Calculate cosmological distances from redshift using Hubble law. Explore lookback time and the expanding universe.
Redshift is the number every galaxy carries: the fraction by which the expansion of the universe has stretched its light since emission. Enter a redshift and this tool converts it into a recession velocity and a distance using the Hubble law, v = H0 d, and reports how far back in time the light left. The thing most people get wrong: redshift does not scale linearly into either distance or time. A galaxy at z = 2 did not emit its light twice as long ago as one at z = 1, because the expansion rate has changed over cosmic history.
A concrete input and expected output from the current implementation.
Input
redshift z = 0.1
Expected output
Recession velocity ~ 29,979 km/s (one-tenth of light speed). Hubble-law distance ~ 428 Mpc, or ~ 1.40 billion light-years, with H0 = 70 km/s/Mpc.
At low redshift the recession velocity is v = cz, so z = 0.1 gives v = 0.1 x c = 29,979 km/s. The Hubble law then gives d = v/H0 = 29,979 / 70 = 428 Mpc, about 1,397 million light-years. The figure assumes H0 = 70; with the Planck value of 67.7 km/s/Mpc the same galaxy comes out at about 443 Mpc.
At small redshift the recession velocity is approximately v = cz, and Hubble's law turns that into a distance: d = cz/H0. The result is linear in redshift, so a galaxy at z = 0.2 comes out twice as far as one at z = 0.1. The only free parameter is the Hubble constant H0, taken in km/s per Mpc; the conventional value is about 70, though measurements still disagree.
Redshift measures the stretch factor: 1 + z is how much larger the universe is now than when the light left, so a galaxy at z = 1 emitted its light when the universe was half its present size. Lookback time is the integral of the expansion rate back to that epoch, which requires assuming a cosmology - matter and dark energy densities as well as H0. Redshift and time are not proportional.
Because it is a coordinate bookkeeping velocity, not a speed that anything travels through space. General relativity allows two well-separated points to recede from each other faster than light, since space itself stretches; nothing locally exceeds c. The distance remains meaningful, but the velocity is not a limit on, say, how fast a signal could cross that separation.
The light has been traveling for about 13 billion years, so the light-travel distance is about 13 billion light-years, and the universe was only about half a billion years old when the light left. But the galaxy is now much farther away - roughly 30 billion light-years - because space kept expanding while the light was en route. Both numbers are correct; they answer different questions.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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