The Expanding Universe

Section 4 of 6

Redshift as a Scale-Factor Record

Part 3: Redshift as a Scale-Factor Record

In Module 1, we learned the Doppler shift: motion along the line of sight stretches or compresses wavelengths. That model still works for many nearby objects. But on cosmological scales, there is a deeper interpretation. Light from a distant galaxy travels while the universe expands. As space expands, the wavelength of that light stretches too.

This equation has three connected ways to read the same redshift, where $z$ is the fractional wavelength increase, $1+z$ is the total stretch factor (the ratio of observed to emitted wavelength), and that stretch equals the scale-factor ratio $a_0/a_{\text{emit}}$ .

The quantity $z$ is defined as

z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}.

If we set today’s scale factor to $a_0 = 1$ , then light emitted when the universe had scale factor $a_{\text{emit}} = 1/2$ is observed with $1+z = 2$ , or $z = 1$ . The universe doubled in scale while the light was traveling.

Schematic timeline of the expanding universe showing galaxies farther apart at later cosmic times, with expansion represented by a widening space-time diagram. — Figure 5What to notice: in an expanding universe, large-scale distances stretch with time. The galaxies are not flying through a pre-existing space in the simple everyday sense; the scale of space changes.

Cosmological redshift is therefore a time-and-scale clue. It tells us that the light we receive has crossed an evolving universe. To turn redshift into a precise distance or age, we need a cosmological model, including the effects of matter, radiation, curvature if present, and dark energy. But the core idea is already enough here: redshift records expansion. We call this stretch the cosmological redshift.

Cosmological redshift

Wavelength stretching caused by the expansion of the universe while light travels — distinct from an ordinary Doppler shift due to motion through space. It records the ratio of the present scale factor to the scale factor at emission.

Multiple choice

If today’s scale factor is $a_0=1$ and light was emitted when $a_\text{emit}=1/4$ , what redshift do we observe?

z = 3

z = 1/4

z = 1

Numeric answer

A galaxy has redshift $z = 3$ . What is the wavelength stretch factor $1+z$ (the ratio of observed to emitted wavelength)?

Your answer