Is cosmological redshift associated with recession velocity when the light left or when it arrived?

Question

Is the cosmological redshift $z$ associated with the recession velocity when the light left, when it arrived, or something in-between?

Answer 1

Neither; the redshift is determined by the ratio of the scale factors at emission and observation.

As an example of this, consider an FRW cosmos whose scale factor $a(t)$ is controlled by a malevolent deity. Initially, the scale factor is constant with respect to time, at some value $a_1$ (and $\dot{a} = 0$.) A photon is emitted from Galaxy A at time $t_1$, and begins traveling to Galaxy B. During the time of flight of the photon, the malevolent deity expands the Universe, so that $a(t)$ increases from $a_1$ to $a_2$. But by the time the photon gets to galaxy B at time $t_2$, the scale factor is now $a(t_2) = a_2 > a_1$, and has $\dot{a} = 0$ again.

Now, Galaxies $A$ and $B$ are at rest with respect to each other at both the emission time and at the reception time of the photon; their proper distances are not increasing with respect to each other at either moment of time. But the photon has also been redshifted during flight; we have $$ 1+z = \frac{a(t_2)}{a(t_1)} = \frac{a_2}{a_1} > 1 $$ and so $z > 0$.

This is a contrived example, of course, but it illustrates an important point: the cosmological redshift is due to the expansion of space and not due to the galaxies "moving away from each other".

Answer 2

Something in between. Relative velocities of distant objects are not uniquely defined in curved spacetimes, so a unique answer isn't possible.

However, you can say that the cosmological redshift is a Doppler shift associated with the velocity difference between the source at the emission time and the receiver at the arrival time, as long as you define that difference by essentially dragging the velocity vectors along the light's spacetime path to bring them together (technically using parallel transport).

Another approach is to think of the cosmological redshift as an accumulation of infinitesimal Doppler shifts along the path of the light. Conceptually, imagine a line of many observers between the source and the receiver, such that each observer is at rest with respect to their local universe. Each time the light passes from one observer to another, it is slightly redshifted because the observers are receding from each other. Successive observers can be taken to be arbitrarily close, so the calculation of the Doppler redshift is unambiguous. Putting all of these redshifts together results in the cosmological redshift.

Importantly, there is no need to invoke expansion of space to accurately describe the cosmological redshift, contrary to what the other answer suggests. There are also lots of reasons why it's misleading to imagine that "expansion of space" has physical effects. These points are discussed at further length in this excellent pedagogical article.