Astronomy & Celestial Mechanics
the sky, computed
Concept

Baryon acoustic oscillations — where it appears

The imprint of sound waves that crossed the coupled photon and baryon fluid before recombination. They leave a preferred separation of about 150 comoving megaparsecs in the galaxy distribution, which serves as a standard ruler at any redshift a survey reaches.

Named by 7 essays across one field — each of them below, with the objects they name alongside 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.

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 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 Every curvature has a Hubble constant that leaves the peaks where they are. The Hubble constant a universe needs in order to put the acoustic peaks of the microwave background at the measured angle, against its spatial curvature Ωₖ, with the physical densities of baryons, matter and radiation held at their fitted values so that the sound horizon is unchanged. Along this line the angle θ* — the sound horizon over the distance to last scattering — is exactly constant and the peak positions do not move. The flat member has H₀ = 67.4 km/s/Mpc; a closed universe with Ωₖ = −0.044 needs 53.5, with a matter density of 0.50; an open one with Ωₖ = +0.010 reaches the locally measured 73. The peaks alone therefore measure a combination of curvature and expansion rate, not either one, and the statement that the universe is flat to a few parts in a thousand needs a second distance measured somewhere else.

A flatness that needs a second distance

The universe is said to be flat to two parts in a thousand, and the figure is usually credited to the microwave background. On its own the background measures one angle, and one angle fixes one distance — so every curvature has a Hubble constant that reproduces the peaks exactly. Flatness is where that line crosses a second measurement, and it is only as good as the crossing.

cosmology · Density parameters 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 One likelihood, read with three priors. A Gaussian likelihood in the effective neutrino mass sum, peaking at −0.08 eV with a width of 0.05 eV — of the shape that combinations of microwave-background and baryon-acoustic data have been reported to give. Read with no prior it says the data prefer a negative sum. Read with the physical prior, sum ≥ 0 (shaded right of zero), its 95 per cent upper limit is 0.059 eV, at the normal-ordering floor. Read with the oscillation prior, sum ≥ 0.059 eV, its limit is 0.102 eV — but only 0.3 per cent of the likelihood lies above the floor at all, which is the more informative number: the data and the oscillation experiments disagree at about 2.8 standard deviations.

A bound below the floor is not a mass

Oscillation experiments prove that neutrinos weigh at least 0.059 eV between them. Cosmology's latest upper bound on their total is 0.064 eV, and a bound computed differently would sit below the floor. The usual reading is that cosmology has nearly weighed the neutrino. The better one is that the fit's likelihood peaks at a negative mass: the data see more clustering than any real neutrino allows, and the bound is a symptom of that tension rather than a measurement of anything.

cosmology · Density parameters

Named alongside it

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

Standard rulerCorrelation functionSound horizonAlcock paczynskiDegeneracyExpansion historyFree-streamingRedshift-space distortionAcoustic scaleAge of the universeAngular-diameter distanceAnisotropic clustering

All concepts