Redshift Calculator: Wavelength, Velocity and Cosmological Distance
This redshift calculator solves three related problems in one tool. In Spectral mode, enter the emitted and observed wavelengths (or frequencies) of a spectral line to get the redshift parameter z. In Velocity mode, enter a known z to find the relativistic recession velocity as a fraction of the speed of light. In Cosmological mode, enter z together with the Hubble constant and density parameters to get lookback time, comoving distance, luminosity distance, and angular diameter distance using a flat Lambda-CDM model with Planck 2018 defaults.
Formula
Worked example
Hydrogen-alpha has a rest wavelength of 656.28 nm. A galaxy spectrum shows this line at 790.0 nm. Then z = (790.0 - 656.28) / 656.28 = 133.72 / 656.28 = 0.2037. The relativistic recession speed is v/c = ((1.2037)^2 - 1) / ((1.2037)^2 + 1) = (1.4489 - 1) / (1.4489 + 1) = 0.4489 / 2.4489 = 0.1834, so v = 54,972 km/s. Using Planck 2018 parameters, the lookback time is about 2.5 Gyr and the comoving distance is about 900 Mpc.
What is redshift?
Redshift is the stretching of light to longer wavelengths (or lower frequencies) when a source is moving away from the observer or when the space between source and observer is expanding. The redshift parameter z = (lambda_obs - lambda_emit) / lambda_emit is dimensionless: z = 0 means no shift, z = 1 means the observed wavelength is double the emitted one. Blueshift (z < 0) occurs when source and observer approach each other. Three distinct physical causes produce apparent redshift: the Doppler effect for moving sources, the cosmological redshift due to the expansion of space, and gravitational redshift where photons lose energy climbing out of a gravitational well.
How to measure redshift from a spectrum
Astronomers identify absorption or emission lines in a spectrum by their pattern. Because atoms produce the same line pattern regardless of where they are, a line found at a different wavelength than its known laboratory value reveals the shift. The most commonly used lines are hydrogen-alpha (656.28 nm), hydrogen-beta (486.1 nm), the Lyman-alpha line (121.6 nm, important at high z when it shifts into the visible), and the calcium H and K lines (393.4 nm and 396.8 nm) in galaxy spectra. Once you have the emitted and observed wavelengths, the formula z = (lambda_obs - lambda_emit) / lambda_emit gives the redshift. The same formula applies to frequencies, with the sign flipped: z = (f_emit - f_obs) / f_obs.
Relativistic Doppler vs non-relativistic approximation
For small redshifts (z much less than 0.1), the simple approximation v = z x c works well. At larger z, special relativity must be used. The relativistic formula v/c = ((1+z)^2 - 1) / ((1+z)^2 + 1) ensures the recession speed can never reach or exceed the speed of light. For example, at z = 1, v_approx = c but the relativistic v = 0.6c. At z = 2 the non-relativistic formula gives 2c (physically impossible) while the correct answer is about 0.8c. This calculator always uses the relativistic formula.
Cosmological distances: comoving, luminosity, and angular diameter
When redshift is cosmological (caused by the expansion of space rather than peculiar motion), it encodes several distinct distance measures. The comoving distance D_C is the separation in the present-day reference frame; it grows as the universe expands and is what you get by integrating c/H(z) from 0 to z using the Friedmann equation. The luminosity distance D_L = D_C x (1 + z) is used to convert an object's apparent brightness to intrinsic luminosity - it is larger than the comoving distance because photons were emitted when the universe was smaller and have been diluted since. The angular diameter distance D_A = D_C / (1 + z) converts an observed angular size to a physical size; it actually decreases again beyond z ~ 1.6 in a standard Lambda-CDM model, meaning objects of the same size appear larger as they get more distant past that redshift. Lookback time is the age difference between the universe now and when the light was emitted. All four quantities are computed here using numerical integration of the Friedmann equation with Planck 2018 parameters.
Notable redshifts in astronomy
| Object / Event | Redshift (z) | Lookback time | Scale factor |
|---|---|---|---|
| Nearby galaxies (Virgo Cluster) | 0.004 | ~56 Myr | 0.9960 |
| Perseus Cluster | 0.018 | ~247 Myr | 0.9823 |
| Distant galaxy - moderate | 0.5 | ~5.1 Gyr | 0.667 |
| Hubble Deep Field era | 1.0 | ~7.7 Gyr | 0.500 |
| Cosmic noon (peak star formation) | 2.0 | ~10.3 Gyr | 0.333 |
| Most distant confirmed galaxies | ~13 | ~13.6 Gyr | 0.071 |
| Cosmic Microwave Background | 1089 | ~13.8 Gyr | 0.00092 |
Reference redshifts and their cosmological context using Planck 2018 parameters.
Frequently asked questions
What does a redshift of z = 1 mean?
A redshift of z = 1 means the observed wavelength is exactly twice the emitted wavelength. In terms of distance, using Planck 2018 cosmological parameters, z = 1 corresponds to a lookback time of about 7.7 billion years and a comoving distance of about 3300 Mpc (10.8 billion light-years). The universe was half its present size when that light left its source.
Can recession velocity exceed the speed of light?
The relativistic Doppler formula used by this calculator never exceeds c, but the cosmological recession speed of distant galaxies can. The non-relativistic approximation v = zc gives velocities above c for z > 1, which is why it must not be used at high z. The correct relativistic result for any Doppler-based velocity is always below c. However, the expansion of space itself can carry objects apart faster than light (for objects beyond the Hubble sphere), and that is a different concept - not captured by the simple relativistic Doppler formula.
What is the difference between Doppler redshift and cosmological redshift?
Doppler redshift arises from relative motion through space, and the exact relativistic formula v/c = ((1+z)^2 - 1) / ((1+z)^2 + 1) applies. Cosmological redshift is caused by the expansion of space: the wavelength of a photon stretches as the universe grows, so by the time it arrives its wavelength has increased by a factor of (1 + z). For nearby galaxies the two descriptions give the same answer, but for distant ones only the cosmological framework with the full Friedmann equations gives physically meaningful distances.
What are the Hubble constant and density parameters used for?
In the cosmological mode, H0 sets the overall scale of the universe's expansion rate (how fast space is stretching per megaparsec). The matter density OmegaM and dark energy density OmegaLambda determine the shape of the expansion history: matter decelerates the expansion, dark energy accelerates it. Changing H0 scales all distances and lookback times proportionally. Changing OmegaM and OmegaLambda shifts the weight between matter-dominated (earlier deceleration) and dark-energy-dominated (later acceleration) epochs. The defaults here are the Planck 2018 best-fit values (H0 = 67.4 km/s/Mpc, OmegaM = 0.315, OmegaLambda = 0.685).
Which spectral lines are best for measuring redshift?
Hydrogen-alpha (656.28 nm) is the most commonly used in optical spectra of galaxies; at higher redshifts it shifts into the near-infrared and requires infrared telescopes. Lyman-alpha (121.6 nm) is used for high-z galaxies and quasars: at z ~ 2 it shifts to about 365 nm (near-UV) and at z ~ 6 it shifts to visible 850 nm. The calcium H and K doublet (393.4/396.8 nm) is useful in absorption-line spectra of elliptical galaxies. Radio astronomers use the 21-cm hydrogen line (1420 MHz) and CO rotational lines. This calculator works with any line as long as you know both the rest and observed wavelengths or frequencies.
What is the highest measured redshift?
As of early 2025, the highest spectroscopically confirmed galaxy redshift is around z = 14 to z = 16, detected by the James Webb Space Telescope. The cosmic microwave background (CMB) comes from z ~ 1089, when the universe was about 380,000 years old and roughly 0.09% of its current size. This calculator handles any redshift from near zero up to and including CMB-scale values, though the numerical integration uses a matter-plus-dark-energy model and ignores radiation density, which becomes significant at z > 3000.