A ruler read in absorption

A ruler measured along and across ends on an asymmetry: a galaxy survey measures the transverse acoustic scale about twice as well as the radial one, because a wide sky holds many more independent modes than a redshift shell does, and because the radial coordinate is contaminated by peculiar velocities and the transverse one is not.

That asymmetry is unfortunate, because the radial measurement is the more valuable. It gives the expansion rate at a redshift rather than an integral of it, and no other observation in cosmology does.

Beyond redshift one the situation changes completely, and it changes in the useful direction.

A map made of pencils #

The tracer is not an object. It is the neutral hydrogen between here and a distant quasar, which absorbs at Lyman-α in the rest frame of whatever gas it happens to be in — the same transition whose complete absorption at higher redshift marks the end of reionisation — so a quasar’s spectrum, blueward of its own Lyman-α emission, is a record of the density along the whole path.

That gives a geometry nothing else in cosmology has. Each sightline is a one-dimensional sample of the three-dimensional density field, finely resolved along its own length and infinitely thin across. A survey of such sightlines samples the field on a set of parallel lines, and the map it builds is made by correlating one line with another.

The consequences follow immediately and all of them are about anisotropy.

The radial sampling is set by the spectrograph. A pixel of seventy kilometres a second is about three tenths of a megaparsec at redshift 2.3, so the radial direction is sampled to wavenumbers of ten inverse megaparsecs — far finer than any structure the acoustic measurement cares about, and fine enough to be limited by the gas’s own thermal broadening rather than by the instrument.

The transverse sampling is set by the sky. Quasars bright enough to yield a usable spectrum number a few tens per square degree, so neighbouring sightlines are separated by fifteen to twenty megaparsecs and the transverse direction is sampled to wavenumbers of about a fifth of an inverse megaparsec.

And the acoustic scale sits between the two. A hundred and fifty megaparsecs corresponds to a wavenumber of about 0.04, which is comfortably within reach in both directions — so the feature is measurable in both, and it is measurable far better along the line of sight — after the displacement-field correction has been applied to what little of it the sparse sampling permits.

Why that is worth having #

An acoustic measurement gives two things: the comoving angular-diameter distance divided by the sound horizon, from the transverse scale, and the expansion rate times the sound horizon, from the radial one.

The first is an integral of the expansion rate from here to the sample’s redshift. The second is the rate itself, at that redshift.

An integral is insensitive to what happens over any short interval, which is why distance measurements constrain the expansion history so much more weakly than a naive count of their error bars suggests. A differential measurement — the rate at several redshifts — constrains it directly, and the forest supplies the only ones beyond redshift two.

That redshift range is where the interesting question is. Dark energy is negligible there: at redshift 2.3 the universe is matter-dominated, and the expansion rate is predicted from the matter density with no free function in it. So a measurement of the rate at that redshift is a test of the model in a regime where the model has nothing to hide behind, and a disagreement would be a disagreement about matter rather than about dark energy.

What a tracer made of absorption costs #

Everything above is the geometry, and the geometry is the good part. The systematics of the forest are of a kind the galaxy measurements do not have at all.

The continuum has to be fitted. What is observed is the quasar’s flux, and what is wanted is the fraction transmitted — which requires knowing what the quasar’s own spectrum would have been without the absorption. That continuum is estimated from the shape of the spectrum redward of Lyman-α and from a mean quasar spectrum, and the estimate is wrong in a way that varies smoothly along each spectrum. A smooth error along a sightline is an error at exactly the large scales the acoustic measurement lives at.

The standard treatment turns that into a statement about the estimator: the continuum fit is performed on the data, and the same fit is performed on simulated spectra with a known continuum, and the resulting distortion of the correlation function is measured and divided out. The distortion is large — it removes most of the power at the largest scales — and it is a linear operation on the field, so it can be inverted.

The relation between flux and density is not linear. Transmission is the exponential of minus an optical depth, and the optical depth is a steep power of the density, so the tracer saturates where the density is high. A dense region absorbs everything and cannot absorb more, which flattens the response exactly where the signal is largest.

And the gas is not the matter. Its temperature, its ionisation state and its pressure smoothing all enter, and all of them depend on the history of reionisation and on the ultraviolet background — none of which is measured independently. The bias of the forest as a tracer is fitted, and it depends on scale.

The one measurement that disagreed #

The forest’s radial measurement produced, for several years, the only significant tension in the acoustic data set, and the way it resolved is worth recording because it is a good example of how such things usually go.

An early analysis of the forest auto-correlation gave an expansion rate at redshift 2.3 that sat about two and a half standard deviations from what the standard cosmology predicted, in the direction of a faster expansion. Combined with the cross-correlation the discrepancy exceeded two sigma, and for a measurement in a regime where dark energy is negligible that was interesting: it could not be explained by any ordinary modification of the late-time expansion.

Three things happened over the following years. The sample grew by a factor of several. The treatment of the continuum distortion was improved, and it moved the answer. And the modelling of high-column-density systems — the rare, saturated absorbers that punch broad holes in the forest — was revised, and that moved it too.

The current measurements agree with the standard cosmology. The tension is gone and it was not resolved by a single identified error; it faded as three systematics were reduced, each of which moved the answer by about a standard deviation.

That is the ordinary fate of a two-sigma result in a measurement with several systematics of comparable size, and it is worth stating because the same measurement’s precision is now better than the old tension was large. A forest measurement today disagreeing with the standard cosmology at the same absolute size would be a five-sigma result, and there is no obvious reason it would be any more real.

Two ways to correlate a forest #

There are two statistics in use and the difference between them is not cosmetic.

The auto-correlation of the forest with itself uses every pair of pixels in every pair of sightlines. It has the most pairs, it is dominated by the radial direction for the reason above, and it carries all of the continuum systematics twice.

The cross-correlation of the forest with the quasars themselves uses every pixel against every quasar position. There are fewer pairs, but the quasars are a point tracer with no continuum problem, so the systematics are different — and being different is the point. The two measurements are made from the same data and combined, and their agreement is the internal check.

That combination is where the strongest forest constraints come from, and the cross-correlation contributes more than its number of pairs suggests because it breaks a degeneracy: the auto-correlation’s amplitude depends on the forest’s bias squared and the cross-correlation’s on the forest’s bias times the quasars’, so the two together separate the bias from the signal.

What was actually measured #

A few hundred thousand quasar spectra, each covering a range of redshift along its own sightline, from a survey that also had to find the quasars in the first place.

Three steps stand between a spectrum and a distance.

Selection. Quasars are identified photometrically and confirmed spectroscopically, and the selection function depends on colour, which depends on the amount of absorption — so the survey’s completeness is correlated with the quantity being measured. That correlation is small and it is not zero, and it is handled by simulating the selection.

Continuum fitting, as above, with its distortion measured and inverted.

And the fit itself, which is a model of the correlation function with the acoustic peak’s position as a free parameter, marginalising over the forest’s bias, the redshift-space distortion parameter, a broad-band shape, the effect of high-column-density absorbers, and the correlated absorption from other transitions of other species that happen to fall in the same wavelength range.

The published measurements give the radial and transverse scales at redshift 2.3 to about two and three per cent respectively — and the ordering is the reverse of every galaxy sample in the same analysis, which is the clearest statement of what this tracer is for.

What the sparse axis costs, and what it does not #

The transverse sampling is coarse by a factor of sixty, and it is worth being precise about what that does and does not prevent.

It does not prevent the transverse measurement. The acoustic feature is at 150 megaparsecs, which is ten times the sightline separation, so pairs of sightlines at the right separation exist in quantity — there are simply far fewer independent transverse modes than radial ones at any smaller scale, and the small scales are not what the measurement uses.

What it costs is precision, in the ratio of the number of modes available in each direction. The published errors reflect it: the radial scale at redshift 2.3 comes out better determined than the transverse, which is the reverse of every galaxy sample.

It also costs something subtler. Reconstruction needs the density field on the smoothing scale — ten to fifteen megaparsecs — in all three dimensions. The forest has it along a sightline and does not have it across, so the displacement field cannot be estimated the way it is for a galaxy survey, and the acoustic feature in the forest is measured essentially unreconstructed.

A measurement can be limited in one direction and unaffected in the quantity it was built for, and distinguishing the two is the difference between a real limitation and a number that sounds bad. Sixty is the ratio of the sampling; the ratio of the error bars on the two acoustic scales is under two.

Where the model stops #

The forest figures here are a lognormal approximation. The relation between optical depth and density used to draw them is a power law, which is the standard analytic description and is a fit to hydrodynamic simulations rather than a derivation. Real analyses use the simulations.

Nothing here has a metal line in it. Absorption systems along the sightline produce lines of other elements at other wavelengths, some of which land inside the forest and correlate with it — which puts spurious peaks into the correlation function at separations set by the wavelength ratios, at scales that happen to be near the acoustic one. They are modelled and subtracted, and the subtraction is one of the leading systematics.

And the sampling argument is an idealisation. Sightlines are not on a regular grid and their density varies across the sky, so the transverse sampling is a distribution rather than a number, and the effective reach is worse than the mean separation implies.

The pixels are not independent. The gas is thermally broadened and pressure-smoothed over tens of kilometres a second, so pixels finer than that carry correlated information. The radial reach quoted here is the spectrograph’s and the physical one is coarser — which does not affect the comparison with the transverse direction, since the transverse sampling is coarser than both by an order of magnitude.

Why the tracer runs out #

The forest works over a range of redshift with a hard edge at each end, and both edges are worth understanding because they bound what the technique can ever do.

At the low end the forest thins out. The universe expands, the gas becomes more diffuse and more highly ionised, and by redshift 1.6 the mean transmitted flux is above ninety per cent — so there is almost no absorption to measure, and what there is has a poor signal-to-noise per pixel. Below that redshift the transition also shifts into the ultraviolet, where it is unobservable from the ground.

At the high end the opposite happens. The mean transmission falls, more of each spectrum saturates, and the tracer’s response to density flattens. By redshift 4 the forest is blotchy rather than continuous, and past redshift 6 it is complete absorption with no continuum left — which is a measurement of when reionisation ended and not of anything geometric.

So the usable window is roughly redshift 2 to 3.5, and the effective redshift of every published measurement sits inside it. That window happens to be where the universe is matter-dominated and where galaxy surveys have run out, which is the coincidence that makes the technique valuable.

A tracer’s redshift range is set by its own physics rather than by the instrument, and that is unusual: almost every other observational limit in cosmology is one of depth or area and moves with the telescope. This one does not, and the only way past it is a different transition of a different species.

The generalisation #

The shape worth carrying is that a measurement’s anisotropy is usually the instrument’s rather than the universe’s, and that choosing an instrument is choosing which direction to be good in.

A galaxy survey is a wide-field instrument with a shallow redshift shell, so it samples the sky finely and the line of sight coarsely. A forest survey is a spectroscopic instrument with a sparse sky, so it does the reverse. Neither anisotropy is a property of the structure being measured — the acoustic feature is a sphere in both cases — and the two are complementary in precisely the sense that means combining them is worth more than either.

When a quantity is measured along two axes with different precision, ask which axis the instrument chose. The answer is almost never the physics, and it is frequently reversible by picking a different instrument rather than a better one.

The second reading is about what makes a tracer useful. The forest is a poor tracer by every conventional standard: it saturates, its bias is unknown, its relation to matter depends on a thermal history nobody has measured, and half of its systematic budget is the shape of a quasar’s own spectrum. It is nonetheless the best measurement of the expansion rate at its redshift, because there is nothing else there — and because the quantity it measures well is set by geometry rather than by any of the things it does badly.

A tracer’s systematics and its geometry are separate questions, and a bad tracer in a good geometry beats a good one that cannot be put where it is needed. The acoustic scale is measurable in the forest not because the gas is well understood but because the feature is at a fixed comoving length and the gas is at the right redshift.

Still open: the same test applied to what is empty #

What comes next swaps the feature for its complement. Voids are the largest structures in the density field, their shapes are measurable, and the Alcock–Paczyński test applied to them is larger in amplitude than the one applied to the acoustic feature — because a void’s distortion is a direct statement about the coordinates rather than a shift in a bump.

Beside it lies the question every acoustic measurement has deferred: what happens when the sound horizon is treated as a free parameter rather than as a calibration. The forest is the measurement best placed to answer it, because it is furthest from the low-redshift calibrations and nearest the epoch the sound horizon was set in.