Cosmology Calculator with BAO Scale
Distances, ages and the angular BAO scale at any redshift, for flat \(\Lambda\)CDM — dark energy and cold dark matter, the standard model of cosmology
Era boundaries are drawn for \(\Lambda\)CDM.
At \(z\) = 0 the universe is 13.791 Gyr old, in the dark-energy era.
Derived Quantities
\(\theta_\mathrm{BAO}\) is the angle the 147.1 Mpc ruler subtends at that redshift — the curve plotted on the BAO page against the 3.2° tile of the Dark Energy Spectroscopic Instrument. 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, \(E(z) = \sqrt{\Omega_\mathrm{r}(1+z)^4 + \Omega_\mathrm{m}(1+z)^3 + \Omega_\Lambda}\), with \(\Omega_\Lambda = 1 - \Omega_\mathrm{m}\) and radiation included, and \(\Omega_\Lambda\) fixed by closure at \(1 - \Omega_\mathrm{m} - \Omega_\mathrm{r}\).
The sound horizon \(r_\mathrm{d}\) is the comoving distance a pressure wave crossed between the Big Bang and the drag epoch, just after the cosmic background radiation was set free at recombination. That radiation is the cosmic microwave background — the CMB — and Planck's map of it fixes the Hubble constant, matter density and sound horizon offered here as defaults. The same ruler, frozen into the distribution of galaxies, is what DESI measures at low redshift, which is why the one number does duty at both ends of the timeline.
\(\Omega_\mathrm{r}\) looks negligible at \(9.2\times10^{-5}\) 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 \(z \approx 3400\) and dominates everything before that. Leaving it out of the closure — writing \(\Omega_\Lambda = 1 - \Omega_\mathrm{m}\) — 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 \(r_\mathrm{d}\). That second column is DESI's convention: BAO measures a ratio of distance to the standard ruler, not a distance, so \(D_\mathrm{M}/r_\mathrm{d}\) and \(D_\mathrm{H}/r_\mathrm{d}\) 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.
The cosmic timeline starts at the Big Bang with the Planck era, and continues with inflation, the quark–gluon plasma, Big Bang nucleosynthesis, the radiation-dominated era, the matter-dominated era, the Dark Ages, star formation, galaxy formation, galaxy evolution, and the dark-energy era currently being probed by DESI.
Elsewhere on this site
- \(\Lambda\)CDM the standard model of cosmology — the six parameters, and the cosmic timeline
- Evolving dark energy calculator the same quantities with \(w_0\) and \(w_a\), against \(\Lambda\)CDM
- Baryon acoustic oscillations where the 147 Mpc ruler comes from
- The Hubble tension why \(H_0\) is contested