The same ruler measured twice, ten billion years apart

Every distance in astronomy up to this point has been measured with a candle. Something of known intrinsic brightness is found, its apparent brightness measured, and the ratio converted into a distance — and the whole difficulty is knowing the intrinsic brightness, which is why the ladder is a chain of calibrations rather than a measurement.

A ruler works differently. Something of known physical length is found, the angle it subtends is measured, and the ratio is a distance. There is exactly one object in cosmology whose length is known from first principles, and this essay is about the second place it shows up.

Why there is a bump #

The mechanism is the same one that makes the acoustic peaks in the microwave background, followed forward in time and looked at from the other side.

Before recombination, a small overdensity in the dark matter launches a spherical pressure wave in the photon–baryon fluid, which runs outward at more than half the speed of light. The dark matter does not go with it — it feels no pressure and stays where it is. So the configuration at any moment is a central dark-matter peak with an expanding shell of baryons and photons around it.

When the photons decouple, the pressure vanishes and the shell stops. What is left is a dark-matter peak at the centre and a baryon overdensity in a shell of radius 147 comoving megaparsecs around it. Gravity then works on both, the two components mix, and by the present day each initial overdensity has become a central concentration of galaxies with a faint excess at that radius.

The excess is small because the baryons are a fifth of the matter. If the universe were all baryons the shell would be as prominent as the centre; as it is, the ratio produces a bump of about one per cent in the correlation function, which is why the measurement needed a survey of a million galaxies and was not made until 2005.

What a ruler buys #

The advantages over a candle are worth being specific about, because they are structural rather than a matter of precision.

The length is computed, not calibrated. It follows from the radiation density, the matter density and the baryon density, all three of which the microwave background measures independently. Nothing about it is fitted to a nearby example.

It does not care about the object. A candle requires that a supernova at $z = 1$ be the same kind of object as one nearby. The acoustic scale is a property of the pattern, not of the galaxies making it, so it survives whatever the galaxies are doing. Galaxies at different redshifts have different masses, colours and clustering strengths, and none of that moves the bump — the bias affects the amplitude of the correlation function and leaves the position alone.

And it works in two directions at once. Measured across the line of sight the bump gives an angle, and therefore $D_M(z)$. Measured along the line of sight it gives a redshift interval, and therefore $H(z)$ directly, since $\Delta z = H(z)\,r_{\rm s}/c$. No other observation in cosmology measures the expansion rate at a redshift rather than a distance integrated over one. Comparing the two — the Alcock–Paczyński test — is a further constraint, because a wrong cosmology distorts a sphere into an ellipsoid and the bump is known to be spherical.

What is actually measured #

A galaxy redshift survey measures two angles and a redshift for each galaxy. Converting that to a three-dimensional position requires a cosmology, which is why every analysis states a fiducial model, computes the correlation function in it, and then reports how much the bump has to be stretched to match — a single number per redshift bin, the dilation parameter. Three effects blur the bump and all three have to be modelled.

Non-linear evolution. Galaxies have moved since the pattern was laid down, by tens of megaparsecs, and that smears a sharp shell into a broadened one. The smearing is largely reversible: the displacement field can be estimated from the observed density field and run backwards, which sharpens the bump and improves the distance precision by a factor of about 1.5. “Reconstruction” is now standard.

Redshift-space distortions. A galaxy’s measured redshift includes its peculiar velocity, so the radial coordinate is systematically wrong — and wrong in a way that depends on the local density, because infall towards an overdensity compresses structures along the line of sight. The effect is large and is itself a measurement of the growth rate of structure, but it has to be separated from the geometry.

Galaxy bias. Galaxies do not trace the matter one for one; they form preferentially in dense regions, so their correlation function is amplified by a factor $b^2$ that depends on which galaxies were selected. Luminous red galaxies are strongly biased, $b \approx 2$; emission-line galaxies less so. On the scales that matter the amplification is a constant, which is why the bump’s position survives it — and why a survey can choose its tracer for convenience rather than for fidelity, which no candle-based method can do.

The insensitivity to all three is the property worth emphasising, and it can be stated as a single sentence: every one of these effects moves the amplitude, the width or the shape of the correlation function, and none of them moves the position of a feature at a hundred megaparsecs. A systematic that changes a distance by one per cent has to be a systematic that translates a bump sideways, and there are very few candidates.

The current state is that the acoustic scale has been measured at redshifts from 0.15 to 2.4 — the last using the Lyman-α forest in quasar spectra rather than galaxies — with the distance to each known to about one per cent.

The inverse ladder #

Putting the ruler together with the light-element abundances produces a determination of the expansion rate with no local distance measurement in it at all, and it is worth spelling out because it is the cleanest illustration of what an absolute ruler is for.

Deuterium gives the baryon density. The microwave background’s damping tail and third peak give the matter and radiation densities — or, in the strictest version of the argument, the matter density comes from the growth of structure and cluster counts instead, so that no part of the chain touches a sky map. Those three fix the sound horizon at the drag epoch — 147 Mpc — from pre-recombination physics. The galaxy surveys then measure the angle that length subtends at a range of redshifts, which gives $D_M(z)$ in megaparsecs rather than in units of anything. Extrapolating to $z = 0$ gives $H_0$.

That chain — “inverse distance ladder”, because it runs downward from a calibration at high redshift instead of upward from one nearby — returns $67.4$, and it uses no Cepheid, no supernova calibration and no microwave-background map of the sky. It is an entirely independent route to the early-universe value, and its existence is why the disagreement with the local determination cannot be attributed to any single experiment.

Two measurements at right angles #

The ruler is a length in space, and space has directions. Measuring the same feature across the line of sight and along it gives two different quantities, and separating them is where most of the method’s power lies.

Across the line of sight, a separation is an angle. Converting the observed angular scale into the known length gives the comoving angular diameter distance, $D_M(z)/r_s$.

Along the line of sight, a separation is a redshift difference. Converting that into the same known length gives the expansion rate directly: $c/(H(z)\,r_s)$.

So one survey at one redshift yields two independent numbers, and the second is not an integral over the expansion history but the rate at that moment — which is why the acoustic scale constrains a time-varying dark energy far better than a distance measurement alone.

There is also a consistency test hidden in the pair, and it is a strong one. The acoustic feature is spherical: the sound waves propagated equally in all directions, so the excess of galaxy pairs at the acoustic separation must be the same across and along. If the assumed cosmology is wrong, the conversion from angles and redshifts into distances is wrong by different factors in the two directions, and the feature comes out elliptical.

Requiring it to be round is the Alcock–Paczyński test, and it constrains the product $D_M(z)H(z)$ with no reference to the ruler’s length at all. A measurement that depends on the feature being isotropic rather than on its size is one the sound horizon’s calibration cannot spoil, which is a useful thing to have in a subject where that calibration is exactly what is disputed.

Putting the galaxies back #

The acoustic feature in the galaxy distribution is broader and shallower than the one in the microwave background, and the reason is that thirteen billion years of gravity have moved the galaxies.

Matter falls towards overdensities, so a pair of regions separated by the acoustic scale drifts, typically by five to ten megaparsecs — which is several per cent of the scale itself. Averaged over the whole survey, that smears the sharp shell into a broad bump, and a broad bump is a less precise ruler.

The remedy is to undo it, and the fact that it can be undone is not obvious. The displacement of each galaxy is, to first order, computable from the density field the galaxies themselves trace: measure the field, solve for the displacement that would have produced it, and move each galaxy back along that vector. The result is a reconstructed catalogue in which the acoustic feature is substantially sharper.

The gain is large — the distance precision improves by up to about forty per cent, which is equivalent to doubling the survey — and it costs nothing but computation on data already taken. The technique works because the motion is dominated by gravity acting linearly on a field the survey has measured, so the smearing is not noise but a known transformation of the signal, and known transformations can be inverted.

It is now standard, applied to every large survey’s acoustic measurement, and it is the reason the quoted precisions have improved faster than the survey volumes.

Both of these are reasons the acoustic scale has overtaken every other route to the expansion history: it yields two numbers per redshift bin rather than one, it carries its own internal test of the geometry, and its principal degradation can be reversed after the fact.

The reconstruction step also has a limit worth naming: it can only undo the motion it can infer, and what it infers comes from the galaxies themselves, which are a biased and sparse tracer of the field. In a survey with few galaxies per unit volume the reconstructed displacement is noisy and the gain is smaller, which is one of the reasons survey design trades area against density rather than simply maximising volume.

What the pictures cannot show #

The hero figure’s curve is a representation, not a derivation. The broad-band shape is the nuisance model that BAO analyses actually fit — a smooth function with no cosmological content — and its amplitude and the bump’s width and height are taken from the measurement. What is computed is the bump’s position, which is the only part the analysis extracts. Reading the drawn amplitude as a prediction would be reading a fit.

The error bars drawn are the error on a survey of a million galaxies and the figure cannot show why. The uncertainty on a correlation function at 100 Mpc is set by how many independent volumes of that size the survey contains, which is a property of the survey’s geometry rather than of its depth. Doubling the number of galaxies in the same volume helps very little; doubling the volume helps a great deal.

And nothing here shows the radial measurement, which is half the information. A correlation function plotted against separation has already averaged over the angle between the pair and the line of sight, and the anisotropy that averaging discards is where $H(z)$ comes from.

How it arrived #

The prediction is old — Peebles and Yu worked out the acoustic oscillations in 1970, and that the same feature must appear in the galaxy distribution was understood by the early 1980s. The difficulty was always the amplitude. A one per cent feature in a correlation function requires enough pairs at 100 Mpc to beat the shot noise, and that means a survey volume of order a cubic gigaparsec with tens of thousands of galaxies in it.

Two surveys got there simultaneously in 2005: the Sloan Digital Sky Survey’s luminous red galaxy sample, with 46,748 galaxies over 3,816 square degrees, and the 2dF Galaxy Redshift Survey. Both found the bump; neither had the precision to do cosmology with it alone.

What made it a precision tool was reconstruction, from 2007 onwards — the realisation that the smearing caused by galaxies having moved could be largely undone using the galaxies’ own positions to estimate the displacement field. That single idea recovered most of the precision that non-linear evolution had destroyed, and it is why a modern BAO distance is good to one per cent rather than two.

It is worth noticing what kind of advance that was, because it is unusual. It required no new telescope, no new detector and no new observation: it was a reanalysis of data already taken, using a physical insight about what the data had been doing since the pattern was imprinted. The information had been there and had been scrambled by a process that was itself measurable from the same catalogue. That is a rarer thing in observational astronomy than a better instrument, and the reason it was possible here is the same reason the whole method works — the quantity being measured is a position, and a position can be corrected for a displacement in a way that a brightness cannot be corrected for an unknown luminosity.

The generalisation #

The distinction this essay turns on is one this collection has met before and never this cleanly: a candle measures a distance by an intensity and needs an absolute calibration; a ruler measures it by an angle and needs only a length. Both are ratios, but the length can be a consequence of physics in a way that a luminosity cannot.

The distance ladder is entirely candles, and its whole error budget is calibration. A parallax is the one rung that is neither — it is pure geometry, with the Earth’s orbit as the known baseline, and it is the reason the ladder has a bottom at all. The acoustic scale is a parallax with the baseline supplied by the early universe, and the reason it took a million galaxies rather than one star is that the baseline is not attached to anything visible.

The same idea appears wherever a known length is available. An eclipse’s umbra ends where similar triangles say it does, which turns a shadow into a distance if either body’s size is known. A galaxy’s angular size against its physical size would be a ruler too, if galaxies had a standard size — and the fact that they do not is exactly why the acoustic scale is valuable.

Where the ladder goes next #

The correlation function used here as a tool has a much older question attached to it: how far out does the clustering go, and is there a scale above which the universe stops being lumpy. That is two essays away.

Later rungs on this anchor: the radial measurement and $H(z)$ directly; the Alcock–Paczyński test as a geometric constraint independent of the ruler’s length; reconstruction, and how much of the non-linear smearing it can undo; the Lyman-α forest measurement at $z = 2.4$, where the tracer is absorption rather than galaxies; and the prospect of measuring the acoustic scale in the twenty-one centimetre line at redshifts no galaxy survey will reach.