Cosmological Redshift
Cosmological redshift is the increase in the wavelength of photons as they travel through an expanding universe, caused by the stretching of space itself rather than by the relative motion of source and observer (Doppler effect). Quantified by the redshift parameter z = (λ_observed − λ_emitted) / λ_emitted, it is directly related to the expansion factor of the universe: 1 + z = a(t_now) / a(t_emit). Cosmological redshift allows astronomers to determine the distance and lookback time to distant galaxies and serves as a primary tool for mapping the large-scale structure of the universe.
Key Formula
z = (λ_obs - λ_emit) / λ_emit = a(t₀)/a(tₑ) - 1
LaTeX: z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}} = \frac{a(t_0)}{a(t_e)} - 1
| Symbol | Meaning | Unit |
|---|---|---|
| z | Redshift parameter (dimensionless) | dimensionless |
| λ_obs | Observed wavelength | nm or m |
| λ_emit | Emitted (rest-frame) wavelength | nm or m |
| a(t₀) | Scale factor of universe today | dimensionless |
| a(tₑ) | Scale factor at time of emission | dimensionless |
Worked Example
Problem
A distant quasar shows the hydrogen Lyman-alpha line (rest wavelength 121.6 nm) observed at 729.6 nm. Calculate the redshift z and the scale factor of the universe at the time of emission.
Solution
Step 1 – Calculate redshift: z = (λ_obs - λ_emit) / λ_emit z = (729.6 - 121.6) / 121.6 z = 608.0 / 121.6 z = 5.0 Step 2 – Scale factor at emission (taking a(t₀) = 1): 1 + z = a(t₀) / a(tₑ) 1 + 5.0 = 1 / a(tₑ) a(tₑ) = 1 / 6.0 ≈ 0.167 This means the universe was about 1/6 of its present size when this light was emitted.
Answer
Redshift z = 5.0; scale factor at emission a(tₑ) ≈ 0.167 (universe was ~1/6 present size).
Redshift Ranges and Corresponding Cosmic Eras
| Redshift z | Lookback Time (Gyr) | Era | Observable Objects |
|---|---|---|---|
| 0 | 0 | Present day | Milky Way, Local Group |
| 0.1 | ~1.3 | Recent universe | Nearby galaxy clusters |
| 1 | ~7.7 | Half-age universe | Active galactic nuclei |
| 3 | ~11.5 | Early galaxy formation | Quasars, Lyman-break galaxies |
| 6 | ~12.9 | Reionisation era | Earliest quasars |
| 1100 | ~13.8 | Recombination | CMB photons |
Interactive Tools
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Related Terms
Expanding Universe
The expanding universe is the observational discovery that the fabric of space itself is stretching over time, causing galaxies that are not gravitationally bound to each other to recede from one another at velocities proportional to their separating distances. First confirmed observationally by Edwin Hubble in 1929 through measurements of galaxy redshifts, this expansion is described by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric in general relativity. The rate of expansion, parameterised by the Hubble constant H₀ ≈ 67–73 km/s/Mpc, has been measured to be accelerating due to dark energy.
Hubble's Law
Hubble's Law is the empirical observation that the recession velocity of a galaxy is directly proportional to its distance from the observer, expressed as v = H₀d, where H₀ is the Hubble constant. First published by Edwin Hubble in 1929 based on measurements of galaxy redshifts, it provided the first direct observational evidence for the expanding universe predicted by general relativity. The Hubble constant H₀, currently estimated at approximately 67–73 km/s/Mpc from different methods, also allows astronomers to estimate the age of the universe as roughly 1/H₀.
Cosmic Microwave Background
The Cosmic Microwave Background (CMB) is the thermal electromagnetic radiation permeating the entire observable universe, representing the afterglow of light released approximately 380,000 years after the Big Bang when the universe cooled enough for protons and electrons to combine into neutral hydrogen atoms. It is observed today as a nearly uniform blackbody radiation at a temperature of approximately 2.725 K, with tiny temperature fluctuations of about 1 part in 100,000 that encode the seeds of large-scale cosmic structure. The CMB is considered one of the strongest pieces of evidence for the Big Bang model and provides precise measurements of cosmological parameters.
"Cosmological" derives from Greek "kosmologikos" (of the world order). "Redshift" combines "red" (longest visible wavelength, from Old English "rēad") and "shift" (Old English "sciftan," to arrange), reflecting the displacement of spectral lines toward longer wavelengths. Distinguished from Doppler redshift by Lemaître and Hubble in the 1920s–1930s.