For a universe described by the RW metric, a relation between the scale factor at the time of emission of light and the redshift can be derived, and yields $$ a(t_e) = \frac{1}{1+z} $$ The above equation depends only on the time of emission and the redshift. The above relation implies that the redshift of any light from any source at any distance is the same, and depends only on the scale factor of the universe at the time of emission. In an isotropic and homogeneous universe (in large scales), the scale factor is only a function of time. How does this settle with Hubble's law, which states that there is a relation between the redshift and the distance between galaxies?
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2 Answers
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The short answer is that, as you said, the redshift depends upon the scale factor at the time of transmission (as compared to the present). Since light travels at a finite speed, light from more distant sources was transmitted at a different time and hence scale factor.
You're redshift equation does NOT imply the same redshift for any distance, I think you were just interpreting, forgetting that light we're currently receiving from distant and near (relatively speaking) stars was released at VERY different times (and hence scale factors). The Hubble relation follows directly from the redshift equation for an expanding universe.
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$\begingroup$ OK so 2 sub-questions: (i) So I basically need to consider $z$ also as a function of the time of emission? (ii) The fact that redshift is not a property characterizing the light-source, but a property that is measured relatively to the light-source, two light source and different distances from the observer emitting light on the same time will not reach the observer on the same meaningful time scale, hence when the light from the more distant object will be more (or less) redshifted? $\endgroup$– JonTrav1Jul 30, 2016 at 11:33
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$\begingroup$ i) z is a function of the time of emission and the time of observation; however since the speed of light is a constant, one can instead determine the distance at emission, they're interrelatable. ii)correct, however things get more complicated if you consider just what kind of function is the scale parameter of t. This is what the redshift surveys are doing, measuring the change in the change of the scale parameter (ie. the acceleration of the universe), which even itself is changing. $\endgroup$ Jul 30, 2016 at 11:40
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Define a galaxy to be at a distance $D$, where $D$ changes with the scale factor $$\frac{D(t)}{D_0} = a(t),$$ where $t$ is the time of light emission and $a_0=1$.
The recession velocity $$ v = \dot{D(t)} = D_0 \dot{a(t)}.$$
If we say $H = \dot{a}/a$, then $$ v = D_0 H a(t) = HD(t)$$
This is the fundamental Hubble relationship. But the linear relationship with $z$ is an approximation for small $z$ and where $H$ does not change greatly with time. $$ z = a(t)^{-1} -1 \simeq (a_0-a_0H_0t)^{-1} -1 \simeq H_0t$$
If we say $t \simeq D/c$ then $$cz = H_0 D$$
However this relationship is not true at very, very small redshift. The objects have to be far enough away that their peculiar velocities are small with respect to the "Hubble flow", so that there is a nearly unique relationship between distance, scale factor and time of emission.
