Cosmic Calculator with BAO Scale
Distances, ages and the angular BAO scale at any redshift, for flat ΛCDM
At z = 0 the universe is 13.791 Gyr old, in the dark-energy era. Era boundaries are drawn for ΛCDM.
Derived Quantities
θBAO is the angle the 147.1 Mpc ruler subtends at that redshift — the curve plotted on the BAO page against DESI's 3.2° tile. Sweep z upward and watch it fall through 3.2° at about z = 0.72, which is where a BAO-sized patch of sky stops spanning many pointings and starts fitting inside one.
What it computes
Four numbers go in: the Hubble constant, the matter density, the sound horizon at the drag epoch, and a redshift. The defaults are the Planck 2018 values. Everything else follows from the Friedmann equation for a flat universe, E(z) = √(Ωr(1+z)4 + Ωm(1+z)3 + ΩΛ), with ΩΛ = 1 − Ωm and radiation included, and ΩΛ fixed by closure at 1 − Ωm − Ωr.
Ωr looks negligible at 9.2×10−5 and is, today. It is not negligible early: radiation scales as (1+z)4 against matter's (1+z)3, so it overtakes matter above z ≈ 3400 and dominates everything before that. Leaving it out of the closure — writing ΩΛ = 1 − Ωm — makes the model sum to 1.000092 rather than 1, which is a curved universe by accident. The value quoted is photons plus three species of massless neutrino.
The distances are given both in megaparsecs and divided by rd. That second column is DESI's convention: BAO measures a ratio of distance to the standard ruler, not a distance, so DM/rd and DH/rd are what the papers quote. Set z = 0.7 and you get 17.6 and 20.2, against the 17.35 and 19.6 DESI measured for its LRG2 sample — the gap between prediction and measurement, readable in one line.
Age integrals use the substitution a = 1/(1+z), which removes the infinite upper limit; integration is Simpson's rule. The method follows cosmic_timeline.py, against which every displayed digit was checked.