Cosmological Redshift Calculator

Cosmological redshift is the stretching of light wavelengths caused by the expansion of the universe. As a photon travels through expanding space, the space it occupies stretches, increasing the photon's wavelength proportionally. For a galaxy at redshift z, every wavelength has been stretched by a factor of (1+z) relative to its emitted value. This is distinct from Doppler redshift, which is caused by an object's velocity through space.

The scale factor a describes the relative size of the universe at a given cosmic time compared to today. Today a = 1 by definition. At redshift z, the scale factor was a = 1/(1+z). At z = 1, the universe was half its current size (a = 0.5). At z = 3, it was one-quarter the current size (a = 0.25). The CMB was emitted at z approximately 1089, when a was about 0.00092.

For cosmological redshift, the concept of recession velocity is frame-dependent, but a useful definition uses the relativistic Doppler formula: v = c[(1+z)^2 - 1] / [(1+z)^2 + 1]. This gives the velocity an object would need in special relativity to produce the same redshift. For small z (z less than 0.1), the linear approximation v = cz is accurate to a few percent. Note that recession velocities can exceed c for z greater than about 1.5.

The CMB temperature scales as T = T0 x (1+z), where T0 = 2.725 K is today's CMB temperature. This is because the CMB photons are stretched by the same factor (1+z) as all wavelengths in an expanding universe. At z = 1089 (recombination), T was approximately 2.725 x 1090 = 2970 K, consistent with the hydrogen recombination temperature of about 3000 K.

Doppler redshift arises from an object moving through space (special relativity). Cosmological redshift arises from space itself expanding between emitter and observer. In the real universe both effects can be present: peculiar velocities (motion through space) add a Doppler component on top of the cosmological redshift from expansion. For distant galaxies, the cosmological component dominates; for nearby galaxies in clusters, peculiar velocities can dominate.

The Virgo Cluster: z approximately 0.004 (1200 km/s). The Coma Cluster: z approximately 0.023. The most distant quasar J1030+0524: z = 6.30. The most distant galaxy JADES-GS-z14-0: z = 14.32. The CMB: z approximately 1089. At z = 1089 the universe was 380,000 years old; at z = 14.32 it was only about 290 million years old.

The calculator includes: H-alpha (Balmer series, 656.28 nm, rest), H-beta (486.13 nm), Lyman-alpha (121.567 nm, UV, used for high-z quasars), Ca K (393.366 nm, used for galaxy surveys), O III (500.684 nm, emission line), and Mg II (279.8 nm, UV doublet). Any custom wavelength can also be entered in the rest wavelength field.

Lyman-alpha (121.567 nm) is the shortest-wavelength photon that neutral hydrogen can emit. At shorter wavelengths (Lyman limit at 91.2 nm), hydrogen becomes opaque. When a galaxy is at redshift z, its Lyman break shifts to 91.2 x (1+z) nm and Lyman-alpha shifts to 121.567 x (1+z) nm. At z = 3, the Lyman break falls at 364.8 nm (visible), making the galaxy disappear in blue filters. This Lyman break technique identifies high-z galaxies photometrically.

Yes. Galaxies with redshift z greater than about 1.5 have recession velocities exceeding c in the sense of Hubble's law v = H0 x d. This is not a violation of special relativity because the recession is due to space expansion, not motion through space. Light from these galaxies can still reach us because the expansion was slower in the past. The observable universe extends to z approximately 1089 (the CMB), well into the superluminal regime.

The lookback time (how long ago the light was emitted) depends on the cosmological model. In the standard flat LCDM model (H0 = 67.4, Omega_m = 0.315, Omega_Lambda = 0.685), z = 1 corresponds to about 7.9 Gyr lookback time (universe was 5.9 Gyr old), z = 3 to about 11.5 Gyr, and z = 10 to about 13.3 Gyr. Precise lookback times require numerical integration of the Friedmann equation.

Every photon emitted at redshift z arrives with its wavelength multiplied by (1+z). If a galaxy emits H-alpha at 656.28 nm and is at z = 0.5, we observe it at 656.28 x 1.5 = 984.42 nm (near-infrared). Photon energy is proportional to 1/wavelength, so photons arrive with 1/(1+z) of their emitted energy. At z = 1, each photon carries half its original energy. The CMB photons were emitted as near-infrared at recombination and are now microwaves: stretched by factor 1090.

Cosmological blueshift would require space to be contracting rather than expanding. In the current accelerating universe, cosmological blueshift does not occur for distant galaxies. However, galaxies in the Local Group (Andromeda, Triangulum) show blueshift due to their peculiar velocities toward the Milky Way being larger than the small cosmological recession at their distance. Andromeda shows a blueshift of about 110 km/s and will merge with the Milky Way in roughly 4.5 Gyr.

As of 2024, the highest-confirmed galaxy redshift is JADES-GS-z14-0 at z = 14.32, observed by the James Webb Space Telescope. This galaxy existed when the universe was only about 290 million years old. Some candidate galaxies at z greater than 16 have been reported but need spectroscopic confirmation. The absolute record is the CMB at z approximately 1089, the surface of last scattering.