Redshift Distance Converter
The light from distant galaxies is stretched to longer, redder wavelengths as the universe expands. Enter a redshift to see how far away the galaxy is and how long its light has been travelling.
Redshift to distance
Velocity, cosmological distances, lookback time and the age of the universe.
z = (observed wavelength ÷ emitted wavelength) − 1.
km/s per megaparsec.
Between 0 and 1. Dark energy is the rest.
Lookback time
7.715 billion years
The light set off when the universe was 5.752 billion years old.
- Comoving distance (now)
- 3,303.8 Mpc10.78 billion ly
- Luminosity distance
- 6,607.7 Mpc21.55 billion ly
- Angular diameter distance
- 1,651.9 Mpc5.388 billion ly
- Hubble law distance (cz ÷ H0)
- 4,282.7 Mpc13.97 billion ly
- Recession velocity (relativistic Doppler)
- 179,875 km/s
- As a fraction of light speed
- 0.6000 c
- Expansion speed now (H0 × distance)
- 231,268 km/s (0.771 c)
- Age of the universe then
- 5.752 billion years
- Age of the universe now
- 13.47 billion years
- Hubble time (1 ÷ H0)
- 13.97 billion years
Above about z = 0.1 the simple Hubble law is no longer a good guide: here it is 30% off the comoving distance. Use the comoving or luminosity distance instead.
Flat universe with matter and dark energy only (ΩΛ = 1 − Ωm). Radiation is ignored, which only matters at very high redshift such as the microwave background. The Doppler speed treats redshift as motion; in an expanding universe the expansion speed can exceed light speed.
How this is calculated
- 1
Expansion history
In a flat universe of matter and dark energy, the expansion rate at redshift z is H(z) = H0 × E(z), with E(z) = √(Ωm(1 + z)³ + ΩΛ) and ΩΛ = 1 − Ωm. Radiation is left out.
- 2
Comoving distance
D_C = (c ÷ H0) × ∫ dz ÷ E(z) from 0 to z, worked out numerically with Simpson's rule. The luminosity distance is (1 + z) × D_C and the angular diameter distance is D_C ÷ (1 + z).
- 3
Lookback time and age
Lookback time = (1 ÷ H0) × ∫ dz ÷ ((1 + z) E(z)). The age at scale factor a = 1 ÷ (1 + z) is (1 ÷ H0) × ∫ √a da ÷ √(Ωm + ΩΛ a³) from 0 to a. With H0 in km/s/Mpc, 1 ÷ H0 is 977.8 ÷ H0 billion years.
- 4
Velocity
The relativistic Doppler formula v = c((1 + z)² − 1) ÷ ((1 + z)² + 1) gives a speed that never reaches c. Hubble's law, d = cz ÷ H0, is only a fair guide below about z = 0.1.
Frequently asked questions
How do you convert redshift to distance?
For small redshifts, multiply z by the speed of light and divide by the Hubble constant: z = 0.01 is about 43 Mpc, or 140 million light years, with H0 = 70. For larger redshifts you need to integrate over the expansion history, which this calculator does for you.
Can galaxies move away faster than light?
Yes, in the sense that the distance to them grows faster than light speed. Nothing outruns light locally; space itself is expanding. With H0 = 70 the crossover is about 4,300 Mpc, roughly z = 1.5.
What is lookback time?
How long ago the light we now see left the object. For a galaxy at z = 1 it is about 7.7 billion years, so we see it as it was when the universe was under half its present age.
Why are there several different distances?
In an expanding universe distance depends on what you mean. Comoving distance is how far away the galaxy is now, luminosity distance sets how bright it looks, and angular diameter distance sets how big it looks. They agree only for nearby objects.