Cosmological Redshift Calculator

Stretch a photon across the expanding universe. Set z, choose an emitted wavelength, and watch the spectrum shift.

Worked example — Cosmological Redshift Calculator

Redshift definition	0.1000
Observed wavelength	721.6 nm
Recession velocity	28487 km/s (9.50% of c)
Scale factor a	0.9091

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Three different horizons, and only one is where v = c 🖖

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, and the photon's wavelength grows in proportion to the scale factor a(t). At low z the distinction is academic, since both pictures give v ≈ H₀d. Past z ≈ 1.5 the recession velocity exceeds c, which is impossible for motion through space and unremarkable for space itself: relativity's speed limit governs travel through spacetime, not the expansion of it. Three separate distances get muddled here. Recession reaches c at the Hubble radius, c/H₀ ≈ 14.5 billion light-years — and we routinely observe galaxies far beyond it, which is the plainest proof that superluminal recession does not hide anything. What we cannot see past is the particle horizon at about 46 billion light-years, the distance light has had time to cover. And what is permanently unreachable is set by a third surface, the event horizon near 16 billion light-years: light leaving a galaxy beyond that today will never arrive, however long we wait.

What redshift actually measures 🖖

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).

Redshift stretches time, not just light 🖖

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.

Problem solved in full

Recession speed when hydrogen's Hα line arrives at 721.6 nm 5 steps
Hydrogen's Hα line leaves a galaxy at 656 nm and arrives at 721.6 nm. Find the recession speed. The obvious formula is out by about 1500 km/s here, and at higher redshift it fails outright.
1. Redshift is defined as the fractional stretch of the wavelength, so it reads straight off the two numbers. z = 0.1: the light arrived 10% redder than it left.
  
  $z = \frac{λ _{obs} - λ _{0}}{λ _{0}} = \frac{721.6 - 656}{656} = 0.1000$
2. The formula almost everyone reaches for is v = cz, inherited from the low-speed Doppler shift. It returns 29 979 km/s, a tenth of light speed.
  
  $v_{naive} = cz = 0.1 \times 299792 = 29979 km/s$
3. But a tenth of light speed is not low speed. The relativistic Doppler relation ties 1 + z to a square root of (1 + β)/(1 − β); solving for β takes one line of algebra and gives 0.09502, not 0.1.
  
  $1 + z = \frac{1 + β}{1 - β} ⟹ β = \frac{( 1 + z ) ^{2} - 1}{( 1 + z ) ^{2} + 1} = \frac{0.21}{2.21} = 0.09502$
4. That is 28 487 km/s. The naive figure overstates it by 5.2% — roughly 1500 km/s, which is larger than the velocity differences astronomers routinely argue about.
  
  $v = 0.09502 c = 28487 km/s (\frac{29 979}{28 487} = 1.052)$
5. Push z to 1 and the discrepancy stops being a correction. v = cz returns exactly c; the relativistic formula returns three-fifths of it.
  
  $a = \frac{1}{1 + z} = 0.9091; z = 1 : v_{naive} = c, β = \frac{3}{5} = 0.600$
Answer

The tool prints 721.6 nm, 28 487 km/s and 9.50% of c, with a scale factor of 0.9091 — the universe was 91% of its present size when that light set out. The line to carry away is step 5: at z = 1 the shortcut returns light speed exactly, while the answer is 0.600c. Click the deep-field preset and check it. Galaxies are catalogued at z = 7 and beyond, where cz would hand you seven times light speed — which is not a paradox about relativity, it is a formula being used a very long way outside the regime it was derived in.

$β = \frac{( 1 + z ) ^{2} - 1}{( 1 + z ) ^{2} + 1} : z = 0.1 \Rightarrow 0.0950 c, z = 1 \Rightarrow 0.600 c (not c)$

References (2)

The distinction between the three horizons the first block separates: T. M. Davis and C. H. Lineweaver, "Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe." Publications of the Astronomical Society of Australia 21(1), 97–109, 2004.
The H₀ value the Hubble radius is computed from: N. Aghanim et al. (Planck Collaboration), "Planck 2018 results. VI. Cosmological parameters." Astronomy & Astrophysics 641, A6, 2020.

By the 42Math team · Updated 2026-07-06

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Local galaxy (z=0.01) - Hα leaves at 656 nm and arrives at 662.6. Those few nanometres are the entire measurement: from them the panel reads 2,983 km/s, a comoving 42.7 Mpc, and light that set out 0.139 Gyr ago, when the universe stood at 0.9901 of its present size. A one per cent stretch is already a hundred and forty million years of travel.
Hubble deep (z=1) - z = 1 is the clean case. The 656 nm line arrives at 1312.0, exactly doubled, and the scale factor reads 0.5000 — the universe was half the size it is now. The light has been in flight 7.655 Gyr, more than half the age of the universe, and the recession velocity comes out at 60.00% of c, where a literal v = cz would have said 100%.
Quasar (z=3) - z=2.5: distant quasar, lookback ~11 Gyr
CMB (z=1089) - z=1089: cosmic microwave background, T~3000 K then