Edwin Hubble found that distant galaxies' spectra are shifted to the red, more so the farther away they are (see Hubble's Law). For nearby galaxies the shift looks like an ordinary Doppler shift. But in an expanding universe there is a cleaner way to think about it, and it is the one that works at any distance.
Why it works
Describe the expansion with a scale factor $a(t)$: every distance between galaxies (far enough apart not to be bound together) is proportional to $a$, with $a = 1$ today. Light traveling between galaxies is stretched along with the space it crosses: each wave crest is carried apart with everything else. So if light left a galaxy when the scale factor was $a$ and arrives today, its wavelength has grown by $1/a$: $$1 + z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}} = \frac{a_{\text{now}}}{a_{\text{then}}}.$$
Key idea
A cosmological redshift tells you how much the universe has expanded while the light was on its way: $1 + z = 1/a$. A galaxy at $z = 1$ sent its light when distances were half of today's; the cosmic microwave background, at $z \approx 1100$, when they were about 1/1100 of today's.
Worked examples
Worked example: A distant galaxy
The Lyman-alpha line (121.6 nm) of a galaxy is seen at 851 nm. Find $z$ and the scale factor when the light left.
- $1 + z = 851/121.6 = 7.00$.
- $z = 6.00$ and $a = 1/7.00 = 0.143$.
Answer: $z = 6.0$: distances were one-seventh of today's. The line, emitted in the ultraviolet, now arrives as infrared light.
Worked example: How much have distances grown?
Light from a quasar has $z = 3.0$. By what factor have distances between galaxies grown since it left?
- $1 + z = 4.0$.
Answer: Four times larger, so the average density of matter was $4^3 = 64$ times higher then.
Worked example: Redshifts multiply
Light from a quasar at $z = 4$ passed a galaxy that we see at $z = 1$. What redshift did that galaxy see?
- Stretch factors multiply: $(1 + z_{\text{total}}) = (1 + z_1)(1 + z_2)$.
- $1 + z_1 = 5/2 = 2.5$.
Answer: $z = 1.5$. Adding the redshifts (to get 3) would be wrong.
Common mistake
Thinking a cosmological redshift is only a Doppler shift from galaxies moving through space. For nearby galaxies the two pictures agree ($v \approx cz$). For distant ones the redshift measures the expansion of space during the trip, $1 + z = 1/a$, and plugging large $z$ into a Doppler formula gives a "speed" with no clear meaning. Distances between faraway galaxies can grow faster than $c$ without anything moving through space faster than light. Check: $1 + z$ is always a ratio of sizes of the universe.