Cosmological redshift

From the previous lecture, every galaxy has fixed comoving coordinates, and physical distances are $\ell_p(t) = a(t)\,\ell_c$. In the idealised model all matter is at rest in these coordinates (real galaxies also have small "peculiar" motions).

Time is measured by cosmic time: clocks at rest relative to the local matter, synchronised by something everyone in the universe can see. Guth's example: agree to set every clock to midnight when the cosmic microwave background has cooled to a chosen temperature. Because the universe is homogeneous, every observer sees the same history, so clocks synchronised once stay synchronised. Any quantity that changes with time and is the same everywhere would do, such as the value of the Hubble rate.

Cosmic time is an idealisation. Near a lump of matter clocks run differently — near the Sun by about one part in a million, near a giant black hole by much more — but for the universe on large scales it works.

Put us at the origin and a galaxy at comoving distance $\ell_c$. Think of each crest of the light wave as a separate pulse travelling toward us. In comoving coordinates every pulse moves at

This speed depends on time but not on position — that is homogeneity. So two successive crests always move at exactly the same coordinate speed at the same moment, and the coordinate distance $\Delta x$ between them never changes. Their physical separation, which is the wavelength, is $a(t)\,\Delta x$, so it grows in step with the universe:

The time between crests is $\Delta t = \lambda/c$, so periods stretch in the same way, and the redshift is simply

Here $t_S$ is when the light left the source and $t_O$ when we observe it. The redshift tells you directly how much the universe has expanded while the light was on its way.

Two differences matter.

It is more general. The derivation used only two facts: the scale factor converts metres to coordinates, and light always moves at $c$ as measured locally. Gravity does not change either fact, so the formula holds with gravity included. Gravity is hidden in $a(t)$, whose evolution it controls. No time-dilation factor appears, because both clocks — at the galaxy and here — are at rest relative to their local matter and read cosmic time.

It uses different variables. The special-relativity formula gives $z$ as a function of a velocity measured in one inertial frame. The cosmological formula gives $z$ in terms of the scale factor at two times. In a real universe with gravity there is no single inertial frame covering both galaxies, so "the velocity" of a distant galaxy is not well defined in the special-relativity sense.

The two must agree when there is no gravity. Then nothing decelerates the expansion, velocities stay constant and $a(t)$ grows linearly; choosing $t = 0$ at the moment when $a = 0$ gives $a \propto t$. Guth announces an extra-credit problem to show that in that case $1 + z = a(t_O)/a(t_S)$ reproduces the relativistic Doppler formula exactly.

• Comoving galaxies and cosmic time give a common description for every observer.
• Light's coordinate speed $c/a(t)$ is the same everywhere at a given time, so crests keep a fixed coordinate separation and wavelengths grow in proportion to $a(t)$.
• $1 + z = a(t_O)/a(t_S)$: redshift measures how much the universe expanded while the light travelled.
• The result includes gravity (through $a(t)$) and reduces to the special-relativity Doppler shift when there is no gravity.