Baryon Acoustic Oscillations

Baryon acoustic oscillations (BAO) are the fossil of a sound wave that rang through the infant universe. Before recombination, photons and baryons were welded by Thomson scattering into a single fluid whose radiation pressure fought gravity; every primordial overdensity therefore did not simply collapse but rang, launching a spherical pressure wave outward at nearly $c/\sqrt3$ . When the universe cooled enough for electrons and protons to combine at $z\simeq1090$ , the photons streamed free and the pressure vanished — but the baryons, released a little later at the drag epoch $z_\mathrm{d}\simeq1060$ , were stranded in a shell of comoving radius equal to the sound horizon $r_\mathrm{d}\approx147\ \mathrm{Mpc}$ . That shell is frozen into the clustering of matter to this day: every galaxy is slightly more likely to have a neighbour at a separation of $r_\mathrm{d}$ than at any nearby distance. The scientific power of the effect is that $r_\mathrm{d}$ is a standard ruler, its length fixed by linear physics we understand exactly and calibrated by the CMB. Lay it across the sky at redshift $z$ and its angular size measures the comoving angular diameter distance $D_M(z)$ ; lay it along the line of sight and the redshift interval it spans measures the Hubble rate $H(z)$ directly — the only clean geometric handle on $H(z)$ we possess. Sweeping the ruler from $z\approx0.1$ in galaxy surveys to $z\approx2.3$ in the Lyman- $\alpha$ forest maps the expansion history and weighs dark energy. This article synthesises the acoustic physics, the sound-horizon integral, and the distance relations into the tool that anchors modern cosmology.

Key takeaways

Before recombination the photon-baryon plasma carried sound at $c_s\approx c/\sqrt3$ ; each overdensity launched a spherical wave that stalled when the photons decoupled.
The wave freezes at the comoving sound horizon $r_\mathrm{d}\approx147\ \mathrm{Mpc}$ , a shell of baryons imprinting a preferred separation around every initial peak.
This appears as a single bump in the galaxy correlation function at $s\approx100\,h^{-1}\,\mathrm{Mpc}$ and as damped wiggles in the matter power spectrum.
Because $r_\mathrm{d}$ is CMB-calibrated, BAO are a standard ruler: the transverse scale gives $D_M(z)$ , the radial scale gives $H(z)$ , jointly pinning dark energy and curvature.
The signal is weak — only the baryon fraction $\Omega_b/\Omega_m\approx0.16$ sits in the shell — yet is measured to sub-percent precision by DESI, BOSS, and the Lyman- $\alpha$ forest.
Key idea: The comoving sound horizon at the baryon drag epoch, $r_\mathrm{d}=\int_{z_\mathrm{d}}^{\infty}c_s(z)\,\mathrm{d}z/H(z)\approx147\ \mathrm{Mpc}$ , is frozen into the clustering of matter and serves as a cosmological standard ruler that measures $D_M(z)$ transversely and $H(z)$ radially.

The core insight

The comoving sound horizon at the baryon drag epoch, $r_\mathrm{d}=\int_{z_\mathrm{d}}^{\infty}c_s(z)\,\mathrm{d}z/H(z)\approx147\ \mathrm{Mpc}$ , is frozen into the clustering of matter and provides a cosmological standard ruler that measures $D_M(z)$ transversely and $H(z)$ radially.

Where this lives today

BAO is now the backbone of the geometric distance ladder above the local universe. The Dark Energy Spectroscopic Instrument (DESI), a 5000-fibre spectrograph on the Mayall telescope, is mapping the acoustic ruler across $0.1<z<2.3$ with tens of millions of redshifts; its first releases deliver $D_M/r_\mathrm{d}$ and $D_H/r_\mathrm{d}=c/(Hr_\mathrm{d})$ at sub-percent precision, and — tantalisingly — hint at dark energy that may not be a pure cosmological constant. At the high-redshift frontier the Lyman- $\alpha$ forest turns the intergalactic medium into a tracer: absorption troughs toward distant quasars sample neutral hydrogen at $z\approx2$ – $3$ , and cross-correlating thousands of sightlines recovers the ruler deep in the matter-dominated era. Combined with the CMB's absolute calibration of $r_\mathrm{d}$ , these measurements furnish an inverse distance ladder that pins $H_0$ without Cepheids or supernovae — at the heart of the "Hubble tension" debate and every consistency test of the $\Lambda$ CDM concordance model.

The physical picture

Drop a stone into a still pond and a ring of ripples spreads at the wave speed. In the early universe each primordial overdensity is such a stone. The cold dark matter, having no pressure, stays put as a central cusp; the photon-baryon fluid rings outward as a spherical sound wave at $c_s\approx c/\sqrt3$ , expanding for as long as photons and baryons stay coupled. At recombination the coupling switches off, the driving pressure vanishes, and the baryon shell stops where it is — at a comoving radius equal to the sound horizon $r_\mathrm{d}$ . The overdensity now has two parts: a central dark-matter peak and a faint baryon shell at $r_\mathrm{d}$ , which gravity later mixes so that all matter carries a whisper of the shell. The first figure freezes this instant: the CDM cusp, the photons streaming away, and the stranded shell that becomes the ruler.

Figure

The acoustic shell freezes at recombination: cold dark matter stays as a central cusp while the photon-baryon sound wave stalls at comoving radius r_d ≈ 147 Mpc, set by r_d = ∫ c_s dz / H with c_s ≈ c/√3. Photons free-stream outward past the shell

In Fourier space the same story is a clock. Every mode was driven from the same phase at the same early time and stopped at the same drag epoch, so a mode of wavenumber $k$ is caught with phase $\propto\cos(k\,r_\mathrm{s})$ . The power spectrum acquires a cosine modulation of period fixed by the single length $r_\mathrm{d}$ — the "baryon wiggles" — just as a plucked string of fixed length has one fundamental. Silk damping smears the shortest wavelengths, so only the first few crests survive; transformed back to real space, the oscillation collapses into one localised bump at the ruler length. The second figure is what survey astronomers plot: the galaxy correlation function, weighted by $s^2$ to expose the bump.

Figure

The measured galaxy two-point correlation function weighted by s²: a broad declining clustering shape carries a single localised excess, the BAO peak, near s ≈ 100 h⁻¹ Mpc, which equals the comoving sound horizon r_d ≈ 147 Mpc after inserting h ≈ 0.68. It is a few-percent feature, not a tall spike

The feature is faint by construction. Cold dark matter outweighs baryons roughly five to one, and only the fraction $\Omega_b/\Omega_m\approx0.16$ ever sat in the acoustic shell; the pressureless CDM built a smooth central concentration with no ring. The BAO bump is therefore only a few-percent excess above the smooth clustering — that it is measured at all across billions of galaxies is a quiet triumph of observational cosmology. Its saving grace is robustness: because the ruler's length is set by pre-recombination physics, nonlinear growth, galaxy bias, and redshift-space distortions smear and shift the peak only slightly, in ways reconstruction partly undoes. The BAO scale is among the cleanest geometric measurements in astrophysics.

The core equations

The ruler is an integral. In the tightly coupled epoch a sound wave travels a comoving distance equal to the sound speed integrated against cosmic time; in redshift this is the sound horizon at the drag epoch, the length that freezes into the clustering.

r_\mathrm{d}=\int_{z_\mathrm{d}}^{\infty}\frac{c_s(z)}{H(z)}\,\mathrm{d}z\approx147\ \mathrm{Mpc} \qquad (1)

where: $r_\mathrm{d}$ — comoving sound horizon at the baryon drag epoch (the BAO ruler); $z_\mathrm{d}\approx1060$ — drag redshift, when baryons release from the photons; $c_s(z)$ — sound speed of the photon-baryon fluid; $H(z)$ — Hubble parameter. Reading it: the ruler lengthens if the fluid is faster (less baryon loading) or the universe expands more slowly. The sound speed is set by the fluid's stiffness and inertia. Photons supply all the pressure; baryons add mass but no pressure, so loading the fluid slows the wave below the pure-radiation value $c/\sqrt3$ .

c_s(z)=\frac{c}{\sqrt{3\left(1+R\right)}},\qquad R\equiv\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. Reading it: at high $z$ , $R\to0$ and $c_s\to c/\sqrt3\approx0.58c$ ; at the drag epoch $R\approx0.6$ , $c_s\approx0.45c$ . To use the ruler at redshift $z$ we need the distances it subtends. The transverse direction is governed by the comoving angular diameter distance.

D_M(z)=\frac{c}{H_0}\int_0^{z}\frac{\mathrm{d}z'}{E(z')},\qquad E(z)\equiv\frac{H(z)}{H_0} \qquad (3)

where: $D_M$ — comoving transverse distance (equals the line-of-sight comoving distance when flat); $E(z)$ — dimensionless expansion rate; $H_0$ — present Hubble constant. Reading it: a fixed ruler's angle shrinks as $1/D_M$ , so measuring it returns $D_M(z)/r_\mathrm{d}$ . The expansion rate follows from the energy budget of the standard model.

E(z)=\sqrt{\Omega_m(1+z)^3+\Omega_r(1+z)^4+\Omega_\Lambda+\Omega_k(1+z)^2} \qquad (4)

where: $\Omega_m,\Omega_r,\Omega_\Lambda,\Omega_k$ — present density parameters of matter, radiation, cosmological constant, and curvature. Reading it: radiation dominates the early integral that sets $r_\mathrm{d}$ , dark energy the late integral that sets $D_M$ . When the data cannot separate the two directions, surveys report a single isotropised distance — the geometric mean of two transverse and one radial mode.

D_V(z)=\left[D_M^2(z)\,\frac{cz}{H(z)}\right]^{1/3} \qquad (5)

where: $D_V$ — volume-averaged (dilation) distance, quoted as $D_V(z)/r_\mathrm{d}$ at low signal-to-noise. Reading it: two powers of the angular scale, one of the radial, cube-rooted. The ruler is read out of the clustering through the two-point correlation function, the real-space Fourier transform of the matter power spectrum.

\xi(s)=\int\frac{k^2\,\mathrm{d}k}{2\pi^2}\,P(k)\,\frac{\sin(ks)}{ks} \qquad (6)

where: $\xi(s)$ — two-point correlation function; $s$ — comoving pair separation; $P(k)$ — matter power spectrum. Reading it: the wiggles $\propto\cos(k\,r_\mathrm{d})$ in $P(k)$ transform to a single bump in $\xi(s)$ at $s\approx r_\mathrm{d}$ . Because surveys assume a fiducial cosmology to turn redshifts into distances, the fit returns two dilation parameters rescaling the observed ruler relative to that assumption — the Alcock-Paczynski split.

\alpha_\perp=\frac{D_M(z)/r_\mathrm{d}}{[D_M(z)/r_\mathrm{d}]_\mathrm{fid}},\qquad \alpha_\parallel=\frac{[H(z)\,r_\mathrm{d}]_\mathrm{fid}}{H(z)\,r_\mathrm{d}} \qquad (7)

where: $\alpha_\perp,\alpha_\parallel$ — transverse and radial scalings of the feature; $[\ldots]_\mathrm{fid}$ — quantities in the assumed fiducial cosmology. Reading it: an isotropic ruler seen through the wrong cosmology looks squashed, and that anisotropy alone constrains the true $D_M$ and $H$ . The ruler's absolute length is calibrated by the CMB, which measures the sister sound horizon at last scattering as a precise angle on the sky.

\theta_*=\frac{r_*}{D_M(z_*)}\approx0.0104\ \mathrm{rad} \qquad (8)

where: $\theta_*$ — acoustic angular scale measured by the CMB; $r_*$ — sound horizon at recombination $z_*\simeq1090$ , a few Mpc shorter than $r_\mathrm{d}$ ; $D_M(z_*)$ — comoving distance to last scattering. Reading it: the CMB fixes $r_*$ (hence $r_\mathrm{d}$ ) from $\Omega_b h^2$ and $\Omega_m h^2$ , handing BAO an absolutely calibrated ruler.

Derivation AThe sound speed of the photon-baryon fluid

The goal is to establish equation (2), that the coupled plasma carries sound at $c_s=c/\sqrt{3(1+R)}$ , and to see why loading the fluid with baryons drags the wave below the pure-radiation speed $c/\sqrt3$ . The adiabatic sound speed is the square root of a restoring pressure divided by the inertia it accelerates; here the pressure comes entirely from the photons while the inertia comes from both photons and baryons, so the single ratio $R$ of their momentum densities controls the whole result.

Step 1. Start from the thermodynamic definition, $c_s^2=(\partial p/\partial\rho)_S$ , at fixed entropy because the tightly coupled fluid oscillates adiabatically — Thomson scattering keeps photons and baryons at one temperature. Why: sound is a restoring-force phenomenon, and $\partial p/\partial\rho$ is exactly how pressure responds to a small adiabatic compression per unit inertia.

Step 2. Insert the two facts of the plasma: the pressure is purely radiative, $p=\tfrac13\rho_\gamma c^2$ , while the inertia is the total, $\rho=\rho_\gamma+\rho_b$ . Why: only the photon gas pushes back when the fluid is compressed, yet the cold baryons must be shoved along too — they weigh the wave down without stiffening it.

Step 3. Differentiate along the adiabat: $c_s^2=\tfrac13 c^2\,\mathrm{d}\rho_\gamma/(\mathrm{d}\rho_\gamma+\mathrm{d}\rho_b)=c^2/[3(1+\mathrm{d}\rho_b/\mathrm{d}\rho_\gamma)]$ . With $\rho_\gamma\propto a^{-4}$ and $\rho_b\propto a^{-3}$ , $\mathrm{d}\rho_b/\mathrm{d}\rho_\gamma=\tfrac34\rho_b/\rho_\gamma\equiv R$ . Why: the baryon mass riding each unit of photon energy is precisely the momentum-density ratio $R$ , which is why that combination appears.

Step 4. Assemble $c_s=c/\sqrt{3(1+R)}$ and check limits. At high $z$ , $R\to0$ and $c_s\to c/\sqrt3\approx0.577c$ ; near the drag epoch $R\approx0.6$ drops $c_s$ to $\approx0.45c$ . Since $R\propto a$ , the sound speed falls monotonically as the fluid grows heavier. Why: this declining $c_s(z)$ makes the sound-horizon integral (1) converge to a definite $\sim147$ Mpc.

Go deeper · Derivation B · 1 minEstimating the sound horizon from the early-universe integral

Derivation BEstimating the sound horizon from the early-universe integral

The goal is to evaluate the sound horizon of equation (1) with an analytic approximation that reproduces the standard $r_\mathrm{d}\approx147\ \mathrm{Mpc}$ and exposes how the ruler depends on just two numbers the CMB measures — the physical baryon density $\Omega_b h^2$ and the physical matter density $\Omega_m h^2$ . Because the integrand peaks near the drag epoch and the pre-recombination universe is a mix of matter and radiation, the whole integral collapses to a closed-form logarithm.

Step 1. Recast the integral in the scale factor $a$ via $\mathrm{d}z=-\mathrm{d}a/a^2$ , giving $r_\mathrm{d}=\int_0^{a_\mathrm{d}}c_s\,\mathrm{d}a/(a^2H)$ with $R=(3\Omega_b/4\Omega_\gamma)\,a\equiv R_0\,a$ growing linearly with $a$ . Why: scale factor is the natural early-universe clock and makes the loading $R\propto a$ manifest inside the integral.

Step 2. Approximate the expansion by its matter-plus-radiation form, $H(a)=H_0\sqrt{\Omega_m}\,a^{-3/2}\sqrt{1+a_\mathrm{eq}/a}$ , with $a_\mathrm{eq}=\Omega_r/\Omega_m$ the epoch of equality. Why: only radiation and matter contribute at $z\gtrsim1000$ , and this two-term $H(a)$ is exact for that mixture, making the integral elementary.

Step 3. The integrand reduces to $\mathrm{d}a/\sqrt{a(1+R_0a)(a+a_\mathrm{eq})}$ , integrating to the Eisenstein-Hu form $r_\mathrm{d}\approx\dfrac{2c}{3H_0\sqrt{\Omega_m a_\mathrm{eq}}\,\sqrt{R_\mathrm{eq}}}\ln\dfrac{\sqrt{1+R_\mathrm{d}}+\sqrt{R_\mathrm{d}+R_\mathrm{eq}}}{1+\sqrt{R_\mathrm{eq}}}$ . Why: the closed form shows the ruler shrinks when $\Omega_m h^2$ rises (faster early expansion) or $\Omega_b h^2$ rises (heavier, slower fluid) — the dependences BAO calibration hinges on.

Step 4. Insert Planck values $\Omega_m h^2\approx0.143$ , $\Omega_b h^2\approx0.0224$ , giving $a_\mathrm{eq}\approx1/3400$ , $R_\mathrm{d}\approx0.6$ , and $r_\mathrm{d}\approx147\ \mathrm{Mpc}$ , matching Boltzmann codes ( $147.1\pm0.3\ \mathrm{Mpc}$ ). Why: reproducing the exact value from a hand integral confirms the ruler is set by clean thermodynamics, not galaxy astrophysics.

Go deeper · Derivation C · 1 minReading distance and expansion from one frozen ruler

Derivation CReading distance and expansion from one frozen ruler

The goal is to show how a single fixed comoving length, viewed at redshift $z$ , splits into two independent measurements: its angular size across the sky returns $D_M(z)$ , and the redshift interval it spans along the line of sight returns the Hubble rate $H(z)$ . This is why BAO are prized — a standard candle yields only an integral of $1/H$ , whereas the acoustic ruler delivers $H(z)$ itself, locally, at every redshift it can be measured.

Step 1. Lay the ruler transverse to the line of sight. A proper transverse length $\mathrm{d}\ell_\perp$ subtends an angle through the angular diameter distance, $\mathrm{d}\ell_\perp=D_A(z)\,\mathrm{d}\theta$ with $D_A=D_M/(1+z)$ . Why: $D_A$ is by definition proper transverse size over subtended angle — the right converter for a ruler seen sideways.

Step 2. Convert to comoving by multiplying by $(1+z)$ : $r_\mathrm{d}=(1+z)\,\mathrm{d}\ell_\perp=D_M(z)\,\theta_\perp$ , hence $\theta_\perp=r_\mathrm{d}/D_M(z)$ . Why: the ruler is fixed in comoving units because it froze into the comoving matter distribution, so cancelling $(1+z)$ leaves a clean relation.

Step 3. Now lay it along the line of sight. A comoving length maps to a redshift interval through the Hubble element $\mathrm{d}r_\mathrm{comoving}=c\,\mathrm{d}z/H(z)$ ; setting this equal to the ruler gives $\Delta z_\parallel=r_\mathrm{d}\,H(z)/c$ . Why: the redshift a fixed length spans is dictated entirely by how fast the universe expands there — that is $H(z)$ .

Step 4. Combine: $\theta_\perp$ delivers $D_M(z)/r_\mathrm{d}$ and $\Delta z_\parallel$ delivers $H(z)\,r_\mathrm{d}/c$ , their ratio being the Alcock-Paczynski test of equation (7). Why: two orthogonal projections of one isotropic ruler carry independent information — transverse geometry and radial expansion — making BAO a two-dimensional probe, not a single distance.

Limiting cases

How the BAO feature presents itself depends on the observable and the viewing direction:

Case	Result	Meaning
Transverse (angular)	$\theta_\mathrm{BAO}(z)=\dfrac{r_\mathrm{d}}{D_M(z)}$	The ruler across the sky subtends an angle set by $D_M$ ; measuring $\theta$ yields $D_M(z)/r_\mathrm{d}$ .
Radial (line of sight)	$\Delta z_\mathrm{BAO}(z)=\dfrac{r_\mathrm{d}\,H(z)}{c}$	Along the sightline the ruler spans a redshift interval set by $H(z)$ ; measuring $\Delta z$ yields $c/[r_\mathrm{d}H(z)]$ .
Isotropic (averaged)	$D_V(z)=\left[D_M^2(z)\,\dfrac{cz}{H(z)}\right]^{1/3}$	At low signal-to-noise only the dilation distance is measured — two transverse and one radial mode; surveys report $D_V/r_\mathrm{d}$ .
Power-spectrum wiggles	$P(k)\simeq P_\mathrm{smooth}(k)\,[1+A(k)\sin(k\,r_\mathrm{d})]$	Fixed period $\Delta k=2\pi/r_\mathrm{d}$ ; the envelope $A(k)$ is cut down by Silk damping, so only a handful of crests survive.
Baryon-free ( $\Omega_b\to0$ )	feature amplitude $\to0$	No baryons, no acoustic shell; wiggles and bump vanish, leaving only smooth CDM clustering.

Common trapThe BAO ruler is the sound horizon at recombination

It is not — and the distinction is a real percent-level error. The photons last scatter at $z_*\simeq1090$ , but the baryons remain dragged by residual photon friction slightly longer, releasing only at the drag epoch $z_\mathrm{d}\simeq1060$ . Because BAO trace baryons, the ruler is the sound horizon accumulated up to $z_\mathrm{d}$ , namely $r_\mathrm{d}$ , a few Mpc longer than the CMB's $r_*$ ; quoting $r_*$ where $r_\mathrm{d}$ belongs biases every inferred distance. A second trap: do not equate the correlation-function scale $s\approx100\,h^{-1}\,\mathrm{Mpc}$ with the ruler $r_\mathrm{d}\approx147\,\mathrm{Mpc}$ — they are the same length in different units, agreeing only after multiplying by $h\approx0.68$ , since $100/0.68\approx147$ .

Worked examples

1 · The ruler at z = 0.57 (BOSS CMASS, fully worked)

A galaxy sample peaks at $z=0.57$ . Take flat $\Lambda$ CDM with $H_0=67.4\ \mathrm{km\,s^{-1}Mpc^{-1}}$ , $\Omega_m=0.315$ , $\Omega_\Lambda=0.685$ , and $r_\mathrm{d}=147.1\ \mathrm{Mpc}$ . Find $H(z)$ , the transverse angle, the radial span, and $D_V/r_\mathrm{d}$ .

Step 1. Expansion rate. $E(0.57)=\sqrt{0.315(1.57)^3+0.685}=\sqrt{1.219+0.685}=\sqrt{1.904}=1.380$ , so $H(0.57)=67.4\times1.380=\mathbf{93.0\ km\,s^{-1}Mpc^{-1}}$ .

Step 2. Comoving distance. With $c/H_0=299792/67.4=4448\ \mathrm{Mpc}$ , integrating $\int_0^{0.57}\mathrm{d}z'/E(z')$ gives $D_M(0.57)=\mathbf{2183\ Mpc}$ , so $D_M/r_\mathrm{d}=2183/147.1=\mathbf{14.8}$ .

Step 3. Transverse angle. $\theta_\mathrm{BAO}=r_\mathrm{d}/D_M=147.1/2183=0.0674\ \mathrm{rad}=\mathbf{3.9^\circ}$ — a several-degree feature on the sky, which is why resolving it needs surveys spanning thousands of square degrees.

Step 4. Radial span. $\Delta z_\parallel=r_\mathrm{d}H(z)/c=147.1\times93.0/299792=\mathbf{0.0456}$ — a redshift slice of only $\sim0.05$ , so redshifts must be measured to better than $\sim0.001$ .

Step 5. Isotropic distance. $cz/H=299792\times0.57/93.0=1837\ \mathrm{Mpc}$ , so $D_V=[(2183)^2\times1837]^{1/3}=2061\ \mathrm{Mpc}$ , giving $D_V/r_\mathrm{d}=2061/147.1=\mathbf{14.0}$ — a pure geometric distance to $z=0.57$ , measured to sub-percent precision by BOSS and calibrated only by early-universe physics.

2 · The sound speed and loading at the drag epoch

Compute the loading $R$ and sound speed $c_s$ at the drag epoch $z_\mathrm{d}=1060$ , using $\Omega_b h^2=0.0224$ , $\Omega_\gamma h^2=2.47\times10^{-5}$ .

Step 1. Loading. The prefactor $3\Omega_b/4\Omega_\gamma=0.0672/(9.88\times10^{-5})=680$ , so $R_\mathrm{d}=680/1061=\mathbf{0.64}$ — well loaded, but photons still dominate the inertia.

Step 2. Sound speed. $c_s=c/\sqrt{3(1+R)}=c/\sqrt{4.92}=\mathbf{0.451\,c}$ . Against the deep-radiation value $c/\sqrt3=0.577c$ , baryon loading has slowed the wave by about 22 percent by the time it freezes.

Step 3. Sanity on the ruler. Taking the mean sound speed $\bar c_s\approx0.5c$ over the conformal time to the drag epoch, $\eta_\mathrm{d}\approx280\ \mathrm{Mpc}/c$ , gives $r_\mathrm{d}\sim\bar c_s\,\eta_\mathrm{d}\approx\mathbf{140\ Mpc}$ , within ten percent of the exact $147\ \mathrm{Mpc}$ — the ruler is genuinely sound speed times the age of the wave.

3 · From correlation-function units to the ruler

Surveys quote the BAO peak at $s\approx100\,h^{-1}\,\mathrm{Mpc}$ . Show this equals $r_\mathrm{d}\approx147\ \mathrm{Mpc}$ , and estimate why the bump is only a few percent.

Step 1. Units. With $h=0.68$ , one $h^{-1}\,\mathrm{Mpc}=1/0.68=1.47\ \mathrm{Mpc}$ , so $s=100\,h^{-1}\,\mathrm{Mpc}=\mathbf{147\ Mpc}=r_\mathrm{d}$ — the same length, the $h^{-1}$ convention simply deferring the choice of $H_0$ .

Step 2. Amplitude. Only the baryon fraction rides in the shell: $\Omega_b/\Omega_m\approx0.0224/0.143=\mathbf{0.157}$ . In the total matter field the feature is diluted by roughly this fraction (and further by projection and damping), leaving a few-percent bump in $\xi(s)$ — small, but coherent across the whole survey volume and thus detectable with enough galaxies. The BAO peak is a real length ( $r_\mathrm{d}$ ) carried by a minority of the mass, which is why it is both robust and faint.

Physics of the portrait

BAO began as a prediction that the CMB's acoustic peaks must shadow the galaxy distribution, and became, over three decades, the sharpest geometric ruler in cosmology. No single person owns that arc, but one name recurs at every turn.

Origins · 1 minDaniel Eisenstein and how this was discovered

Daniel Eisenstein

born 1970 · large-scale structure and observational cosmology

Daniel Eisenstein is the physicist most tightly bound to baryon acoustic oscillations as a working tool of cosmology. In 1998 he and Wayne Hu published the fitting formulae for the transfer function and the sound horizon that let anyone compute $r_\mathrm{d}$ from $\Omega_b h^2$ and $\Omega_m h^2$ without a Boltzmann code — the equations of this article's second derivation. The theory had deep roots: Sakharov noted plasma oscillations in 1968, and Peebles and Yu, with Sunyaev and Zel'dovich, predicted the acoustic imprint on matter clustering in 1970. But it was Eisenstein who, in 2005, led the Sloan Digital Sky Survey team that first saw the bump — a few-percent excess in the correlation function of 46,000 luminous red galaxies at exactly $100\,h^{-1}\,\mathrm{Mpc}$ (announced alongside the 2dF Galaxy Redshift Survey detection by Cole and collaborators). He went on to devise density-field reconstruction and to direct the BOSS and DESI surveys that turned BAO from a detection into precision cosmology — the effect's whole journey, from a formula, to a photograph, to a percent-level map of cosmic expansion.

History

1968

— Sakharov notes acoustic oscillations in the primordial plasma

1970

— Peebles and Yu, and Sunyaev and Zeldovich, predict the acoustic imprint on matter clustering

1998

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

2005

— Eisenstein and the SDSS team, and the 2dFGRS team, detect the BAO peak at 100 per-h Mpc

2014

— BOSS measures BAO in the Lyman-alpha forest at redshift 2.3

2024

— DESI delivers percent-level BAO distances across redshift 0.1 to 2.3

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 8–9 · Dodelson, Modern Cosmology, 2nd ed., Ch. 8–9 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 4 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 27–30 · Binney & Tremaine, Galactic Dynamics, 2nd ed. (linear perturbation background). Calibrated values used throughout: $r_\mathrm{d}=147.1\pm0.3\ \mathrm{Mpc}$ , $z_\mathrm{d}\approx1060$ , $z_*\approx1090$ , $\theta_*\approx0.0104$ (Planck 2018). The Eisenstein–Hu closed-form integral evaluated at $\Omega_m h^2\approx0.143$ , $\Omega_b h^2\approx0.0224$ reproduces $r_\mathrm{d}\sim147\ \mathrm{Mpc}$ ; the survey convention $100\,h^{-1}\,\mathrm{Mpc}$ equals $147\ \mathrm{Mpc}$ at $h=0.68$ ; and the $z=0.57$ benchmark gives $H=93.0\ \mathrm{km\,s^{-1}Mpc^{-1}}$ , $D_M=2183\ \mathrm{Mpc}$ , $D_V/r_\mathrm{d}=14.0$ .

Prerequisites. Acoustic Oscillations in the Photon-Baryon Fluid · The Sound Horizon and the Standard Ruler · Transfer Function and the Matter Power Spectrum · Density Perturbations and the Density Contrast · Silk Damping. Leads to. Dark Energy and the Cosmological Constant · The ΛCDM Concordance Model · The CMB Anisotropy Power Spectrum.

Ask the tutor: Ask me to walk through how the CMB's angle $\theta_*$ actually calibrates the ruler $r_\mathrm{d}$ , or to work a DESI-style $D_M/r_\mathrm{d}$ and $H r_\mathrm{d}$ measurement at any redshift you choose.

Why is the ruler the drag-epoch horizon r_d and not the recombination horizon r_star?

Because BAO trace baryons, and baryons decouple a little later than the photons do. Photons last scatter at $z_*\simeq1090$ , setting the CMB's ruler $r_*$ ; but so long as a few free electrons remain, residual photon drag keeps nudging the baryons, which release at the drag epoch $z_\mathrm{d}\simeq1060$ . The baryon shell therefore freezes at the sound horizon up to $z_\mathrm{d}$ — that is $r_\mathrm{d}$ , a few Mpc longer than $r_*$ . Using the wrong one biases inferred distances at the percent level, which matters for precision cosmology.

Why does BAO appear as one bump in galaxies but many peaks in the CMB?

Two projections of the same oscillation. In the matter distribution we see a single spherical baryon shell of radius $r_\mathrm{d}$ around each overdensity, giving one localised excess in $\xi(s)$ . The CMB instead captures the entire oscillating field at a single instant — last scattering — and projects it onto the sky, so modes caught at different phases produce a whole harmonic series of peaks. Silk damping then wipes out all but the first crest in the matter case, leaving one clean bump, while the CMB retains many peaks at scales the diffusion length has not yet erased.

How can BAO measure H(z) directly when distances are always integrals?

A standard candle such as a Type Ia supernova gives only the luminosity distance, an integral of $1/H$ that smears the expansion history together. The BAO ruler seen along the line of sight does something different: a fixed comoving length spans a redshift interval $\Delta z=r_\mathrm{d}H(z)/c$ , so the radial acoustic scale returns $H(z)$ at that redshift locally, not as an integral. That is the unique power of a standard ruler over a candle — the instantaneous expansion rate at each redshift it can be resolved, exactly what dark-energy studies need.

What are the Alcock-Paczynski parameters, and why are two needed?

To turn observed angles and redshifts into distances, a survey must assume a fiducial cosmology. The dilation parameters $\alpha_\perp$ and $\alpha_\parallel$ measure how much the real transverse and radial scales differ from that assumption, tracking $D_M/r_\mathrm{d}$ and $Hr_\mathrm{d}$ . Two are needed because the ruler is isotropic in true comoving space: if the assumed cosmology is wrong, a spherical shell appears squashed, and that anisotropy is itself a measurement. One parameter captures only an overall scale; two capture the full geometry, separating distance from expansion.

Why is the BAO peak so robust to messy galaxy physics?

Because the ruler's length was fixed before any galaxy existed, by clean linear thermodynamics of the photon-baryon plasma. Nonlinear growth, galaxy bias, and redshift-space distortions all shift and broaden the peak — but only by a few percent, in ways perturbation theory and reconstruction largely remove. A single galaxy's history changes nothing about $r_\mathrm{d}$ ; it merely blurs the statistical feature. That separation of scales is why BAO distances are among the least systematics-prone in cosmology.

What is reconstruction, and why does it sharpen the peak?

Over cosmic time, bulk flows and nonlinear collapse drag galaxies around, smearing the sharp acoustic shell into a broader, lower bump. Reconstruction uses the observed density field to estimate the displacement of each galaxy from its initial position via the Zel'dovich approximation, then shifts the galaxies back. The undone bulk flows re-sharpen the peak, recovering close to the linear-theory precision the survey would have had if structure never went nonlinear. It is now routine in every major BAO analysis, improving distance errors by tens of percent.