Calculate redshift z from velocity or wavelengths. Compute recession velocity, Hubble distance, lookback time, scale factor, and CMB temperature at any epoch.
Cosmological redshift (z) quantifies how much the universe has expanded since light left a distant object. A galaxy at z = 1 emitted its light when the universe was half its current size — wavelengths have been stretched by a factor of (1 + z) = 2 during the journey. The redshift directly encodes the recession velocity, distance, and lookback time.
This calculator converts between redshift z, recession velocity (using the full relativistic Doppler formula), and observed/emitted wavelengths. From the redshift, it computes the Hubble distance, lookback time (using a simplified flat ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7), cosmic scale factor, and the CMB temperature at that epoch.
Preset buttons load famous redshifts: Andromeda (z = −0.001, blueshifted!), the Virgo Cluster, quasar 3C 273, the CMB surface of last scattering (z = 1089), and the most distant known galaxy. A reference table of notable objects connects redshift values to physical meaning.
The tool bridges observational astronomy (measured z) and cosmological models (distance, age), making it invaluable for students, amateur astronomers, and researchers.
Redshift is the fundamental observable in extragalactic astronomy. Converting it to physically meaningful quantities (distance, lookback time, cosmic epoch) requires relativistic formulas and cosmological models.
This calculator handles the math instantly, from z = −0.001 (Andromeda) to z = 1089 (the CMB). Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain.
z = (λ_obs − λ_emit) / λ_emit = √((1+β)/(1−β)) − 1. Recession: β = ((1+z)²−1) / ((1+z)²+1), v = βc. Hubble distance: d = v/H₀ (approximate for low z). Scale factor: a = 1/(1+z). CMB temperature: T(z) = 2.725 × (1+z) K.
Result: v = 44,700 km/s, d ≈ 640 Mpc ≈ 2.1 Gly, lookback ≈ 1.9 Gyr
Quasar 3C 273 at z = 0.158. β = (1.158²−1)/(1.158²+1) = 0.1491. v = 44,700 km/s. d = 44,700/70 = 639 Mpc = 2.08 Gly. Light left 1.9 billion years ago.
Use consistent units, verify assumptions, and document conversion standards for repeatable outcomes.
Most mistakes come from mixed standards, rounding too early, or misread labels. Recheck final values before use. ## Practical Notes
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Check assumptions and units before interpreting the number. ## Practical Notes
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Yes. z > 1 does not mean the object moves faster than light relative to us locally. It means the metric expansion stretches wavelengths by more than 2×. The most distant galaxies have z > 10.
Cosmological redshift is caused by the expansion of space stretching photon wavelengths. Doppler redshift is caused by relative motion through space. For nearby objects (z < 0.01), they give the same result.
Andromeda is close enough that its peculiar (gravitational) velocity toward the Milky Way (~110 km/s) exceeds the Hubble expansion velocity at its distance. It will merge with us in ~4.5 billion years.
The CMB (cosmic microwave background) has z = 1089, meaning the universe was 1/(1+1089) = 1/1090 its current size. The CMB was emitted 380,000 years after the Big Bang at T ≈ 3000 K (glowing orange-hot plasma).
For z < 0.1, Hubble law (d = v/H₀) is accurate to ~5%. For higher z, you need the full Friedmann equation integral, accounting for the decelerating/accelerating expansion history.
CMB measurements give H₀ ≈ 67.4 km/s/Mpc. Local ladder measurements (Cepheids + supernovae) give ≈ 73. This "Hubble tension" is one of the biggest mysteries in modern cosmology.