Redshift to Distance Calculator

There is no single distance to a redshift. The comoving distance computed here, D_C = (c/H0) times the integral of dz/E(z), is the separation you would measure today if the expansion were frozen, where E(z) = sqrt(Omega_M (1+z)^3 + Omega_Lambda) encodes how the expansion rate changed over cosmic time. The Hubble distance c/H0 sets the scale and equals about 4,283 Mpc at H0 = 70 km/s/Mpc. Two further measures follow directly: luminosity distance D_L = D_C (1+z), which preserves the inverse-square law once photon energy loss and arrival-rate dilation are absorbed into it, and angular diameter distance D_A = D_C/(1+z), which governs apparent angular size. Their ratio of (1+z)^2 is why cosmological surface brightness dims as (1+z)^4, a strong observational test of expansion. The angular diameter distance behaves counterintuitively: for standard parameters it peaks near z = 1.6 at roughly 1,750 Mpc and then decreases, so a galaxy of fixed physical size at z = 10 subtends a larger angle than the same galaxy at z = 2. Realistic parameter choices are Omega_M near 0.31 and Omega_Lambda near 0.69, with Omega_M + Omega_Lambda = 1 for a flat universe. The main limitations are structural: the integral assumes flatness and a cosmological constant with equation of state exactly w = -1, it omits the radiation term that becomes important above about z = 100, and it inherits the roughly 8 percent spread between the Planck value of 67.4 and the local distance-ladder value near 73 km/s/Mpc.

The mathematical framework predates the observations. Alexander Friedmann derived the expanding and contracting solutions of general relativity in 1922, Georges Lemaitre rederived them with an explicit velocity-distance prediction in 1927, and Howard Robertson and Arthur Walker established in the 1930s that the metric is the unique form for a homogeneous isotropic universe. Converting a measured redshift into a distance within that metric required solving the integral, and Wolfgang Mattig published the closed-form matter-only solution in 1958, which remained the standard working formula for decades because a universe with a cosmological constant was widely considered unnecessary. That changed in 1998 when the High-z Supernova Search Team led by Adam Riess and Brian Schmidt and the Supernova Cosmology Project led by Saul Perlmutter independently found distant Type Ia supernovae fainter than a decelerating universe allowed, implying accelerated expansion and a nonzero Lambda. The three researchers shared the 2011 Nobel Prize in Physics. With Lambda restored, the integral generally has no elementary closed form and is evaluated numerically, exactly as this calculator does. David Hogg standardized the notation and definitions of the competing distance measures in a widely cited 1999 arXiv note, Ned Wright published a public web cosmology calculator the same year, and successive CMB missions - COBE, WMAP, and Planck through its 2018 release - progressively tightened the parameters the integral depends on.