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The standard ruler: one length, two epochs

Sound waves in the first 370,000 years left a length of about 150 megaparsecs imprinted on the universe. The cosmic microwave background shows it at one distance and galaxy surveys show it at others. Here is where it comes from, how it is seen, and what it says about the Hubble constant.

The page has three interactive scenes that use the calculator’s sliders: the sound shell forming around one dense spot (computed with CAMB), every sighting of the ruler at the DESI redshifts and at the CMB for your model, and the ruler against the distance ladder.

A length written into the early universe

For about its first 370,000 years the universe was a hot plasma. Light and ordinary matter (“the gas” here) were tied together and behaved as one fluid with a very high pressure. Dark matter felt only gravity. Take one spot that starts slightly denser than its surroundings in every component. The dark matter stays where it is. The pressure of the light drives the gas and the light outwards as a spherical sound wave, at about half the speed of light. The first scene shows this for the Planck 2018 parameters, calculated with CAMB: the gas shell is 46 Mpc from the centre after 31,000 years, 80 Mpc after 103,000 years, and 133 Mpc when the CMB is released, 367,000 years after the Big Bang. These are positions of the top of the shell, which trails the front of the wave by 5 to 8 %. Distances are comoving: measured on today’s scale.

Then the plasma turns into neutral gas. The light escapes, and we see it today as the CMB. The gas is left without pressure and the wave coasts to a halt. The distance the front of the wave has travelled by then is the sound horizon, r_d = 147 Mpc, about 480 million light years. Gravity does the rest: the shell of gas attracts dark matter, the dark matter at the centre attracts gas, and by today both have the same pattern, a central concentration with a faint shell around it. In the profile drawn here the top of the shell ends at about 150 Mpc, about 2 % outside r_d: the shell is tens of megaparsecs wide, the curve shows mass per unit radius, which favours its outer side, and the pattern shifts outwards by a few megaparsecs after the gas is released.

Every dense spot did this at the same time, and the shells overlap, so no ring can be seen in a map of galaxies. What remains is a statistical preference: two galaxies are slightly more likely to lie about 150 Mpc apart than 130 or 170. It was first detected in 2005 in the Sloan Digital Sky Survey (Eisenstein et al.) and in the 2dF Galaxy Redshift Survey (Cole et al.). These are the baryon acoustic oscillations, BAO.

One ruler, seen at two epochs

The same wave is seen twice. In the CMB it sets the size of the hot and cold spots: for the Planck model the sound horizon covers 0.60° of sky at redshift z ≈ 1090. In galaxy surveys it is the preferred separation, and the same length covers 4.3° at z = 0.51, 2.6° at z = 0.93 and 1.5° at z = 2.33. A known length and a measured angle give a distance, D_M(z) = r_d/θ. That is how DESI measures distances in units of the ruler. Strictly, the CMB shows the wave at the moment the light was released (r_* = 144 Mpc) and the galaxies remember the moment, slightly later, when the gas was released (r_d, 1.8 % longer).

A galaxy survey has depth as well. Along the line of sight the ruler spans an interval of redshift, Δz = r_d H(z)/c: 0.044 at z = 0.51 and 0.116 at z = 2.33 for the Planck model. So the same ruler measures the expansion rate H(z) directly. The CMB is a single shell around us and shows the ruler only across the sky.

The second scene draws every sighting for your model. H₀ moves all the galaxy sightings together; Ω_m and Ω_Λ change the near and the far ones by different amounts; curvature acts most on the CMB sighting, the most distant. One sighting cannot fix the model; several can. This is why the CMB alone can hardly tell a slightly curved universe from a flat one (task 5 of the CMB lab), and why adding BAO settles it.

What sets the length

The sound horizon depends on how fast the sound travelled and for how long. That comes down to two densities the CMB measures well: the baryon density Ω_bh² (baryons weigh the fluid down and slow the wave) and the matter density Ω_mh² (which sets how fast the universe was expanding). Once those two are fixed, H₀, dark energy and curvature play no part: the ruler was finished before they mattered. Change Ω_bh² or Ω_m in the panel and the first scene shows the new length. For Ω_bh² = 0.02242 and Ω_mh² = 0.1424, iCosmos gives r_d = 147.1 Mpc (CAMB gives 147.21 for the same parameters; the 0.1 % difference comes from how the neutrino mass is treated). Planck’s value from the CMB alone, at slightly different densities, is 147.09 ± 0.26 Mpc. Raising Ω_bh² to 0.024 shortens the ruler to 145.7 Mpc (−0.9 %). The baryon density can also be taken from the deuterium made in the first minutes, with no CMB data at all.

Ruler against candle

There are two ways to fix the scale of cosmic distances. A standard candle is calibrated outwards: parallaxes of nearby stars, those stars in galaxies that host supernovae, then supernovae far enough away to measure the expansion. A standard ruler is calibrated inwards: early-universe physics gives its length, and surveys measure distances in units of it. BAO surveys measure the product of H₀ and the ruler. DESI DR2 finds h·r_d = 101.54 ± 0.73 Mpc for a flat universe with a cosmological constant, where h = H₀/100. With Planck’s ruler this gives H₀ = 69.0 ± 0.5 km s⁻¹ Mpc⁻¹. The Local Distance Network measures 73.50 ± 0.81 by the outward route. For the BAO distances to give that value, the ruler would have to be 138.1 Mpc, 6.1 % shorter than standard physics makes it.

A shorter ruler means that something was different before the CMB was released, for example an extra form of energy that made the young universe expand faster for a while (“early dark energy”). No such model is established. Changing the ruler also changes the heights of the CMB peaks and the damping tail, and it has proved hard to shorten the ruler without spoiling the fit to the CMB (Knox & Millea 2020). The other possibility is an unrecognised error in the distance ladder or in the surveys. The third scene lets you change the length of the ruler and see where the two routes meet. It does not say which is right.

How the scenes are made

Scene 1 is computed: CAMB transfer functions for the Planck 2018 parameters, turned into the response to one point-like dense spot, as in Eisenstein, Seo & White (2007). The curves show mass per unit radius, and each frame is scaled to its own maximum, so the growth of the pattern with time is not shown. Photons and neutrinos are multiplied by 3/4, the factor that makes all four equal at the very beginning; by the first frame they have already separated. When you change Ω_bh² or Ω_m only the radius is rescaled to your model’s r_d; the shapes remain those of the Planck model. The face-on view and the scattered points are illustrations.

Questions for a class

1. Scene 1. How far from the centre is the gas shell when the CMB is released, and how old is the universe then?
2. Scene 1. Where does the shell come to rest, and what is the sound horizon r_d for the Planck model?
3. Scene 2, preset “Planck 2018 + BAO”. What angle does the ruler cover at z = 0.51, and what angle does the sound horizon cover at the CMB? What distance D_M does the first angle correspond to?
4. Scene 2. What interval of redshift does the ruler span along the line of sight at z = 2.33, and what expansion rate H(z) does that correspond to?
5. With the same preset, open “Survey & CMB parameters” and raise Ω_bh² from 0.02242 to 0.024. What happens to the ruler? What happens if you change H₀ instead?
6. Scene 3. DESI measures h·r_d = 101.54 Mpc. What H₀ follows for Planck’s ruler, r_d = 147.09 Mpc? What ruler would give H₀ = 73.50?

Answers

1. About 133 Mpc (comoving), 367,000 years after the Big Bang.
2. The top of the shell ends at about 150 Mpc; r_d = 147 Mpc. The shell is wide and the curve shows mass per unit radius, so its highest point lies about 2 % outside r_d.
3. 4.26° at z = 0.51 and 0.596° at the CMB. D_M = r_d/θ = 1.98 Gpc (θ in radians).
4. Δz = 0.116, so H = c Δz/r_d = 236 km s⁻¹ Mpc⁻¹.
5. r_d falls from 147.1 to 145.7 Mpc (−0.9 %): more baryons slow the sound. Changing H₀ at fixed Ω_m changes Ω_mh², and with it r_d; at fixed Ω_mh² and Ω_bh² the ruler does not depend on H₀.
6. H₀ = 100 × 101.54 ÷ 147.09 = 69.0 km s⁻¹ Mpc⁻¹. For 73.50 the ruler would have to be 138.1 Mpc, 6.1 % shorter.

References

• D. J. Eisenstein, H.-J. Seo & M. White (2007), On the robustness of the acoustic scale in the low-redshift clustering of matter, ApJ 664, 660.
• D. J. Eisenstein et al. (2005), Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies, ApJ 633, 560.
• S. Cole et al. (2005), The 2dF Galaxy Redshift Survey: power-spectrum analysis of the final data set and cosmological implications, MNRAS 362, 505.
• DESI Collaboration (2025), DESI DR2 Results II: measurements of baryon acoustic oscillations and cosmological constraints, Phys. Rev. D 112, 083515.
• L. Knox & M. Millea (2020), Hubble constant hunter’s guide, Phys. Rev. D 101, 043533.
• Planck Collaboration (2020), Planck 2018 results. VI. Cosmological parameters, A&A 641, A6.

Related: BAO distances for your model, the angular scale of the CMB, the Hubble tension, the CMB lab, how big is the universe?

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iCosmos 4.11.1 · updated 12 November 2026