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

Sten
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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".

Michael Seifert
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    I gave another answer detailing this, but I disagree that expansion of space is necessary or appropriate to explain the cosmological redshift. – Sten Jan 19 '24 at 23:10
  • @Sten Interesting perspective in the Bunn & Hogg paper. I'll have to mull it over further, but I suspect I'm probably one of the "purists" mentioned in Section III of the paper. – Michael Seifert Jan 20 '24 at 03:05
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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.

Sten
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    Ah, at last an explanation. The parallel transport argument makes perfect sense. – Lee Mosher Jan 20 '24 at 03:57
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    So, in order not to invoke the “expansion of space” you are invoking “many [imaginary] observers between the source and the receiver”. But those observers have positive expansion (in the precise technical sense), moreover this set of imaginary observers is uniquely determined by spacetime (as maximizing proper time since BB). So what you are really invoking is expansion of unique congruence associated to spacetime. So why is it important not to refer to this expansion as “expansion of space”? – A.V.S. Jan 20 '24 at 18:36
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    @A.V.S. What is problematic is the reification of expanding space. It's just a coordinate choice. Expanding space doesn't stretch things -- light or anything else. Distant galaxies are not at rest while space expands between them -- that is only one possible coordinate choice. All that is real about cosmic expansion is that the contents of the universe are expanding. The imaginary observers are comoving with those contents (nothing fancier than that). – Sten Jan 20 '24 at 19:06
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    @A.V.S. Also note that both of the calculations I suggest are completely general, not specific to FLRW spacetimes as you seem to suggest. For the first (parallel transport along the light path) this is obvious. For the second, the observer velocities just need to smoothly interpolate between the source and the receiver (although it's a bit more natural to picture in fluid-filled spacetimes, so that each observer can be at rest with respect to their local fluid element). – Sten Jan 20 '24 at 19:18