11.3 Baryon acoustic oscillations

Baryon Acoustic Oscillations

Origin of baryon acoustic oscillations

Before recombination, the universe was hot and dense enough that baryons (ordinary matter) and photons were tightly coupled through Thomson scattering, forming a single baryon-photon fluid. Quantum fluctuations generated during cosmic inflation seeded tiny density perturbations throughout this fluid, and those perturbations launched acoustic (sound) waves.

Two competing forces drove the oscillations:

• Gravity pulled the fluid inward, compressing it into overdense regions (potential wells).
• Radiation pressure from photons pushed back against that compression, causing the fluid to bounce outward.

This tug-of-war produced spherical sound waves expanding outward from each initial overdensity. At recombination (redshift $z \approx 1100$), the universe cooled enough for electrons and protons to combine into neutral hydrogen. Photons decoupled from baryons and streamed freely (becoming the CMB), and without radiation pressure to sustain them, the oscillations froze in place. The baryon overdensity stalled at whatever radius the sound wave had reached by that moment, permanently imprinting a preferred separation scale into the matter distribution.

BAO imprint on galaxy distribution

That frozen shell of slightly excess baryonic matter became a seed for later gravitational collapse. The result is a small but measurable excess probability of finding pairs of galaxies separated by a characteristic distance: the sound horizon at decoupling, roughly 150 comoving Mpc (~490 million light-years).

This signal shows up in two complementary statistical tools:

• Two-point correlation function — measures the excess probability of finding a galaxy pair at a given separation compared to a random distribution. BAO appears as a bump (the "BAO peak") near 150 comoving Mpc.
• Matter power spectrum — the Fourier transform of the correlation function, describing how clustering strength varies with spatial scale. BAO appears here as a series of wiggles (oscillatory features) superimposed on the smooth broadband shape of the spectrum.

Think of the correlation function as showing the bump directly in real space, while the power spectrum reveals the same information as a harmonic pattern in Fourier space.

BAO as a cosmic standard ruler

Because the sound horizon is set by well-understood early-universe physics (the baryon-to-photon ratio, the expansion rate before decoupling, and the recombination redshift), its physical size can be calculated precisely from CMB observations. That makes it a standard ruler: an object of known physical length that you can observe at various redshifts to extract distance information.

Cosmologists use the BAO scale in two directions:

1. Transverse (across the sky) — Comparing the known physical size of the BAO scale to its apparent angular size on the sky at redshift $z$ yields the angular diameter distance $D_A(z)$. This quantity depends on the integrated expansion history between us and that redshift.

2. Radial (along the line of sight) — Measuring the BAO scale in the redshift direction gives the Hubble parameter $H(z)$ at that redshift, since a known physical length maps to a redshift interval that depends on the local expansion rate.

Together, $D_A(z)$ and $H(z)$ at multiple redshifts trace out the universe's expansion history and geometry, much like measuring the same ruler at different distances on a map tells you about the map's projection.

BAO in cosmological constraints

BAO measurements are valuable partly because they are geometrical and relatively clean: the signal depends on large-scale clustering where nonlinear astrophysical effects are small. This makes BAO highly complementary to other probes like the CMB and Type Ia supernovae, each of which has different systematic uncertainties.

By tracking the BAO scale across a range of redshifts, surveys constrain:

1. Matter density parameter $\Omega_m$ — governs how much matter contributes to the total energy budget and how expansion decelerates at early times.
2. Dark energy equation of state $w$ — characterizes the pressure-to-density ratio of dark energy. A cosmological constant has $w = -1$; deviations from this would signal more exotic physics.

BAO data have been central to establishing the $\Lambda$CDM model as the standard cosmological framework, in which the universe is dominated by cold dark matter (CDM) and a cosmological constant $\Lambda$ responsible for accelerated expansion. The consistency of BAO results with independent CMB and supernova constraints is one of the strongest pieces of evidence for dark energy.

Current and upcoming spectroscopic surveys are pushing BAO precision further. DESI (Dark Energy Spectroscopic Instrument) is already collecting data, and ESA's Euclid mission launched in 2023. These programs will map tens of millions of galaxies and quasars across a wide redshift range, tightening constraints on $w$ and testing whether dark energy evolves with time.