Cosmological Redshift Calculator

Cosmological redshift is often described as a Doppler effect, but it is more subtle. A galaxy is not moving through space — the space between us and it is stretching. The photon wavelength grows proportionally to the scale factor a(t) as the photon travels. At low z the distinction is academic (both give v ≈ H₀d), but at z > 1 the recession velocity exceeds c — impossible for ordinary motion, but consistent with expanding space. Special relativity's speed limit applies to motion through spacetime, not to the expansion of spacetime itself. This also means there is a particle horizon: galaxies beyond ~46 billion light-years are receding faster than c today and are permanently beyond our observational reach, even though we can still see the ancient light they emitted when the universe was smaller.

Light from a distant galaxy arrives with its wavelengths stretched toward red. Divide the observed wavelength by the emitted one, subtract 1, and you have z. The bigger z is, the longer the light has been travelling — so z is a cosmic clock and ruler in one. At z = 1 the light left its galaxy about 8 billion years ago, when the universe was half its present size (scale factor a = 1/(1+z) = 0.5).

Expansion stretches every wavelength by a factor (1+z) — but it stretches the duration of events by the same factor. A Type Ia supernova at z = 1 is seen to brighten and fade over twice as many days as an identical one nearby. Astronomers measured exactly this stretching, which killed the old 'tired light' idea: only genuine expansion slows the clock as well as reddening the light.