Cosmological Parameters
Redshift
Physics & numerical method
All distances are computed via numerical integration of the FLRW comoving line element. For a flat universe (Ωk = 0), the comoving distance is:
d_C(z) = (c/H₀) ∫₀ᶻ dz' / E(z')
where E(z) = √(Ωr(1+z)⁴ + Ωm(1+z)³ + Ωk(1+z)² + ΩDE(z)) and for a dark energy equation of state w ≠ −1:
ΩDE(z) = ΩΛ (1+z)^(3(1+w))
The integration uses Gaussian quadrature (50 nodes) for accuracy across the full z range. Curvature Ωk = 1 − Ωm − ΩΛ; the calculator handles curved geometries correctly via the sinh/sin angular diameter distance relation.
Radiation: Ωr = 2.469×10⁻⁵ h⁻² (CMB + neutrinos with N_eff = 3.04) — relevant only for z > 100.
Distance relations: D_A = D_C / (1+z) (angular diameter); D_L = D_C × (1+z) (luminosity); D_M = D_C (comoving transverse, curved case differs).
Lookback time: t_look = ∫₀ᶻ dz' / [(1+z') H(z')] where H(z)=H₀E(z).
Age of universe: t_age = ∫₀^∞ dz / [(1+z) H(z)] — computed to z=1100.
Wright 2006, PASP 118:1711 (cosmological calculator — original Ned Wright tool)
Hogg 1999, arXiv:astro-ph/9905116 (distance measures in cosmology — formula reference)
Peebles 1993, Principles of Physical Cosmology (textbook)
WMAP-9: Hinshaw et al. 2013, ApJS 208:19