Redshift Calculator
Compute redshift using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Formula
z = (lambda_obs - lambda_emit) / lambda_emit | v = c × ((1+z)² - 1) / ((1+z)² + 1)
Redshift z measures how much a wavelength has been stretched by cosmic expansion. The recession velocity uses the special relativistic Doppler formula. Distances are calculated by numerically integrating the Friedmann equation for a flat Lambda-CDM cosmology with H0 = 67.4 km/s/Mpc, Omega_M = 0.315, Omega_Lambda = 0.685.
Worked Examples
Example 1: Distant Galaxy Observation
Problem: A galaxy has a measured redshift of z = 2.0. Calculate its recession velocity, comoving distance, and lookback time.
Solution: z = 2.0\nRelativistic velocity: v = c × ((1+z)² - 1)/((1+z)² + 1)\nv = 299792 × (9-1)/(9+1) = 299792 × 0.8 = 239,834 km/s\nv/c = 80%\nComoving distance ≈ 5,200 Mpc (numerical integration)\nLookback time ≈ 10.3 Gyr (universe was 3.5 Gyr old)
Result: v = 239,834 km/s (80% c) | Distance: ~5.2 Gpc | Lookback: ~10.3 Gyr
Example 2: Hydrogen-Alpha Line Shift
Problem: The hydrogen-alpha emission line (656.3 nm) is observed at 722 nm from a galaxy. Calculate the redshift and distance.
Solution: z = (722 - 656.3) / 656.3 = 65.7 / 656.3 = 0.1001\nVelocity ≈ 29,553 km/s\nComoving distance ≈ 420 Mpc ≈ 1.37 billion light-years\nLookback time ≈ 1.29 Gyr\nScale factor at emission: a = 1/1.1 = 0.909
Result: z = 0.1001 | Distance: ~420 Mpc | Lookback: ~1.29 Gyr
Frequently Asked Questions
What is cosmological redshift?
Cosmological redshift is the stretching of light wavelengths caused by the expansion of the universe. As light travels from a distant galaxy toward us, the space through which it travels expands, stretching the wavelength of the light. This makes the light appear redder (shifted toward longer wavelengths) compared to when it was emitted. The redshift parameter z is defined as z = (lambda_observed - lambda_emitted) / lambda_emitted. A redshift of z = 1 means the wavelength has doubled during its journey, indicating the universe has expanded by a factor of 2 since that light was emitted. Cosmological redshift is distinct from Doppler redshift (caused by relative motion) and gravitational redshift (caused by strong gravitational fields), though all three produce similar observational effects.
What is the Hubble constant and why does it matter for redshift?
The Hubble constant (H0) describes the current rate of expansion of the universe, typically expressed in kilometers per second per megaparsec (km/s/Mpc). The current best estimate from the Planck satellite is approximately 67.4 km/s/Mpc, meaning a galaxy 1 Mpc (3.26 million light-years) away is receding from us at about 67.4 km/s. For nearby objects (z < 0.1), Hubble's Law provides a simple linear relationship: velocity = H0 × distance. For higher redshifts, the relationship becomes non-linear and requires integrating over the expansion history of the universe. The precise value of H0 is actively debated in modern cosmology, with local measurements yielding about 73 km/s/Mpc, creating the so-called Hubble tension.
What is the highest redshift ever observed?
The highest spectroscopically confirmed galaxy redshifts have been pushed beyond z = 13 by the James Webb Space Telescope (JWST), with JADES-GS-z14-0 confirmed at z = 14.32 in 2024, meaning its light was emitted when the universe was only about 290 million years old. The cosmic microwave background (CMB) has a redshift of z = 1089, representing the earliest observable light from when the universe became transparent about 380,000 years after the Big Bang. At z = 1089, the CMB photons have been stretched by a factor of 1,090, shifting from visible/infrared light at ~3,000 K to the microwave radiation at 2.725 K we observe today. Gravitational waves could potentially detect signals from even earlier epochs.
Can redshift exceed z = 1, and does that mean faster than light?
Yes, redshift values routinely exceed z = 1 for distant galaxies. Thousands of galaxies have been observed with z > 1, and the most distant known objects have z > 10. While z > 1 corresponds to recession velocities exceeding the speed of light when calculated using the relativistic Doppler formula naively, this does not violate special relativity. The recession is due to the expansion of space itself, not motion through space. General relativity allows space to expand at any rate; it is only local motion through space that cannot exceed c. The recession velocity at z = 1.5 is approximately 0.724c using the special relativistic formula, but the cosmological recession velocity (Hubble flow) actually exceeds c for objects beyond the Hubble sphere, approximately 4,400 Mpc from us.
What is redshift and how does it indicate motion?
Redshift occurs when light from an object moving away from us is stretched to longer (redder) wavelengths. It is measured as z = (observed wavelength - emitted wavelength) / emitted wavelength. Cosmological redshift indicates the expansion of the universe. A galaxy at z=1 is seen as it was when the universe was half its current size.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.