The Redshift

We define doppler redshift to be the shift in spectral lines due to motion:

which, in the case of v<<c reduces to the familiar

The cosmological redshift is something different, although we are often sloppy and refer to it in the same terms of the doppler redshift. The cosmological redshift is actually due to the expansion of space itself.

How do we relate the cosmological redshift to the expansion of the Universe? Start with the R-W metric again

and watch what happens to a light ray moving through space. In this case, ds²=0 (why?)

Again orient the coordinate system so that theta=phi=0, then integrate along the path length, from the time of emission (t_e) to now (t₀). Look at two wavecrests in the light ray, separated in time by .

(sheesh!)

Okay, now we can pull R(t) out of the integral and treat it as a constant (why?). Then we get

Now since wavelength is equal to c we have

Or, by our definition of redshift:

We get:

(the Lemaitre Equation)

So redshift is related to the expansion factor of the Universe. If we measure a redshift of z=2, the Universe is 3x bigger now than it was when that photon was emitted.

Also, this gives rise to an expression of cosmological time dilation:

(1+z) = dt(0) / dt(e)

The event as observed takes longer (is stretched) than as it happens in the rest frame.

(from Goldhaber et al 1996)