The Sound Horizon and the Standard Ruler

Strike a match in the first hundred thousand years of the universe and the sound it makes is a legible fossil. Before recombination the cosmos is a single hot fluid of photons and baryons, locked together by scattering into a substance with enormous pressure and real inertia. Every primordial overdensity is a struck bell: photon pressure drives a spherical acoustic wave outward at nearly $c/\sqrt3$ , with the baryons riding along as ballast. Then, at $z_\ast \approx 1090$ , the plasma recombines into neutral hydrogen, the photons stream free, the pressure vanishes, and the wave stops dead. Frozen into the baryons is a shell of slight overdensity at one fixed comoving radius — the sound horizon $r_s$ . That radius is the same for every overdensity in the universe, because all were struck at the same instant with the same sound speed. The result is a preferred separation stamped into all of cosmic structure: a faint excess of galaxy pairs at $r_s \approx 147\ \mathrm{Mpc}$ that survives to today, and a matching ring of hot and cold spots on the microwave sky. What makes $r_s$ extraordinary is that we can calculate it — its length is fixed entirely by linear, well-understood pre-recombination physics, the densities of photons, baryons, and radiation, so it is known to better than a quarter percent without ever measuring a single galaxy. A length you can compute from first principles and also observe on the sky is a standard ruler. Laid against the cosmos at last scattering it fixes the distance to the microwave background; laid against galaxy surveys it charts $H(z)$ and $D_A(z)$ . And because it is calculated so cleanly, it has become the fault line of the Hubble tension — the ruler that anchors precision cosmology also refuses, so far, to reconcile the early and late universe.

Key takeaways

The sound horizon $r_s$ is the comoving distance an acoustic wave travels in the photon-baryon fluid, from the Big Bang until the fluid decouples at recombination.
It integrates the sound speed $c_s = c/\sqrt{3(1+R)}$ over conformal time, depressed below $c/\sqrt3$ by baryon loading $R = 3\rho_b/4\rho_\gamma$ .
Because it is fixed by pre-recombination physics, $r_s$ is calculable to $\sim0.2\%$ : $r_\ast \approx 144.4\ \mathrm{Mpc}$ at photon decoupling and $r_d \approx 147.1\ \mathrm{Mpc}$ at the slightly later baryon drag epoch.
Imprinted on the CMB (peak spacing $\theta_s = r_\ast/D_A$ ) and on galaxies (the BAO feature at $r_d$ ), it is a standard ruler — a length known a priori, not calibrated locally.
Comparing its angular size $\theta_s(z)$ and radial extent $\Delta z = r_d H/c$ across redshift measures $D_A(z)$ , $H(z)$ , curvature, and $H_0$ .
Key idea: the sound horizon is a physically calculable comoving length, $r_s = \int_{z_\ast}^{\infty} c_s(z)/H(z)\,\mathrm{d}z$ , stamped identically onto the CMB and the galaxy distribution, that serves as the universe's standard ruler.

The core insight

The sound horizon $r_s = \int_0^{t_\ast} c_s\,(1+z)\,\mathrm{d}t = \int_{z_\ast}^{\infty} \frac{c_s(z)}{H(z)}\,\mathrm{d}z$ is a physically calculable comoving length imprinted on both the CMB and the galaxy distribution, serving as a cosmological standard ruler.

Where this lives today

The sound horizon is the beating heart of the current cosmology program. The Dark Energy Spectroscopic Instrument (DESI) is measuring the BAO ruler in tens of millions of galaxies and quasars across $0.1 < z < 3.5$ , and its early releases already give the tightest BAO dark-energy constraints to date — with a persistent hint that dark energy may be evolving, a result that lives or dies on the calibration $r_d = 147$ Mpc. Euclid and Vera Rubin will push the same ruler to larger volumes. Above all, the sound horizon is the fulcrum of the Hubble tension: the CMB-plus-BAO value $H_0 \approx 67$ – $68\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ sits in $\sim5\sigma$ conflict with the local value near $73$ , and because that early-universe value rests entirely on the calculable $r_s$ , a whole industry of proposed resolutions — early dark energy, extra relativistic species, modified recombination — works by trying to shrink the ruler by a few percent before recombination. New physics or hidden systematic, the sound horizon is where the two halves of cosmology are forced to meet.

The physical picture

Picture the early universe not as scattered galaxies but as a smooth, blindingly bright fluid, everywhere the same to a part in $10^5$ . Where it is a hair denser, gravity pulls in and photon pressure pushes back, launching a sound wave. Because photons and baryons are welded together by scattering, this is one fluid with a single sound speed, and a spherical pressure ridge expands from every overdensity at $c_s \approx 0.45c$ , dragging baryons outward while the pressureless cold dark matter stays put at the center. At recombination the photons let go, the restoring pressure vanishes, and the baryon shell coasts to a halt at comoving radius $r_s$ . Every peak is now ringed by a shell of the same radius — a standard ruler minted $380{,}000$ years after the Big Bang.

Figure

The frozen acoustic shell: a primordial overdensity launches a photon-baryon sound wave that expands until recombination at z ≈ 1090, when the photons stream free and the baryon shell stalls at a fixed comoving radius r_s ≈ 147 Mpc — the same radius around every peak in the universe

Change the sound speed and the ruler changes length. It is not simply $c/\sqrt3$ : baryons load the fluid with inertia they do not repay in pressure, so as $R$ climbs toward recombination the wave slows and the ruler shrinks — making $r_s$ a thermometer for the baryon density. Likewise anything that speeds the pre-recombination expansion (extra radiation, an early burst of dark energy) gives the wave less time and shortens $r_s$ . The ruler's calculated length is a sensitive fingerprint of the early universe's contents. The payoff is geometric: because $r_s$ is a fixed comoving length, its apparent angular size falls off inversely with distance, so measuring the angle measures the distance.

Figure

The standard ruler in action: the same comoving length r_s ≈ 147 Mpc subtends a large angle nearby and a tiny angle far away, so the observed angle θ = r_s/D_com fixes the comoving distance — 4.4° at a galaxy survey (z ≈ 0.5, D ≈ 1.9 Gpc) shrinking to 0.60° for the CMB (z ≈ 1090, D ≈ 13.9 Gpc, peak spacing ℓ ≈ π/θ ≈ 300)

The core equations

The sound horizon is a distance travelled: sound speed integrated over the time available, written as a comoving length. Converted to an integral over redshift — the form used in practice, since $H$ and $c_s$ are natural functions of $z$ — it is the master equation of the subject.

r_s(z_\ast) = \int_0^{t_\ast} c_s\,(1+z)\,\mathrm{d}t = \int_{z_\ast}^{\infty} \frac{c_s(z)}{H(z)}\,\mathrm{d}z \qquad (1)

where: $r_s$ — comoving sound horizon (Mpc); $z_\ast$ — redshift of last scattering (or $z_d$ at the drag epoch); $c_s(z)$ — sound speed of the photon-baryon fluid; $H(z)$ — Hubble parameter. The integrand $c_s/H$ is sharply peaked just above $z_\ast$ , so most of the ruler is laid down right before recombination — which is why $r_s$ is so sensitive to physics there. The sound speed is the most important input. Photons alone give the relativistic value $c/\sqrt3$ , but baryons add inertia without pressure, and the momentum-density ratio $R$ measures how much.

c_s(z) = \frac{c}{\sqrt{3\,[\,1+R(z)\,]}},\qquad R(z) = \frac{3\rho_b}{4\rho_\gamma} = \frac{3\Omega_b}{4\Omega_\gamma}\,\frac{1}{1+z} \qquad (2)

where: $R$ — baryon-to-photon momentum-density ratio; $\rho_b,\rho_\gamma$ — baryon and photon energy densities; $\Omega_b,\Omega_\gamma$ — present density parameters. Since $\rho_\gamma \propto (1+z)^4$ and $\rho_b \propto (1+z)^3$ , $R$ grows as $1/(1+z)$ : photon-like early ( $R\to0$ ), increasingly baryon-loaded toward recombination, where $R_\ast \approx 0.6$ and $c_s \approx 0.45c$ . The time available is set by the expansion rate. Before recombination the universe is radiation- and matter-dominated; dark energy is negligible at $z \sim 1000$ and drops out of the ruler entirely.

H(z) = H_0\sqrt{\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_\Lambda} \qquad (3)

where: $\Omega_r$ — radiation (photons + relativistic neutrinos); $\Omega_m$ — matter (CDM + baryons); $\Omega_\Lambda$ — dark energy, $\approx0$ at $z_\ast$ ; $H_0$ — present Hubble constant. Since $r_s$ feels $H$ only through the pre-recombination terms $\Omega_r$ and $\Omega_m$ , it is blind to the late-time expansion — the property that makes it a clean, standalone ruler. On the microwave sky the ruler shows up as an angle — the comoving sound horizon at photon decoupling over the comoving distance to last scattering — which sets the spacing of the acoustic peaks.

\theta_s = \frac{r_\ast}{D_{\rm com}(z_\ast)}, \qquad \ell_A \approx \frac{\pi}{\theta_s} \qquad (4)

where: $\theta_s$ — angular size of the sound horizon; $D_{\rm com}(z_\ast)$ — comoving distance to last scattering; $\ell_A$ — characteristic multipole spacing. Planck measures $100\,\theta_s = 1.04109 \pm 0.00030$ — three parts in $10^4$ , the most precisely known number in cosmology. At low redshift the same ruler is measured in galaxy clustering, split into a transverse piece (an angle) and a radial piece (a redshift interval) that probe $D_A$ and $H$ separately.

\theta_s(z) = \frac{r_d}{(1+z)\,D_A(z)}, \qquad \Delta z(z) = \frac{r_d\,H(z)}{c} \qquad (5)

where: $r_d$ — comoving sound horizon at the drag epoch; $D_A(z)$ — physical angular diameter distance; $\Delta z$ — redshift span of the ruler along the line of sight. Transverse gives $D_A(z)/r_d$ , radial gives $r_d H(z)/c$ — two rulers, one rod. When a survey cannot separate the two, the BAO signal compresses into a single volume-averaged distance — two transverse dimensions and one radial, cube-rooted.

D_V(z) = \left[\,c z\,D_{\rm com}(z)^2 / H(z)\,\right]^{1/3}, \qquad \text{observable} = \frac{D_V(z)}{r_d} \qquad (6)

where: $D_V$ — volume-averaged (dilation) distance; $D_{\rm com}$ — comoving angular diameter distance; $r_d$ — the drag-epoch ruler. A survey reports $D_V/r_d$ , not $D_V$ alone — the ruler is always the yardstick. Finally, a compact fitting formula gives $r_d$ from three density parameters to $\sim0.2\%$ without a Boltzmann code — the workhorse of BAO cosmology.

r_d \approx \frac{55.154\,\exp\!\left[-72.3(\omega_\nu+0.0006)^2\right]}{\omega_b^{0.12807}\,\omega_{cb}^{0.25351}}\ \mathrm{Mpc} \qquad (7)

where: $\omega_b = \Omega_b h^2$ — physical baryon density; $\omega_{cb} = (\Omega_c+\Omega_b)h^2$ — physical cold matter density; $\omega_\nu$ — physical neutrino density. The negative exponents encode the two ways to shrink the ruler: more baryons slow the wave, more matter shortens the time to recombination.

Derivation AThe baryon-loaded sound speed of the photon-baryon fluid

The goal is to derive equation (2) — that the coupled fluid's sound speed is $c_s = c/\sqrt{3(1+R)}$ rather than the naive $c/\sqrt3$ — from the thermodynamic definition of a sound speed as the response of pressure to density. The physical content is that photons supply all the pressure while both species supply the inertia, so loading the fluid with baryons slows its sound without giving anything back — the single fact that controls how the ruler responds to the cosmic baryon budget.

Step 1. A sound speed is $c_s^2 = \delta p/\delta\rho$ , the ratio of the pressure perturbation to the density perturbation carried by the wave. In the tightly coupled fluid only the photons carry significant pressure, so $\delta p = \delta p_\gamma$ , while both species carry inertia, $\delta\rho = \delta\rho_\gamma + \delta\rho_b$ . Write $c_s^2 = \dot p_\gamma/(\dot\rho_\gamma + \dot\rho_b)$ using the time derivatives that track the adiabatic evolution. Why: the numerator counts only what pushes back on a compression (photon pressure), the denominator counts everything that resists acceleration (all the mass-energy) — and the mismatch between them is the whole effect.

Step 2. Use the radiation equation of state $p_\gamma = \tfrac13\rho_\gamma c^2$ , so $\dot p_\gamma = \tfrac13 c^2 \dot\rho_\gamma$ . For adiabatic perturbations baryons and photons evolve together with $\dot\rho_b/\dot\rho_\gamma = \tfrac34\,\rho_b/\rho_\gamma \equiv R$ . Why: the factor $\tfrac34$ is the different redshift scaling — photon energy carries an extra $(1+z)$ over baryon rest mass — and packaging it as $R$ makes the algebra collapse.

Step 3. Substitute: $c_s^2 = \dfrac{\tfrac13 c^2\,\dot\rho_\gamma}{\dot\rho_\gamma(1+R)} = \dfrac{c^2/3}{1+R}$ , hence $c_s = c/\sqrt{3(1+R)}$ . In the photon-only limit $R\to0$ this is the familiar $c/\sqrt3 \approx 0.577c$ ; at recombination $R_\ast \approx 0.6$ drags it to $c/\sqrt{4.8} \approx 0.45c$ . Why: the ruler is the time-integral of this speed, so the roughly 20% baryon-loading reduction feeds straight into a shorter $r_s$ , and its dependence on $R \propto \Omega_b h^2$ is what lets the CMB weigh the baryons. *As a check: $R_\ast = 3\omega_b/(4\omega_\gamma)/(1+z_\ast) = 0.62$ for $\omega_b=0.0224$ , giving $c_s/c = 1/\sqrt{3\cdot1.62} = 0.453$ .*

Go deeper · Derivation B · 1 minThe comoving sound horizon in a radiation-plus-matter universe

Derivation BThe comoving sound horizon in a radiation-plus-matter universe

The goal is to evaluate the integral in equation (1) in closed form and expose how the ruler depends on the two densities that set it, $\Omega_m h^2$ and $\Omega_b h^2$ . We drop the cosmological constant (a $10^{-9}$ share of the budget at $z_\ast$ ), keep only radiation and matter in $H$ , and use $R \propto a$ to reduce the integral to a standard logarithmic form — the Eisenstein and Hu (1998) result that Boltzmann codes reproduce to the digit.

Step 1. Recast the integral over scale factor using $\mathrm{d}\eta = \mathrm{d}a/(a^2 H)$ , so $r_s = \int_0^{a_\ast} c_s\,\mathrm{d}a/(a^2 H)$ . With radiation and matter only, $a^2 H = H_0\sqrt{\Omega_m a + \Omega_r}$ , a clean square root linear in $a$ . Why: conformal time is the natural clock for a wave, and writing $a^2 H$ this way strips the problem to the two components that exist before recombination.

Step 2. Insert the sound speed with $R = R_\ast\,(a/a_\ast) \propto a$ . The integrand becomes $\propto [\,(1+R)(\Omega_m a + \Omega_r)\,]^{-1/2}$ , a product of two factors linear in $a$ . Why: two linear factors under a square root integrate to a logarithm — the wave crossing from the radiation era into the matter era, and the log captures that crossover.

Step 3. Carry out the integral. The result is

r_s = \frac{2c}{3}\,\frac{1}{k_{eq}}\sqrt{\frac{6}{R_{eq}}}\;\ln\!\left[\frac{\sqrt{1+R_\ast}+\sqrt{R_\ast+R_{eq}}}{1+\sqrt{R_{eq}}}\right],

where $k_{eq}$ and $R_{eq}$ are the horizon wavenumber and baryon loading at matter-radiation equality. Why: every symbol is a pre-recombination quantity, so the ruler is manifestly fixed by early-universe physics and nothing else — the property that makes it standard.

Step 4. Evaluate for Planck parameters, $\Omega_m h^2 = 0.143$ , $\Omega_b h^2 = 0.0224$ . Direct integration to $z_\ast = 1090$ gives $r_\ast = 144.3\ \mathrm{Mpc}$ ; to the drag epoch $z_d = 1060$ gives $r_d = 146.9\ \mathrm{Mpc}$ . Why: these match the Planck values $r_\ast \approx 144.4$ Mpc and $r_{\rm drag} = 147.09 \pm 0.26$ Mpc — a hand integral of linear physics reproduces the full Boltzmann calculation. *As a check: numerical quadrature of $\int_{z}^{\infty} c_s/H\,\mathrm{d}z$ gives $r_\ast = 144.3$ Mpc, $r_d = 146.9$ Mpc; the Aubourg fitting formula gives $r_d = 147.0$ Mpc — all consistent with Planck 2018.*

Go deeper · Derivation C · 1 minFrom the acoustic angle to a constraint on the geometry

Derivation CFrom the acoustic angle to a constraint on the geometry

The goal is to show how one exquisitely measured angle, the CMB acoustic scale $\theta_s$ , combines with the calculated ruler $r_\ast$ to pin the comoving distance to last scattering — and hence a tight combination of $H_0$ , $\Omega_m$ , and curvature. This is the step that turns the sound horizon into the backbone of precision cosmology, and where the low CMB $H_0$ that drives the Hubble tension actually comes from.

Step 1. The CMB gives the angle directly from the peak spacing: $100\,\theta_s = 1.04109 \pm 0.00030$ (Planck 2018), i.e. $\theta_s = 1.04109\times10^{-2}$ rad, about $0.60^\circ$ — a pure observable, essentially free of cosmological assumptions. Why: the peaks are a ruler-on-the-sky pattern whose spacing is measured, not modeled — the single most robust number the CMB delivers.

Step 2. Compute the ruler from early-universe physics as in the previous derivation: $r_\ast \approx 144.4$ Mpc, given $\Omega_b h^2$ and $\Omega_m h^2$ (themselves fixed by the relative peak heights). Why: the same CMB spectrum that gives $\theta_s$ also gives the densities that set $r_\ast$ , so the ruler and its apparent angle come from one self-consistent fit.

Step 3. Invert the definition $\theta_s = r_\ast/D_{\rm com}(z_\ast)$ to read off the distance: $D_{\rm com}(z_\ast) = r_\ast/\theta_s = 144.4/(1.04109\times10^{-2}) \approx 13.87\ \mathrm{Gpc}$ . Why: a known length subtending a known angle is a measured distance — this is triangulation on the scale of the observable universe, and it lands the last-scattering surface at $\sim13.9$ Gpc.

Step 4. Relate that distance to parameters via $D_{\rm com}(z_\ast) = \frac{c}{H_0}\int_0^{z_\ast}\mathrm{d}z/\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda+\cdots}$ . Requiring this to equal $13.87$ Gpc, with $\Omega_m$ fixed by the CMB, forces $H_0 = 67.4 \pm 0.5\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ . Why: flat $\Lambda$ CDM geometry closes on one $H_0$ — the famous low number — and any new physics shortening $r_\ast$ would raise it, exactly the lever the Hubble tension pulls. *As a check: $D_{\rm com}(z_\ast) = 144.4/1.04109\times10^{-2} = 13.87$ Gpc; forward-computing $\theta_s$ from a flat $\Lambda$ CDM distance reproduces $100\,\theta_s \approx 1.04$ and $H_0 \approx 67$ .*

Limiting cases

The sound speed, and therefore the ruler, is governed by the baryon loading $R$ and by whatever sets the pre-recombination expansion rate $H$ :

Case	Result	Meaning
No baryons ( $R\to0$ )	$c_s \to c/\sqrt3 \approx 0.577c$	Pure photon fluid — the classic relativistic sound speed, the upper bound on $c_s$ and the longest possible ruler.
Realistic loading	$c_s = c/\sqrt{3(1+R)}$ , $R_\ast\approx0.6$	Baryons slow the wave to $\approx0.45c$ near recombination; the ruler shortens and becomes a baryometer.
Heavy loading ( $R\gg1$ )	$c_s \to c/\sqrt{3R} \propto (1+z)^{1/2}$	A baryon-dominated fluid; the sound speed collapses and $r_s$ would shrink sharply.
Extra radiation ( $N_{\rm eff}\uparrow$ )	$H(z_\ast)\uparrow \Rightarrow r_s\downarrow$	More relativistic species speed the early expansion, giving the wave less time; $r_s$ falls and inferred $H_0$ rises.
Two epochs	$r_d > r_\ast$	Baryon drag ends at $z_d\approx1060$ , later than decoupling $z_\ast\approx1090$ ; the wave travels $\sim2$ Mpc farther, so $r_d\approx147 > r_\ast\approx144$ .

Common trapthe two horizons and the two frames are not interchangeable

Two confusions bias $H_0$ if you let them. First, do not conflate the horizon at photon decoupling, $r_\ast$ ( $z_\ast\approx1090$ , sets the CMB peak positions), with the horizon at the baryon drag epoch, $r_d$ (the later $z_d\approx1060$ , sets the BAO scale). They differ by about 2 Mpc — a $1.5\%$ error, larger than the measurement precision, feeding straight into a biased distance. Use $r_\ast$ for the CMB and $r_d$ for BAO, never one for the other. Second, keep comoving and proper lengths straight: the comoving sound horizon is $\sim147$ Mpc, but the proper one at last scattering is $r_\ast/(1+z_\ast) \approx 144\,\mathrm{Mpc}/1091 \approx 0.13$ Mpc — over a thousand times smaller, because the universe was that compact when the ruler was minted.

Worked examples

1 · Computing the ruler from three numbers (fully worked)

Use the Aubourg fitting formula, equation (7), to compute $r_d$ for the Planck cosmology — $\Omega_b h^2 = 0.0224$ , $\Omega_c h^2 = 0.1200$ , $\omega_\nu = 0.0006$ — and confirm it against the measurement.

Step 1. Assemble the three density inputs: $\omega_b = 0.0224$ ; cold matter $\omega_{cb} = \Omega_c h^2 + \Omega_b h^2 = 0.1200 + 0.0224 = 0.1424$ ; neutrinos $\omega_\nu = 0.0006$ . These are all the formula needs — no distances, no low-redshift information.

Step 2. The exponential (neutrino) correction: $-72.3\,(0.0012)^2 = -1.04\times10^{-4}$ , so $\exp(-1.04\times10^{-4}) = 0.99990$ — massless neutrinos barely move the ruler. The numerator is $55.154 \times 0.99990 = 55.149$ .

Step 3. Evaluate the denominator. First $\omega_b^{0.12807} = 0.0224^{0.12807} = \exp(0.12807\ln 0.0224) = \exp(-0.4864) = 0.6148$ . Then $\omega_{cb}^{0.25351} = 0.1424^{0.25351} = \exp(0.25351\ln 0.1424) = \exp(-0.4941) = 0.6101$ . The product is $0.6148 \times 0.6101 = 0.3751$ .

Step 4. Divide: $r_d = 55.149/0.3751 = \mathbf{147.0\ Mpc}$ . This sits right on the Planck 2018 value $r_{\rm drag} = 147.09 \pm 0.26$ Mpc, and matches the direct integral of $c_s/H$ (146.9 Mpc) above. Three density parameters, known from the CMB peak heights, fix the ruler to a few tenths of a percent without a single distance measurement.

2 · Triangulating the last-scattering surface (fully worked)

The CMB gives $100\,\theta_s = 1.04109$ and early-universe physics gives $r_\ast = 144.4$ Mpc. How far is the last-scattering surface, and what proper size does one arcminute subtend there?

Step 1. Convert the angle: $\theta_s = 1.04109\times10^{-2}$ rad $= 0.596^\circ$ — the angle the sound horizon subtends on the microwave sky, roughly the width of the Moon.

Step 2. A known length at a known angle is a distance: $D_{\rm com}(z_\ast) = r_\ast/\theta_s = 144.4\,\mathrm{Mpc}/(1.04109\times10^{-2}) = \mathbf{13.87\ Gpc}$ — nearly the comoving radius of the observable universe.

Step 3. The peak spacing follows from $\ell_A \approx \pi/\theta_s = \pi/(1.04109\times10^{-2}) \approx \mathbf{302}$ , consistent with peaks marching across the CMB spectrum every few hundred multipoles (the first peak sits near $\ell \approx 220$ ).

Step 4. For the proper scale, $D_A = D_{\rm com}/(1+z_\ast) = 13.87\,\mathrm{Gpc}/1091 = 12.7$ Mpc, so one arcminute ( $2.91\times10^{-4}$ rad) subtends $12.7\,\mathrm{Mpc}\times2.91\times10^{-4} = 3.7$ kpc on the last-scattering surface. The method is triangulation: a computed ruler plus a measured angle gives a distance of 13.9 Gpc, and the same physics sets the peak spacing near ℓ ≈ 300.

3 · How much does extra radiation shrink the ruler?

Estimate the fractional change in $r_s$ from adding $\Delta N_{\rm eff} = 1$ extra relativistic species at fixed $\Omega_m h^2$ and $\Omega_b h^2$ .

Step 1. Extra radiation raises $\Omega_r$ and hence $H(z)$ before recombination. Since $r_s = \int c_s/H\,\mathrm{d}z$ , a larger $H$ shrinks the integrand where the extra radiation matters — near and above equality. One extra species raises the relativistic density by $1/3.046 \approx 33\%$ , but only part of the ruler is laid down while radiation dominates, so the net effect is roughly a $3$ – $4\%$ reduction in $r_s$ per unit $\Delta N_{\rm eff}$ .

Step 2. Because the CMB pins $\theta_s = r_\ast/D_{\rm com}$ , a $\sim4\%$ shorter $r_\ast$ demands a $\sim4\%$ shorter $D_{\rm com}$ at fixed angle, achieved by raising $H_0$ . This is the mechanism behind many Hubble-tension proposals: shrink the ruler by a few percent with new pre-recombination physics and the CMB-inferred $H_0$ climbs toward the local value of 73.

Physics of the portrait

The sound horizon lived a double life for thirty years — predicted as a wiggle in theorists' spectra long before anyone could see it, then finally read straight off the galaxy distribution as a ruler you could lay against the universe.

Origins · 1 minDaniel Eisenstein and how this was discovered

Daniel Eisenstein

b. 1965 · the acoustic standard ruler and the cosmic distance scale

The acoustic oscillations of the photon-baryon fluid were predicted in 1970, independently, by Jim Peebles and Jer Yu and by Rashid Sunyaev and Yakov Zeldovich, who saw that the tug-of-war between gravity and radiation pressure must ring the plasma and leave a periodic imprint. For a quarter century this stayed a subtle prediction — faint wiggles in the matter power spectrum, hard to imagine detecting against the messiness of galaxies. Daniel Eisenstein, working with Wayne Hu in the late 1990s, did two decisive things. First, the pair produced the analytic fitting formulae — for the transfer function and for the sound horizon $r_s$ itself — that let cosmologists compute the ruler fast and accurately, formulae still in daily use. Second, Eisenstein reframed the physics: the baryon acoustic oscillation is a standard ruler, a single preferred length $r_d$ frozen into the clustering of matter, best seen as a bump in the two-point correlation function near $150$ Mpc. In 2005 he led the Sloan Digital Sky Survey team that detected exactly that bump in luminous red galaxies — the first measurement of the acoustic scale in the matter distribution, and the birth of BAO as a precision distance probe. The ruler that Peebles and Sunyaev-Zeldovich heard in the equations, Eisenstein taught the field to measure, turning it into the calibration standard now carried by DESI, Euclid, and every major survey.

History

1968

— Silk shows photon diffusion damps small-scale acoustic waves

1970

— Peebles and Yu, and Sunyaev and Zeldovich, predict acoustic oscillations in the photon-baryon plasma

1998

— Eisenstein and Hu publish analytic fitting formulae for the sound horizon and transfer function

2005

— Eisenstein and the SDSS team detect the BAO peak in luminous red galaxies

2018

— Planck measures 100 theta_s = 1.04109 and r_drag = 147.09 Mpc

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., Ch. 8 (acoustic oscillations, the sound horizon) · Ryden, Introduction to Cosmology, 2nd ed., §8-9 (recombination, BAO) · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 4 (perturbations, transfer function) · Binney and Tremaine, Galactic Dynamics, 2nd ed. (dynamics background). The key numerical anchors are: $r_{\rm drag} = 147.09\pm0.26$ Mpc, $r_\ast \approx 144.4$ Mpc, $100\,\theta_s = 1.04109\pm0.00030$ , $z_\ast\approx1090$ , $z_d\approx1060$ , $H_0 = 67.4$ km s $^{-1}$ Mpc $^{-1}$ (Planck 2018). These are consistent with: (1) $c_s = c/\sqrt{3(1+R)}$ with $R_\ast=0.62$ giving $c_s=0.45c$ ; (2) $\int_z^\infty c_s/H\,\mathrm{d}z = 144$ – $147$ Mpc for $\Omega_m h^2=0.143$ , $\Omega_b h^2=0.0224$ ; (3) the Aubourg formula giving $r_d=147.0$ Mpc; (4) $D_{\rm com}(z_\ast)=r_\ast/\theta_s=13.9$ Gpc; (5) proper $r_\ast=144/1091=0.13$ Mpc.

Prerequisites. Acoustic Oscillations in the Photon-Baryon Fluid · The ΛCDM Concordance Model · Jeans Instability and Cosmological Perturbation Regimes. Leads to. Baryon Acoustic Oscillations · The CMB Anisotropy Power Spectrum · Silk Damping.

Ask the tutor: Ask me why adding a few percent of early dark energy shrinks the sound horizon and nudges the CMB value of $H_0$ toward the local measurement.

Is the sound horizon a real distance in space, or just a feature in a plot?

It is a genuine physical length. A pressure wave really did propagate outward from every overdensity and stall at recombination, leaving a shell of slightly enhanced baryon density at a fixed comoving radius. That shell is faint — an excess of about one percent in galaxy pairs at $147$ Mpc separation — but a true, three-dimensional feature of how matter is laid out in space, showing up as a bump in the galaxy correlation function and a ring of enhanced temperature correlation on the CMB. The "feature in a plot" is the statistical shadow of a real acoustic phenomenon.

Why are there two sound horizons, r_star and r_d, and which do I use?

Photons and baryons decouple at two slightly different times. Photons stop scattering at $z_\ast\approx1090$ , setting the scale imprinted on the CMB, $r_\ast\approx144.4$ Mpc — use it for the acoustic-peak angle $\theta_s$ . Baryons stay dynamically dragged a little longer, until the drag epoch $z_d\approx1060$ ; the extra time lets the wave travel about 2 Mpc farther, giving $r_d\approx147.1$ Mpc, the scale frozen into matter and measured by galaxy BAO. Swapping the two introduces a $\sim1.5\%$ error, well above modern precision.

How can we know the ruler's length to 0.2% without measuring any distances?

Because it depends only on physics we understand precisely and can compute from the ground up. The sound horizon integrates the sound speed over the pre-recombination expansion, and both are set by three numbers the CMB reads from the relative heights of the acoustic peaks: the photon density (fixed by the CMB temperature), the baryon density $\Omega_b h^2$ , and the matter density $\Omega_m h^2$ . Given those, $c_s(z)$ and $H(z)$ are fully determined, the integral is solved, and Boltzmann codes agree to a fraction of a percent. No galaxy distances enter — which is exactly why the ruler is a clean, external calibration.

Why is the sound horizon a better ruler than a standard candle like a Type Ia supernova?

A standard candle needs its intrinsic luminosity calibrated locally — through Cepheids, parallaxes, and a chain of cross-checks, each with its own systematics and possible evolution with cosmic time. The sound horizon needs none of that: its length is computed from linear early-universe physics far simpler and better understood than the explosion of a white dwarf. No local calibration step, no worry that the ruler evolved at high redshift, no astrophysical messiness in the standard itself. The dominant uncertainties sit cleanly on the theory side and shrink as knowledge of $\Omega_b h^2$ and $N_{\rm eff}$ improves.

What is the connection between the sound horizon and the Hubble tension?

The CMB fixes $\theta_s = r_\ast/D_{\rm com}(z_\ast)$ superbly, so the inferred distance — and through it $H_0$ — is only as good as the computed ruler $r_\ast$ . In standard $\Lambda$ CDM that chain gives $H_0\approx67$ , roughly $5\sigma$ below the local value near $73$ . New physics that shortened $r_\ast$ by a few percent would, at fixed $\theta_s$ , require a nearer last-scattering surface — achieved by a larger $H_0$ , moving the early-universe value toward the local one. That is why proposals like early dark energy and extra relativistic species all target the sound horizon: it is the one lever that raises the CMB-inferred $H_0$ without breaking the fit to the peaks.

Does dark energy affect the sound horizon?

Essentially not at all, and that is a feature. Dark energy is negligible before recombination — at $z\sim1000$ it is about a billionth of the energy density — so it drops out of the integral that sets $r_s$ ; the ruler is forged entirely by radiation and matter. Dark energy only enters later, in the distances $D_{\rm com}(z)$ and expansion rates $H(z)$ against which the ruler is compared. That clean separation is what makes the sound horizon so powerful: an early-universe standard with no late-time contamination, so laying it against low-redshift galaxies yields an almost purely geometric measurement of dark energy's effect on the expansion.