A redshift calculator for velocity, distance and cosmic time
This redshift calculator turns a redshift z into the numbers you actually want: the equivalent velocity, where spectral lines land, how far away the source is and how long its light has been travelling. You can start from z itself, from a velocity, or from an emitted and observed wavelength — for example the Hα line at 656.281 nm measured at 1,312.6 nm — and the page updates as you type.
Below the result you get the full set of cosmological distances — comoving, transverse comoving, luminosity, angular-diameter and light-travel — together with the lookback time, the age of the universe when the light set out, the distance modulus, the physical size of one arcsecond and the Hubble parameter H(z). The defaults are the Planck 2018 parameters, and every parameter can be edited. All of it is computed in your browser by numerical integration; nothing is sent anywhere.
How to use the Redshift Calculator
- Choose what you know: Redshift z, Velocity (as v/c or in km/s), or Wavelengths (pick a spectral line or type a rest wavelength, then the observed one).
- Read the headline: the redshift, how long ago the light left and how old the universe was then.
- Check the cards for the Doppler velocity, today’s recession velocity, distances, distance modulus and angular scale.
- Use the two tables for every distance in Mpc and billions of light-years, and for where Lyman-α, Hα, [O III] or the 21 cm line appear at this redshift.
- Change H₀, Ωm or ΩΛ under Cosmology, untick Flat universe to allow curvature, or press Planck 2018 to restore the defaults.
Redshift Calculator features
Three ways in
Redshift, velocity or a pair of wavelengths (nm, Å or µm) with six built-in spectral lines; switching mode keeps the same object.
Every distance measure
DC, DM, DL, DA and light-travel distance, following Hogg’s “Distance measures in cosmology”.
Cosmic time
Lookback time, age at z and age today, integrated with the same model.
Editable ΛCDM
H₀, Ωm and ΩΛ, optional curvature and radiation; a check stops parameter sets with no big bang.
Checked against a reference
For Wright’s CosmoCalc default case (H₀ 69.6, Ωm 0.286, ΩΛ 0.714, z = 3) this page reproduces DC = 6,481.3 Mpc, DA = 1,620.3 Mpc, DL = 25,924.3 Mpc and an age at z of 2.171 Gyr.
How the calculator works
Velocity uses the special-relativistic Doppler formula: 1 + z = √((1 + β)/(1 − β)), so β = v/c = ((1+z)² − 1)/((1+z)² + 1). A redshift of 1 is v = 0.6c; the velocity approaches but never reaches c. Wavelengths simply scale: λobserved = (1 + z) λemitted.
Distances come from the expansion rate E(z) = √(Ωr(1+z)⁴ + Ωm(1+z)³ + Ωk(1+z)² + ΩΛ). The comoving distance is DC = (c/H₀) ∫ dz′/E(z′) from 0 to z; DM equals DC when space is flat and picks up a sinh or sin correction otherwise; then DL = (1+z) DM and DA = DM/(1+z). Lookback time is (1/H₀) ∫ dz′/((1+z′)E(z′)). The integrals are evaluated with Simpson’s rule in the scale factor, which stays accurate up to z ≈ 1,100.
Radiation (Ωr = 4.165 × 10⁻⁵/h², photons plus three massless neutrino species) is included by default, as in Wright’s calculator; it matters only at very high redshift. With “flat” ticked, ΩΛ is set to 1 − Ωm − Ωr.
Worked example: z = 1 with Planck 2018 parameters
Notice that the luminosity distance is four times the angular-diameter distance — the factor (1+z)² — and that the light has travelled 7.95 billion light-years while the galaxy is now 11.1 billion light-years away, because space kept expanding while the light was on its way. For distances in other units, try the light-year converter.
| Quantity | Value |
|---|---|
| Doppler velocity | 0.6c = 179,875 km/s |
| Comoving distance DC | 3,402 Mpc (11.10 billion ly) |
| Luminosity distance DL | 6,805 Mpc |
| Angular-diameter distance DA | 1,701 Mpc — 1″ spans 8.25 kpc |
| Lookback time | 7.95 Gyr |
| Age of the universe then | 5.84 Gyr (13.80 Gyr today) |
Doppler velocity or recession velocity?
Cosmological redshift is not really a Doppler shift: light is stretched by the expansion of space while it travels. The relativistic Doppler velocity is the speed that would produce the same shift in flat space, and it never exceeds c. Today’s recession velocity, H₀ × DC, is the rate at which the proper distance grows now; for z above about 1.5 it exceeds the speed of light, which relativity allows for expansion. For nearby galaxies (z below about 0.05) both agree with v ≈ cz and Hubble’s law — use the Hubble’s law calculator for that regime.
Sources and limits
Formulas follow D. W. Hogg (1999), “Distance measures in cosmology”, arXiv:astro-ph/9905116; the radiation term and test case come from E. L. Wright (2006), PASP 118, 1711. Default parameters are from Planck 2018 results VI (A&A 641, A6), Table 2, TT,TE,EE+lowE+lensing: H₀ = 67.36 km/s/Mpc, Ωm = 0.3153, ΩΛ = 0.6847. Dark energy is a cosmological constant (w = −1); neutrino masses, peculiar velocities and the H₀ tension (local measurements give about 73 km/s/Mpc) are not modelled, so treat the last digit as model-dependent.
Redshift Calculator FAQ
How do I convert redshift to distance?
Type z into the redshift calculator. It integrates the ΛCDM expansion history and reports comoving, luminosity and angular-diameter distances. There is no single “distance” at high z, so pick the one that fits your use: DL for brightness, DA for angular size.
How do I calculate redshift from wavelength?
z = λobserved/λemitted − 1. Choose Wavelengths, pick the line (or type its rest wavelength) and enter the observed value.
What velocity does z = 1 correspond to?
With the relativistic Doppler formula, 0.6c (179,875 km/s). The simple v = cz gives c, which is why it only works for small z.
Can galaxies recede faster than light?
Yes. The distance to galaxies beyond z ≈ 1.5 is growing faster than c today. This is the expansion of space, not motion through space, and does not break relativity.
Why does the result change with H₀?
All distances scale as c/H₀ and all times as 1/H₀. Planck gives 67.4 km/s/Mpc, Cepheid-calibrated supernovae about 73; the calculator lets you use either.
What redshift is the cosmic microwave background?
About z = 1,100. Enter it to see that the light left when the universe was roughly 370,000 years old.