Astronomy & Celestial Mechanics
the sky, computed
Concept

Sound horizon — where it appears

How far a pressure wave in the photon–baryon fluid travelled before the photons let go, and the one cosmological length fixed by physics rather than calibration. It is about 147 megaparsecs comoving, and it appears both as the spacing of the microwave background's acoustic peaks and as a bump in the galaxy correlation function.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

H₀: nine determinations in two families. Published determinations of H₀, each with its quoted one-sigma interval, sorted into two families — measured locally, in the late universe, against inferred from z ≈ 1100 through a model. The shaded band behind each family is that family's inverse-variance weighted mean: 72.66 ± 0.75 across 5 of them, against 67.40 ± 0.41 across 4. The difference is 5.26 ± 0.85 km/s/Mpc, which is 6.2 standard deviations, computed here from the quoted errors alone. That number is an upper bound on the significance rather than the significance: the determinations within each family share calibrations, samples and in two cases the same supernovae, so they are not independent, and a correlated pair combines to something wider than the formula used here gives. What the figure does establish is that the split is not one discrepant measurement against a consensus — it is two internally consistent groups, and the grouping is by method rather than by result.

The same constant, measured twice, five sigma apart

Cepheids and supernovae give an expansion rate of about 73 kilometres per second per megaparsec. The microwave background gives 67.4. Both quote errors near one per cent, both have been rebuilt from scratch by rival teams, and the gap between them has grown as the measurements have improved.

cosmology · Hubble constant 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 The damping tail measures a thickness: ℓ_D = 1400 for a shell 80 deep in redshift. The suppression of small-scale structure in the microwave background, against multipole, on logarithmic axes. Every other feature of the power spectrum measures the last-scattering surface as a surface — its distance, the sound horizon written on it, the ruler it provides. This one measures how thick it is. Recombination takes time: the ionised fraction falls over a range of redshift rather than all at once, and while it is falling the photons are still scattering, so each one random-walks. A photon taking N steps of a mean free path λ diffuses √N λ, which is much further than a single step and much less than the whole interval, and any temperature fluctuation smaller than that distance is mixed away before it can be frozen in. What is left is a Gaussian cut-off, drawn here for three shell thicknesses. A thicker shell means more steps and a longer walk, so it damps at a smaller multipole: Δz = 40 gives ℓ_D = 1980, Δz = 80 gives ℓ_D = 1400, Δz = 160 gives ℓ_D = 990. The observed cut-off is near ℓ = 1400, corresponding to a diffusion length of about 0.03 proper megaparsecs at the time — a scale the reader should compare with the sound horizon, some hundred and fifty comoving megaparsecs, which is what the peaks measure. The tail is therefore a genuine probe of the inside of the transition rather than of its position, and because the damping depends on the free-electron density it is also one of the cleanest constraints on anything that changes it: extra relativistic species, a varying fine-structure constant, or an early energy injection all move ℓ_D while leaving the peak positions nearly alone. The normalisation here is set once by the observed value, so what the figure claims is the scaling with thickness and the shape of the cut-off, not the absolute number.

A blur that measures a depth

Every other feature of the microwave background measures the last-scattering surface as a surface — its distance, the ruler written on it, the geometry between. The damping of its small-scale structure measures how thick it is, because a photon random-walking through a finite transition smears away anything smaller than its walk.

cosmology · Microwave background 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 clumped plasma recombines earlier. The fraction of hydrogen still ionised against redshift, decreasing to the right, on a logarithmic axis, from Peebles's three-level atom: for a smooth plasma (solid), and for one whose baryons are clumped into two equal volumes at 1.71 and 0.29 times the mean density, a clumping factor of 0.5 (thick dashed, averaged by mass; thin dashed, each zone alone). Recombination goes as the density squared, so the dense zone recombines first and carries most of the mass; the average ionisation falls through one half at redshift 1292 instead of 1273, and at redshift 1100 it is 0.123 against 0.145. The photons cannot see the individual patches, which are smaller than their mean free path; they see only the average, and the average recombines earlier.

A field that would have moved the ruler

A magnetic field left over from the early universe, too weak to matter to any galaxy, could still have pushed the baryons into small clumps before the universe became transparent. Clumped hydrogen recombines earlier, the sound waves in the plasma stop sooner, and the ruler that the microwave background is read with comes out shorter — which raises the Hubble constant the background implies, in the direction of the disagreement that has resisted every other explanation.

cosmology · Primordial fields 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.

Standard rulerHubble tensionRecombinationBaryon acoustic oscillationsHubble constantInverse distance ladderAcoustic peaksAlcock paczynskiCorrelation functionDamping tailDegeneracyEarly dark energy

All concepts