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BAO distance calculator

Baryon acoustic oscillations measure distances in units of the sound horizon r_d, a standard ruler set in the early universe.

The interactive calculator loads here: change the redshift and the cosmological parameters (Ω_m, Ω_Λ, Ω_r, w₀, w_a, H₀) and the result and plot update instantly.

What it is

For roughly the first 400 000 years, sound waves travelled through the hot plasma of photons and baryons. When the baryons were released (the drag epoch, z ≈ 1060), the waves froze in, leaving a slight excess of galaxy pairs separated by the comoving distance sound had travelled: the sound horizon r_d ≈ 147 Mpc. This is the baryon acoustic oscillation (BAO) scale.

Galaxy surveys measure the BAO scale across the line of sight, giving the transverse comoving distance D_M/r_d, and along it, giving D_H/r_d = c/(H(z) r_d). When only an average over directions can be measured they quote D_V/r_d. Because the results are ratios, BAO alone constrain the shape of the expansion history and the product H₀ r_d; the value of r_d comes from the physics of the early universe.

iCosmos computes r_d from Ω_bh² and Ω_mh² (set them under “Survey & CMB parameters”) with a fit to the CAMB Boltzmann code, within 2 × 10⁻⁵ of CAMB when curvature and dark energy are negligible at early times (up to about 0.07 % at the edge of its validity range), assuming the standard radiation content and massless neutrinos. Planck and DESI include neutrinos with Σm_ν = 0.06 eV, which iCosmos treats as ordinary matter; for their models this makes the BAO ratios about 0.1 % different from a full calculation.

Formula

• D_H(z) = c / H(z)
• D_M(z) = transverse comoving distance (D_C when flat)
• D_V(z) = [z D_M²(z) D_H(z)]^1/3
• r_d = ∫ c_s(z) dz / H(z), integrated from the drag epoch z_d to z = ∞, with c_s = c / √(3(1 + R)) and R = 3ρ_b/(4ρ_γ)

Worked example

For the Planck 2018 model (H₀ = 67.66, Ω_m = 0.3111, Ω_bh² = 0.02242) the sound horizon is r_d = 147.05 Mpc. Planck itself quotes 147.21 Mpc; the small difference comes from Planck’s massive neutrinos (Σm_ν = 0.06 eV), which iCosmos treats as ordinary matter.

At z = 0.934, the effective redshift of DESI’s combined luminous red galaxy and emission-line galaxy sample, this model predicts D_M/r_d = 21.955 and D_H/r_d = 17.575. DESI DR2 measures D_M/r_d = 21.576 ± 0.152.

Common questions

What did DESI DR2 find?

DESI’s second data release (DESI Collaboration 2025, arXiv:2503.14738) measured BAO in more than 14 million galaxies and quasars. On its own it is consistent with flat ΛCDM with Ω_m = 0.2975 ± 0.0086, although the parameters it implies are in mild (2.3σ) tension with those from the CMB. Combined with the CMB, the evolving dark energy model w₀w_a is preferred over ΛCDM at 3.1σ, and at 2.8σ to 4.2σ when supernovae are added, depending on the supernova sample. These are model comparisons, not detections. The four published w₀w_a fits are presets in iCosmos.

Can I judge a model by how close the curve is to the points?

Only roughly. The D_M and D_H measurements at the same redshift are correlated (correlation coefficients of about −0.4 to −0.5), and r_d depends on Ω_bh² and Ω_mh². A proper comparison uses the full published covariance, as in the DESI papers.

Why is the sound horizon not defined for some models?

The standard sound horizon assumes a hot Big Bang with standard radiation in which dark energy and curvature are negligible at z ≈ 1060. If your model has no Big Bang, non-standard Ω_r, Ω_mh² outside 0.03–0.6, or significant dark energy at early times, iCosmos explains why instead of showing a number.

Does r_d depend on H₀?

Not at fixed Ω_bh² and Ω_mh²: it depends on the physical densities only. Changing H₀ at fixed Ω_m changes Ω_mh² and therefore r_d.

Related calculators

• Hubble parameter H(z) calculator
• Angular diameter distance calculator
• Comoving distance calculator
• The full iCosmos cosmology calculator

About iCosmos

iCosmos was created by Mihran Vardanyan. Mihran Vardanyan is a cosmologist who earned his DPhil in cosmology at the University of Oxford, supervised by Joseph Silk and Roberto Trotta. He developed iCosmos during his research years at Oxford Astrophysics, where his work applied Bayesian model selection to the curvature and size of the Universe.

How to cite: Vardanyan, Mihran. iCosmos V4: Cosmology Calculator. https://icosmos.co.uk/. Include your access date, model parameters and exported data when reporting calculations.