Redshift z to Recession Velocity

Redshift to recession velocity: v = c·z

When the redshift is small, recession velocity follows v = c·z. Here c = 299,792 km/s and z is the fractional wavelength shift (z = Δλ/λ). Pair that with Hubble's law, v = H₀·d and H₀ ≈ 70 km/s/Mpc, and you can turn an observed spectrum into a cosmological distance. Take a galaxy at z = 0.01: its velocity comes out around 3,000 km/s, which places it at roughly 43 Mpc. Once z gets large you have to switch to the relativistic version, 1+z = √((1+v/c)/(1−v/c)), and z starts to read directly as the cosmic scale factor. A quasar at z = 1, for instance, sent its light out when the universe was half its present size. The Cosmic Microwave Background sits way out at z ≈ 1,100.

Applications

This shows up across modern cosmology: pinning down how fast the universe is expanding, JWST surveys of high-z galaxies (objects at z > 10, seen as they looked within 500 Myr of the Big Bang), estimating the universe's age (~13.8 Gyr, from H₀ and ΛCDM), mapping large-scale structure with SDSS and DESI, and using baryon acoustic oscillations as a standard ruler.

FAQ

When does v = c·z break down? Once z gets past about 0.1, v is creeping up toward c, so you need the relativistic Doppler formula or a full cosmological model.

Is cosmological redshift a Doppler effect? Not really. It comes from space itself stretching between the galaxies rather than from anything moving through space. At low z, though, the numbers from the two pictures match closely enough that you can't tell them apart.

What is the Hubble tension? H₀ measured from the CMB comes out around 67 km/s/Mpc, while local distance ladders give about 73 km/s/Mpc. That gap runs to roughly 5σ, which points either to new physics or to systematics nobody has tracked down yet.