Redshift Calculator: z to Velocity, Distance and Lookback Time
Convert a redshift z, a pair of observed and rest wavelengths, or a velocity into recession velocity (relativistic and v = cz), Hubble-law distance and flat ΛCDM comoving, luminosity and angular-diameter distances, lookback time and the age of the universe at that redshift.
How it works
z = (λ_obs − λ_rest) ÷ λ_rest · Low z: v ≈ c·z · Relativistic Doppler: v = c × ((1+z)² − 1) ÷ ((1+z)² + 1) · Inverse: z = √((1+β) ÷ (1−β)) − 1 · Flat ΛCDM: E(z) = √(Ωr(1+z)⁴ + Ωm(1+z)³ + ΩΛ), D_C = (c ÷ H₀) ∫₀ᶻ dz′ ÷ E(z′), D_L = (1+z) D_C, D_A = D_C ÷ (1+z), lookback time = (1 ÷ H₀) ∫₀ᶻ dz′ ÷ ((1+z′) E(z′))
Redshift z measures how much a spectral line is stretched toward longer wavelengths: you can enter z directly, work it out from a line’s observed and rest wavelengths, or turn a velocity back into z. For small z the recession velocity is close to c·z, but that overshoots as z grows and passes the speed of light at z = 1, so the relativistic Doppler formula is used for the main result; it stays below c for any finite redshift. For galaxies, though, redshift is caused mostly by the expansion of space, not motion through it. The calculator therefore also integrates a flat ΛCDM model (Planck 2018 defaults: H₀ = 67.4 km/s/Mpc, Ωm = 0.315, radiation included) to give the comoving distance (where the galaxy is now), the luminosity distance (for brightness and supernova work), the angular-diameter distance (for apparent size), the light-travel time and the age of the universe when the light left. The proper recession speed today, H₀ × comoving distance, is the physically meaningful expansion rate and exceeds c beyond about z ≈ 1.5. Change H₀ to about 73 to see the effect of the Hubble tension. Peculiar velocities of a few hundred km/s make results for very nearby galaxies (z below about 0.01) uncertain.
Worked example
For z = 0.1: (1 + z)² = 1.21, so β = 0.21 ÷ 2.21 = 0.09502 and v = 0.09502 × 299,792.458 = 28,487.07 km/s, a little below the low-z estimate c × 0.1 = 29,979.25 km/s. With H₀ = 67.4 the Hubble-law distance is 29,979 ÷ 67.4 = 444.8 Mpc. Integrating flat ΛCDM (Ωm = 0.315) gives a comoving distance of 434.1 Mpc (1.416 billion light-years), a luminosity distance of 477.5 Mpc and a lookback time of 1.350 billion years: the light left when the universe was 12.44 of its present 13.79 billion years old. H-alpha at 656.28 nm is observed at 721.91 nm.
Sources: Hogg, Distance measures in cosmology (arXiv:astro-ph/9905116) · Lambda-CDM model: Planck 2018 parameters — Wikipedia · Relativistic Doppler effect — Wikipedia · Distance measure — Wikipedia
Frequently asked questions
When should I use the relativistic formula instead of v = cz?
Below about z = 0.05 the two agree within a few percent. Above that v = cz overstates the Doppler velocity and gives faster-than-light values beyond z = 1. For galaxies at large z neither formula is a true speed; use the ΛCDM distances and lookback time instead.
How do I find z from a spectrum?
Identify a known line, such as H-alpha at 656.28 nm, measure where it appears, and compute z = observed ÷ rest − 1. Choose the Wavelengths mode, pick the line and enter the observed wavelength in nm, Å, µm or cm.
Can galaxies recede faster than light?
Yes. Special relativity limits motion through space, not the expansion of space itself. In the standard model the proper distance to galaxies beyond about z = 1.5 grows faster than c today, even though their light still reaches us.
Which Hubble constant should I use?
The Planck 2018 cosmic microwave background analysis gives 67.4 km/s/Mpc, while Cepheid and supernova distance-ladder measurements give about 73. This gap is the Hubble tension. Distances scale roughly as 1 ÷ H₀, so switching to 73 shortens them by about 8%.
What does a negative z mean?
It is a blueshift: the source approaches and its light is compressed to shorter wavelengths. The calculator gives the approach velocity but no cosmological distance, since blueshifts occur only for nearby objects such as stars in our galaxy or the Andromeda galaxy.
What redshift is the cosmic microwave background?
About z = 1,100. Its light was released roughly 380,000 years after the Big Bang, and the calculator gives a lookback time of about 13.79 billion years for it.
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This calculator is for educational purposes. Double-check important results before acting on them.