Astronomy & Celestial Mechanics
the sky, computed
Concept

Standard ruler — where it appears

An object whose physical length is known, so that the angle it subtends becomes a distance with no brightness calibrated anywhere. The acoustic scale in the galaxy distribution is the one that matters at cosmological distances, and it is measured in the same survey that supplies the redshifts.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has.

A standing wave frozen at one instant

The temperature of the microwave background varies across the sky by one part in a hundred thousand, and the sizes of the patches are not random. There is a preferred angular scale near one degree, and it is a sound wave that stopped ringing four hundred thousand years after the beginning.

cosmology · Microwave background 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.

cosmology · Baryon acoustic oscillations 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.

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.

cosmology · Baryon acoustic oscillations Take 8 per cent off the horizon and the tension is gone. The Hubble constant against the sound horizon at recombination, along the locus the microwave background's measured angular scale fixes. What is measured is an angle — the angular size of the horizon, to a part in three thousand — and an angle is a length divided by a distance, so extracting an expansion rate requires the length. That length is computed from the physics of the first four hundred thousand years: the baryon density, the radiation density, the number of relativistic species and the recombination history. Change any of those and the locus is unchanged while the point on it moves. The horizontal band is the late-universe measurement from the distance ladder, and it meets the locus at 136 megaparsecs — 8 per cent shorter than the standard model gives. That is the arithmetic behind every proposal to resolve the disagreement by changing the early universe rather than the late one.

A constant that is an angle divided by a length

The microwave background does not measure an expansion rate. It measures one angle — the apparent size of the sound horizon at recombination — to a part in three thousand, and converting that angle into a rate requires the horizon's physical length, which is computed from a model of the first four hundred thousand years rather than observed.

cosmology · Hubble constant A blur undone with the field that made it. The acoustic feature in the correlation function, as it was laid down and as bulk flows leave it, with three reconstructions. The broad band is left out entirely — an analysis divides it away and fits the position of what remains — so what is drawn is the feature itself. Every galaxy has moved since recombination, by several megaparsecs, and the two ends of a pair separated by 150 megaparsecs move almost independently: the pairwise dispersion of those displacements, Σ = 8.5 Mpc here, convolves the feature and broadens it from 35 to 43 megaparsecs. Reconstruction estimates the displacement from the observed density and moves every galaxy back along it. The estimate has to be smoothed, because the relation between density and displacement is only linear on large scales, and whatever the smoothing removes survives as a residual — Σ falls to 1.1 at a 10-megaparsec kernel and would fall further at a finer one, until the noise the kernel is there to suppress took over. The feature is sharpened by the same field that blurred it, and what limits the sharpening is not the idea but how well the field can be measured.

A blur undone with the field that made it

The acoustic feature is smeared by the same bulk flows whose displacements can be estimated from the density they produced. Moving every galaxy back along that estimate sharpens the peak and removes most of its second-order shift — and what limits the procedure is not the idea but how well a field can be measured from the objects it moved.

cosmology · Baryon acoustic oscillations Fine along one axis, coarse across the other two. The smallest scales a forest survey can measure, against how many quasars a square degree it has sightlines to, at redshift 2.33. Along a sightline the sampling is set by the spectrograph: a 69 km/s pixel is 0.29 comoving megaparsecs at this redshift, so the radial wavenumber reaches 10.8 Mpc⁻¹ and it does not depend on the survey's size at all. Across the sky it is set by how densely the quasars sit: at 30 a square degree the mean separation is 18 megaparsecs and the transverse wavenumber reaches 0.17. The ratio is 63, and improving it means finding more quasars, which are a finite population on the sky. That anisotropy is the inverse of a galaxy survey's, where the transverse direction is sampled by a wide sky and the radial one is limited by a redshift shell and contaminated by peculiar velocities — so the forest measures the expansion rate at its redshift better than it measures the distance to it, which is the reverse of every galaxy survey.

A ruler read in absorption

Beyond the reach of any galaxy survey the only tracer left is the hydrogen between here and a quasar, sampled finely along each sightline and coarsely across the sky. That inverts the balance every galaxy measurement is built on — the forest measures an expansion rate better than a distance, which is the harder of the two to get any other way.

cosmology · Baryon acoustic oscillations The Hubble constant the microwave background implies, with an early component. The Hubble constant that keeps the microwave background's measured angular sound horizon fixed, against the peak fraction of early dark energy at z = 3500, with the baryon density held at its standard value and the matter fraction at the 0.315 that late-time distances measure, so that the dark-matter density rises with the Hubble constant. With none, it is 67.4 km/s/Mpc and the horizon 144.3 Mpc. At 5 per cent it is 68.7; at 10 per cent 70.2, with the horizon at 138.5 Mpc; the local value of 73 is reached at 19 per cent. The shrinkage of the horizon is what raises the constant: a shorter ruler at the same angle must be nearer, and nearer at a fixed matter fraction means faster expansion. The extra dark matter this requires is the cost the published fits also find: at the value that reaches 73, the physical matter density is 18 per cent higher than in the standard model.

A ruler shortened by something already gone

The microwave background gives the Hubble constant only through a computed length, the distance sound travelled before the universe became transparent. Add a component of energy that exists for a few thousand years before recombination and then vanishes faster than radiation, and the universe expands a little faster while the sound is travelling, the ruler comes out a few per cent shorter, and the inferred expansion rate rises toward the local value. The price is paid in dark matter and in the finest details of the oldest light.

cosmology · Hubble constant The same ruler, and a wave that arrives early. The acoustic oscillation in the galaxy power spectrum — the ratio to a smooth spectrum, minus one — against wavenumber, for no free-streaming species and for three neutrinos, with the sound horizon held at 99.0 h⁻¹ Mpc in both so that only the phase differs. The two agree on the largest scales and separate as the wavenumber grows: by 0.3 h/Mpc the wave with neutrinos runs 0.21 radians ahead, so its crests sit at smaller wavenumber — near 0.2 h/Mpc by 2.2 thousandths of an h/Mpc. The amplitude 0.05 and the damping of the feature by bulk flows over 7 h⁻¹ Mpc are illustrative; the phase is computed.

A wave pulled ahead by particles that outran it

The neutrinos of the early universe moved at the speed of light while the sound wave that became the acoustic feature moved at little more than half of it, and their gravity dragged the wave forward. The drag survives as a phase in the galaxy power spectrum, a shift that no change in the ruler's length can copy, so galaxy surveys can count the species that streamed freely without asking the microwave background.

cosmology · Baryon acoustic oscillations

Named alongside it

The objects these essays reach for when they reach for this one.

Sound horizonBaryon acoustic oscillationsCorrelation functionAlcock paczynskiDegeneracyExpansion historyHubble constantHubble tensionInverse distance ladderRecombinationRedshift-space distortionAcoustic peaks

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