The BAO

The early universe was a sphere of sound. A plasma of photons and baryons, hot and dense, filled with pressure waves rippling outward from every overdensity, like a cosmic drum that had been struck at the moment of creation. And imprinted on the distribution of galaxies — on the largest scales we can observe — is the fossil of those sound waves.

Baryon acoustic oscillations. The sound of the early universe, frozen into the structure of the cosmos.

The plasma as a fluid

In the first 380,000 years of the universe, matter and radiation were locked together in a hot, dense plasma. Electrons and protons were ionized — not bound into atoms — and photons could not travel freely because they scattered off electrons via Thomson scattering. The mean free path of a photon was microscopic. Photons and baryons formed a single coupled fluid: the photon-baryon fluid.

This fluid had two forces acting on it. Gravity pulled it inward toward overdensities. Radiation pressure pushed it outward. The result was oscillation. The fluid compressed, expanded, compressed, expanded — a standing wave in a medium that filled all of space.

Every primordial density perturbation acted as a center of oscillation. Gravity pulled the photon-baryon fluid inward. As it compressed, the radiation pressure increased. Eventually the pressure exceeded gravity, and the fluid bounced outward. As it expanded, gravity reasserted itself, and the fluid fell back inward. The cycle repeated.

The oscillation frequency was set by the speed of sound in the photon-baryon fluid, which was:

$$c_s = \frac{c}{\sqrt{3(1 + R)}}$$

where $R = 3\rho_b/4\rho_\gamma$ is the ratio of baryon to photon density. In the early universe, $R \ll 1$, so $c_s \approx c/\sqrt{3} \approx 0.577c$. The speed of sound was half the speed of light. Sound waves propagated outward from every overdensity at 57% the speed of light.

The sound horizon

Over 380,000 years, these sound waves propagated outward. The maximum distance any wave could travel by the time the plasma recombined — the time when electrons and protons combined into neutral hydrogen, photons decoupled, and the oscillation ended — is called the sound horizon:

$$r_s = \int_0^{t_{\text{rec}}} c_s(t),dt \approx 147,\text{Mpc}$$

approximately 147 megaparsecs, or about 480 million light-years. This is the radius of the spherical shell that sound waves traveled from every primordial overdensity before recombination.

When recombination happened, the photon-baryon fluid decoupled. Photons streamed freely — becoming the CMB. Baryons stopped oscillating. The acoustic waves froze. At that moment, every primordial overdensity was surrounded by a shell of enhanced baryon density at exactly the sound horizon radius.

Those shells persisted. Even after recombination, the baryons remained slightly enhanced at that characteristic scale. Dark matter, which never felt radiation pressure, had been collapsing gravitationally all along, but the baryons carried the acoustic signature. When galaxies eventually formed, they formed preferentially at the locations of those shells.

The signature in the galaxy distribution

If you take a catalog of millions of galaxies and measure the separation between every pair, you get a two-point correlation function $\xi(r)$. In a perfectly random distribution, $\xi(r) = 0$ everywhere. In a distribution with BAO, you see a peak at $r = r_s$ — the sound horizon.

This peak was first detected in the Sloan Digital Sky Survey in 2005, in the distribution of luminous red galaxies. Since then, it has been measured by BOSS, eBOSS, DESI, and other surveys, with increasing precision. The BAO peak is a standard ruler: a feature of known physical size that can be used to measure distances in the universe.

The BAO measurement works because the sound horizon is calculable from first principles. Given the densities of baryons, dark matter, and photons — all independently constrained by the CMB — the sound horizon is predicted with sub-percent precision. If you measure the apparent angular size of the BAO feature in the sky and its apparent size in redshift space, you get the angular diameter distance $D_A(z)$ and the Hubble parameter $H(z)$ at the redshift of the galaxy sample. These are direct probes of the expansion history of the universe.

Why BAO matters

BAO is one of the cleanest probes of dark energy. Unlike supernovae, which require careful calibration of absolute brightness, BAO is a geometric measurement. The sound horizon is a physical length scale determined by the physics of the early universe. Its absolute size is known from CMB data. Measuring its apparent size at different redshifts tells you the expansion history $H(z)$ and $D_A(z)$ without any astrophysical calibration.

BAO also constrains the nature of dark energy. If dark energy is a cosmological constant ($w = -1$), the expansion history follows a specific curve. If dark energy is dynamical ($w \neq -1$), the curve is different. BAO measurements at multiple redshifts can distinguish between these scenarios. Current BAO data from eBOSS and DESI are consistent with a cosmological constant but leave room for dynamical dark energy. The precision is improving.

The acoustic peaks in the CMB

BAO is not only visible in the galaxy distribution. It is also visible — in a more direct form — in the CMB. The acoustic oscillations of the photon-baryon fluid before recombination created the acoustic peaks in the CMB power spectrum. The positions and heights of these peaks encode the same physics as the BAO feature in galaxy surveys, but at redshift $z \approx 1100$ instead of $z \approx 0.5$ to $2.5$.

The first peak is at angular scale $\ell \approx 220$, corresponding to the angular size of the sound horizon on the sky. Its position tells us the universe is flat. The ratio of odd to even peaks tells us the baryon density. The damping of peaks at small scales tells us the diffusion length of photons before recombination. The entire CMB power spectrum is an acoustic spectrum — the spectrum of sound waves in the early universe.

The BAO feature in the galaxy distribution is the late-time remnant of those same acoustic waves. The CMB shows the waves at the moment they stopped. The galaxy distribution shows the scars they left behind.

The evidence

The BAO signal has been measured to high significance:

• SDSS LRG (2005): first detection, $5.1\sigma$
• BOSS DR12 (2017): $7.6\sigma$ detection, constraining $H(z)$ and $D_A(z)$ at $z = 0.38, 0.51, 0.61$
• eBOSS (2020): BAO at $z = 2.33$ from Lyman-alpha forest, extending the redshift range
• DESI (2024): BAO measurements at 7 redshift bins from $z = 0.1$ to $z = 3.5$, with percent-level precision

The BAO feature is real. It is the fossil of sound waves in the early universe. And it is one of the most powerful tools for measuring the geometry and expansion history of the cosmos.

The universe sang. We can still hear the echo.