comes out with roughly twice the error of DM/rdD_M/r_d from the same data. That matters more than the factor suggests: the radial measurement is the one that carries information a distance cannot supply, so the more valuable observable is the noisier one, and increasing survey depth improves it faster than increasing survey area does.

The currency is volume

Everything about how these surveys are designed follows from one counting argument, and it explains why the field’s instruments look the way they do.

The precision of a clustering measurement is set by the number of independent Fourier modes the survey samples at the relevant scale. A mode is a wave of a given wavelength and direction, and the number available at wavelength λ\lambda in a volume VV is roughly V/λ3V/\lambda^3 — so measuring a feature at 150 megaparsecs needs a volume of many cubic gigaparsecs before the counting statistics become tolerable.

That is a hard floor and it has nothing to do with the telescope. A perfect instrument observing every galaxy in a small volume still measures the acoustic scale badly, because there are only so many independent 150-megaparsec waves in a small box. The limit is called cosmic variance and it is the reason survey design is a question about volume rather than about depth or aperture.

Two consequences follow. Increasing the number of galaxies in a fixed volume helps only until the measurement stops being shot-noise limited and starts being sample-variance limited — beyond that, more spectra of the same volume buy nothing. And the optimal tracer is not the most numerous one but the one that samples the largest volume per unit of observing time, which for the highest redshifts means quasars rather than galaxies despite their being far rarer.

The same argument sets the ceiling. The observable universe contains a finite number of 150-megaparsec modes, so there is a best possible measurement of the acoustic scale, and the current surveys are within an order of magnitude of it at low redshift. Improving further means going to higher redshift, where the volume per unit redshift is larger and the structure is more linear — which is why the next generation of instruments is designed around tracers at redshift two and beyond.

There is a second ceiling worth naming and it is the one the Alcock–Paczyński ratio runs into rather than the distances. The ratio is measured from the anisotropy of the clustering, and the anisotropy has to be separated from the velocity distortion — so its precision is limited by how well the velocity model is known rather than by how many modes are available. Adding volume improves the distances indefinitely and improves the ratio only until the modelling floor is reached.

That reverses the usual ordering. For most of the field’s history the statistics were the limitation and the modelling was comfortably ahead; for the Alcock–Paczyński measurement specifically, the modelling reached its limit first, which is why the effort has moved from building larger surveys to characterising the small-scale velocity field the analysis has to marginalise over.

The same reversal has happened once before in this subject, on the microwave background, where the statistical limit was reached for temperature and the remaining information moved to polarisation. Which quantity is limiting is worth checking before an instrument is designed around improving the wrong one.

A measurement whose precision is set by how much universe is available is unusual, and the awareness that the limit is in sight rather than hypothetical has shaped the field’s planning for two decades.

The ruler being laid down. The comoving sound horizon against the scale factor, integrated from the big bang forwards. A pressure wave in the photon–baryon fluid travels at c/√(3(1+R)), which begins at 0.577c while the photons dominate and falls to 0.444c by the time the baryons have caught up; the curve is the distance it has covered, in comoving megaparsecs. It stops when the photons stop pushing, which is the drag epoch at z = 1059.94, and the value it has reached there is 143.0 Mpc. That is the length of the ruler, and it is set entirely by physics before recombination — by the radiation density, the matter density and the baryon density, all three of which the same sky map measures. Nothing about the later universe enters it. The dashed mark at z = 1089.92 is last scattering, forty thousand years earlier in redshift and 2.5 Mpc shorter, which is why the microwave background's ruler and the galaxy survey's ruler are two slightly different numbers rather than one.
Fig. 8 Where the ruler’s length comes from, at H0=70H_0 = 70 rather than 67.36. The sound horizon is fixed entirely by physics before recombination — the radiation density, the matter density and the baryon density — so changing the expansion rate today changes the distance the ruler is seen at and not the ruler. That separation is the whole reason this method constrains H0H_0: the length is calibrated by the microwave background and the angle is measured by the survey, and the ratio is the distance.

The generalisation

Two ideas here are worth more than their application.

A shape is easier than a size. Measuring the ellipticity of something known to be round requires no calibration; measuring its diameter requires a standard. That is why the Alcock–Paczyński ratio is systematically cleaner than either of the distances it is built from, and it is the same reason a flux ratio is easier than a flux, a colour is easier than a magnitude, and a period is easier than a brightness. Every ratio removes whatever multiplies both halves.

And a coordinate that is not a length behaves differently from one that is. A redshift survey has two sky coordinates that are angles and one that is a velocity, and treating all three as positions is the source of both the opportunity and the contamination in this essay. The same structure appears wherever a map is built from a mixture of observables — a stellar position with a parallax and a proper motion, a spacecraft’s range and its plane-of-sky angle — and in each case the error ellipsoid is elongated along whichever axis was inferred rather than measured.

Where this ladder goes next

Later rungs on this anchor: reconstruction in detail, and why moving galaxies backwards along an estimated displacement field sharpens a feature that was blurred by the same field; the BAO measured in the Lyman-α forest at redshift 2.3, which is the only tracer available beyond the reach of galaxy surveys and gives the radial measurement more cleanly than the transverse one; the void–galaxy cross-correlation, in which the Alcock–Paczyński test is applied to voids and the distortion is larger; the consistency between the acoustic scale measured in the microwave background and in galaxies, which is a test of the expansion history over ten billion years; and the sound horizon as a free parameter, which is what a measurement of H0H_0 that does not assume the early universe requires.

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. 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. Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround. One number that is an age, a size and a density The slope of the velocity–distance relation has units of inverse time, so it can be read as an age, multiplied by c to give a length, or squared to give a density. All three readings are natural, all three are quoted, and not one of them is the quantity it appears to be.

About the same objects

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

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Alcock paczynskiAngular-diameter distanceAnisotropic clusteringBaryon acoustic oscillationsCorrelation functionDegeneracyExpansion historyFiducial cosmologyHubble parameterRedshift space distortionSound horizonStandard ruler