Astronomy & Celestial Mechanics
the sky, computed
Concept

Expansion history — where it appears

The scale factor's run with time, or equivalently the expansion rate as a function of redshift. Distance measurements constrain an integral of it and are correspondingly insensitive to any short interval, which is why a rate measured at one redshift is worth so much.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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 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

Named alongside it

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

Baryon acoustic oscillationsStandard rulerAlcock paczynskiAngular-diameter distanceAnisotropic clusteringAnisotropic samplingContinuum fittingCorrelation functionDegeneracyFiducial cosmologyHubble parameterIntergalactic medium

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