Cosmological Parameters
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At Redshift z
Instructions
- Adjust H₀ to see degeneracy
- Change Ωₘ for curvature effects
- Vary ΩΛ to see dark energy impact
- Compare with SNe Ia constraints
Use the cosmic sound horizon as a standard ruler—measure angular diameter distance from BAO peak positions
Baryon acoustic oscillations (BAO) are the imprints of sound waves in the early universe, frozen into the large-scale distribution of galaxies. These oscillations create a characteristic scale—the sound horizon at the drag epoch—that serves as a "standard ruler" for measuring cosmological distances and constraining dark energy.
Before recombination (z ~ 1100), photons and baryons were tightly coupled through Thomson scattering, forming a photon-baryon fluid. Perturbations in this fluid propagated as sound waves with speed c_s ≈ c/√3. At the drag epoch (z ~ 1020), the fluid decoupled from dark matter, freezing the sound horizon at:
r_s = ∫₀^(t_drag) c_s dt / a ≈ 147 Mpc
This scale appears as a peak in the galaxy correlation function at ~150 Mpc separation.
The BAO scale appears as an angular feature in galaxy surveys. If we know the physical size (r_s) and measure the angular scale (θ), we can infer the angular diameter distance:
D_A(z) = r_s / θ_BAO(z)
In a flat ΛCDM universe: D_A(z) = D_c(z) / (1+z), where D_c is the comoving distance that depends on H(z) = H₀ E(z) with E(z) = √[Ωₘ(1+z)³ + ΩΛ].
BAO measurements break the degeneracy between matter and dark energy in two ways:
Combined with CMB measurements (which fix r_s) and SNe Ia (which measure D_L), BAO provides powerful constraints on Ωₘ, ΩΛ, and H₀.
The BAO feature was first detected in 2005 by SDSS and 2dFGRS. Modern surveys like BOSS, eBOSS, and DESI measure the BAO scale at multiple redshifts (0.1 < z < 2.5) with ~1% precision. These measurements confirmed accelerated expansion and constrain dark energy models independent of supernovae.
BAO is particularly valuable because it measures absolute distances (unlike SNe Ia magnitudes) and is less sensitive to systematic effects like dust extinction or stellar evolution. The combination of BAO + CMB + SNe Ia provides the tightest constraints on cosmological parameters, reducing uncertainties by factors of 2-3 compared to any single probe.