Baryon Acoustic Oscillations

Baryon acoustic oscillations (BAO) are regular, periodic spatial fluctuations in the density and clustering of visible baryonic matter (galaxies and intergalactic neutral hydrogen) across the large-scale structure of the universe, originating from spherical acoustic pressure waves that propagated through the relativistic photon–baryon plasma during the first 380,000 years after the Big Bang. Predicted theoretically by Andrei Sakharov (1966), P. J. E. Peebles and J. T. Yu (1970), and Rashid Sunyaev and Yakov Zel'dovich (1970), these primordial sound waves traveled outward from early dark-matter gravitational overdensities at nearly 57% of the speed of light until the universe cooled sufficiently for protons and electrons to combine into neutral hydrogen atoms at the epoch of recombination (z_* ≈ 1090). At that moment, photons decoupled to stream freely as the cosmic microwave background (CMB), abruptly dropping the radiation pressure to zero and freezing an spherical shell of enhanced baryonic density at a characteristic comoving sound horizon radius r_d ≈ 147.09 ± 0.26 megaparsecs (~480 million light-years). In modern observational cosmology, this frozen acoustic scale serves as a geometric standard ruler for mapping the expansion history of the universe H(z) and the equation of state of dark energy.

During the radiation- and early matter-dominated eras prior to recombination (t < 380,000 years, temperature T > 3,000 K), the universe was filled with a hot, ionized plasma of protons, helium nuclei, free electrons, relativistic photons, and non-baryonic cold dark matter (CDM):

1. Tight Thomson Coupling: Free electrons scattered photons continuously via Thomson scattering (mean free path smaller than the Hubble horizon), while Coulomb electromagnetic forces bound protons and helium nuclei tightly to the electrons. Consequently, photons and baryons behaved mechanically as a single relativistic photon–baryon fluid, whereas collisionless cold dark matter interacted solely through gravity.
2. Competing Gravitational Infall and Radiation Pressure: Consider an isolated primordial gravitational potential perturbation (seeded by quantum fluctuations during cosmic inflation) containing an overdensity of both cold dark matter and the photon–baryon fluid. Gravity pulled matter inward toward the center of the potential well, while the intense thermal radiation pressure of the trapped photons pushed the photon–baryon fluid outward, launching a spherical compressional sound wave.
3. Relativistic Sound Speed (c_s): The adiabatic sound speed c_s of the coupled photon–baryon fluid is determined by the ratio of the baryon momentum density to the photon momentum density, R(z) = (3ρ_b) / (4ρ_γ) ≈ 0.65 [1,090 / (1 + z)]:
   • Sound Speed Formula: c_s(z) = c / √[3(1 + R(z))].
   • In the very early universe (R → 0), the sound speed approached the relativistic limit c_s = c / √3 ≈ 0.577*c* (~173,000 km s⁻¹), slowing gradually to ~0.45*c* as baryon inertia grew near recombination.
4. Recombination and the Baryon Drag Epoch (z_d ≈ 1059.9): As cosmic expansion cooled the plasma below ~3,000 K, electrons combined with protons (p + e⁻ → H + γ), and the photon scattering rate dropped below the Hubble expansion rate. Shortly after photon decoupling (z_* ≈ 1089.8), baryons were released from photon Compton drag at the drag epoch (z_d ≈ 1059.9). Without radiation pressure to drive the wave forward, the acoustic compression wave stalled at the comoving sound horizon:
   • Sound Horizon Integral: r_d = ∫_zd^∞ [c_s(z) / H(z)] dz = 147.09 ± 0.26 Mpc (Planck 2018 ΛCDM calibration).

Because the collisionless cold dark matter remained at the center of the initial perturbation while the baryonic acoustic shell stalled at radius r_d ≈ 147 Mpc, subsequent gravitational instability grew galaxies both at the central dark-matter core and along the 147 Mpc spherical shell—creating a subtle (~1%) statistical excess of galaxy pairs separated by ~147 comoving megaparsecs compared to adjacent separation scales.

Baryon acoustic oscillations manifest differently depending on whether cosmologists analyze the universe in Fourier space (wavenumber k in Mpc⁻¹) or configuration space (comoving pair separation s in Mpc):

• Two-Point Correlation Function ξ(s): In real three-dimensional space, the two-point galaxy correlation function ξ(s) measures the excess probability above random of finding two galaxies separated by comoving distance s. The stalled spherical sound shell appears as a single localized BAO peak centered near s ≈ 105 h⁻¹ Mpc (~147 Mpc for h = 0.674), broadened to a width of ~10–15 Mpc by Silk damping (photon diffusion out of small-scale perturbations prior to recombination) and late-time nonlinear gravitational bulk flows.
• Matter Power Spectrum P(k): Taking the Fourier transform of a delta-function shell of radius r_d yields a damped sinusoid, sin(k r_d) / (k r_d). Consequently, in the matter power spectrum P(k) (and in the angular power spectrum C_ℓ of the CMB temperature anisotropies), the single real-space sound horizon scale appears as a harmonic series of acoustic peaks and troughs at wavenumbers k_n ≈ nπ / r_d.

Because the baryonic density of the universe (Ω_bh² ≈ 0.0224) is roughly one-fifth of the cold dark matter density (Ω_ch² ≈ 0.120), the BAO feature in the post-recombination matter distribution has a much smaller fractional amplitude (~5–10% wiggle in P(k)) than the acoustic peaks in the CMB, requiring three-dimensional redshift maps of hundreds of thousands to millions of galaxies across cubic-gigaparsec volumes.

In 2005, the BAO peak was detected simultaneously and independently by two landmark spectroscopic galaxy redshift surveys:

1. Sloan Digital Sky Survey (SDSS): Daniel J. Eisenstein and collaborators measured the two-point correlation function of 46,748 luminous red galaxies (LRGs) over 3,816 deg² at redshifts 0.16 < z < 0.47, detecting the 100 h⁻¹ Mpc acoustic peak at 3.4σ significance.
2. Two-Degree-Field Galaxy Redshift Survey (2dFGRS): Shaun Cole and collaborators detected the harmonic baryon oscillations in the Fourier power spectrum P(k) of 221,414 galaxies at z ≈ 0.15.

Subsequent generations of spectroscopic surveys have mapped the BAO standard ruler across 11 billion years of cosmic history:

Because the comoving sound horizon r_d is calibrated to 0.18% precision by Planck CMB observations, measuring the apparent size of the BAO sphere at redshift z along two orthogonal axes decomposes the cosmic expansion rate into transverse and radial components (Alcock–Paczynski test):

• Transverse Angular Separation (Δθ): Measuring the angle Δθ = r_d / D_M(z) subtended by the BAO scale perpendicular to the line of sight directly yields the comoving angular diameter distance D_M(z) = (1 + z)D_A(z) = c ∫₀^z dz' / H(z').
• Line-of-Sight Redshift Separation (Δz): Measuring the redshift interval Δz = r_d H(z) / c spanned by the BAO shell parallel to the line of sight directly yields the instantaneous Hubble expansion parameter H(z) (expressed as the Hubble distance D_H(z) = c / H(z)).
• Density-Field BAO Reconstruction: Because late-time peculiar velocities and gravitational clustering smear the BAO peak by ~50%, cosmologists apply Eisenstein–Seo–White density-field reconstruction—inverting the linearized continuity equation (Zel'dovich approximation) from the observed galaxy density field to shift galaxies back along their displacement vectors by ~5–10 Mpc, restoring the sharp linear-regime BAO peak and halving distance uncertainties.

1. Peebles, P. J. E., & Yu, J. T. (1970). Primeval adiabatic perturbation in an expanding universe. The Astrophysical Journal, 162, 815–836. https://doi.org/10.1086/150713
2. Sunyaev, R. A., & Zel'dovich, Y. B. (1970). Small-scale fluctuations of relic radiation. Astrophysics and Space Science, 7(1), 3–19. https://doi.org/10.1007/BF00653471
3. Eisenstein, D. J., Zehavi, I., Hogg, D. W., Scoccimarro, R., Blanton, M. R., Nichol, R. C., ... & York, D. G. (2005). Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies. The Astrophysical Journal, 633(2), 560–574. https://doi.org/10.1086/466512
4. DESI Collaboration, Adame, A. G., Aguilar, J., Ahlen, S., Alam, S., Alexander, D. M., ... & Zou, H. (2025). DESI 2024 VI: Cosmological constraints from the measurements of baryon acoustic oscillations. Journal of Cosmology and Astroparticle Physics, 2025(02), 021. https://doi.org/10.1088/1475-7516/2025/02/021