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 (te) to now (t0). Look at two wavecrests in the light ray, separated in time by .
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
So redshift is related to the
expansion factor of
Universe. If we measure a redshift of z=2, the Universe is 3x
now than it was when that photon was emitted.
Also, this gives rise to an expression of cosmological time