H₀
Ωm
ΩΛ
Ωb
z
Advanced
Ωk
Ωr
n
Integrator: midpoint rule. Includes Wright's high-z particle-annihilation age correction (active at z > 1000).
At redshift z
t(z)Age at z
—
Gyr
tLLookback
—
Gyr
t0Age now
—
Gyr
DCComoving
—
Mpc
DAAngular
—
Mpc
DLLuminosity
—
Mpc
DA·θProper scale
—
Mpc / arcmin (proper)
DM·θComoving scale
—
Mpc / arcmin (comoving)
VCVolume (< z)
—
Gpc³
Sound horizon & BAO
Eisenstein & Hu (1998) fits · at your z
zdDrag epoch
—
rsrs(zd)
—
Mpc
θBAOBAO angle
—
deg
ℓBAOBAO multipole
—
Inverse solver
Find the redshift at which …
→
z = —
→
z = —
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DC comoving
DA angular
DL luminosity
tL lookback (Gyr)
Ωm(z) matter
Ωr(z) radiation
ΩΛ(z) dark energy
|Ωk(z)| curvature
Crossover z’s (when components are equal) are marked with dashed lines.
Rules of thumb at z = —
Multipole → angle
ℓ = π / θ
→
Physical size
Rp = DA(z) · θ
→
Comoving size
Rc = DM(z) · θ
→
Wavenumber
k = 2π / Rc
→
Comoving size vs z (at fixed ℓ)
Angle subtended by fixed physical size vs z
DA(z) turns over around z ≈ 1–2 in ΛCDM, so objects of fixed physical size appear larger on the sky beyond that.