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There are several existing threads on the difference between cosmological and Doppler redshifts. However, I don't see that any of them answered the question below.

@pela gave the following example: "In principle you could have a universe ... that were static when a distant galaxy emitted a photon, then at some point expanded quickly by a factor of 2, and then again is static. In this hypothetical case, the observer would still measure the photon to have been redshifted by a factor of 2 (i.e. 𝑧=1)."

That very nicely made a clear distinction between cosmological and Doppler redshifts. However, suppose we modify this scenario so that the universe is already expanding when the light is emitted from some distant galaxy. Then the distance between that galaxy and ours is increasing when the light is emitted, and therefore we should expect a Doppler shift. Then, while the light is on its way to us, the universe is still expanding, and the light is getting further redshifted due to the expansion. So then it seems like the redshift we observe should be a combination of both a Doppler shift AND a cosmological redshift. Is that correct? And, if so, how does one disentangle the two and determine the contribution from each redshift mechanism?

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Firstly, I think it's important to note that in general relativity there is only one redshift, and that a cosmological redshift can be interpreted as the result of infinitely many infinitesimal redshifts between infinitesimally small inertial frames receding from one another. But I still think it makes sense to distinguish between "regular" Doppler shifts and redshifts due to expanding space, for the reason you quote me for.

If we make that distinction, then it seems to me that your scenario is mixing two things: If space is expanding when a photon leaves a distant galaxy, then that photon experiences a cosmological redshift from the beginning. But the galaxy is not moving through space away from us, so it does not experience a Doppler shift.

In reality, the Universe is in fact expanding from the beginning (although it doesn't expand as fast close to the galaxy, because the mass density is above average).

Peculiar velocities

Also in reality, the galaxy does in factmove through space, so there is actually a Doppler shift, albeit not because of the effect you mention. This motion is called the peculiar velocity of the galaxy, and in principle there is no way to distinguish, although various methods exist to estimate it. For instance, you can deduce the peculiar motion by measuring the redshifts of the galaxies in a group or a cluster, all assumed to be at the same cosmological distance and hence have the same cosmological redshift.

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  • $\begingroup$ I understand what you've written. Isn't it true that as long as the distance between us and a distant galaxy is increasing, we will observe a Doppler shift? Even if the galaxy is receding solely due to the expansion of space, I'd still expect to observe a Doppler shift, because its distance from us is increasing. Am I missing something? If the universe continues to expand while the light is on its way to us, of course there would be an additional cosmological redshift. $\endgroup$
    – Matt
    Feb 15 at 16:38
  • $\begingroup$ @Matt In my answer I'm distinguishing between a Doppler redshift arising from a difference in velocity between emitter and observer, and a cosmological redshift arising from the photon traveling through expanding space. The former only occurs if there's a difference in velocity between [the emitter at the time of emission] and [the observer at the time of observation]. The latter only occurs if space expands at any of the positions that the photon passes, and — and this is important — at the time the photon passes that location [cont'd below]. $\endgroup$
    – pela
    Feb 16 at 12:34
  • $\begingroup$ That means that, in principle, you could have a wacky universe where space expands in some place between the galaxy and us, but for some reason not exactly where the photon is. In that case, the distance between the galaxies would increase, but the photon would not experience any redshift. Our universe isn't so wacky, we believe, but seems to be expanding roughly the same everywhere [cont'd below]. $\endgroup$
    – pela
    Feb 16 at 12:35
  • $\begingroup$ On the other hand, you could in principle also have a static Universe — the observed redshift would then only be a result of 1) the velocity of the distant galaxy at the moment it emitted the photon, and 2) the velocity of us at the moment we receive it. $\endgroup$
    – pela
    Feb 16 at 12:36
  • $\begingroup$ Let's look at your wacky universe where space is expanding only in a small region between the distant galaxy and us (and assume that neither that galaxy nor us is moving through space). Because of that expansion, the distance between that galaxy and us is increasing. So, because of that increasing distance, when the photon is emitted, I'd expect a Doppler shift -- just from the fact that the distance between the two galaxies is increasing (even though that distance increase is coming from an expansion where the photon is not present). Am I missing something? $\endgroup$
    – Matt
    Feb 26 at 20:54

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