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I am somewhat confused regarding the factors that cause cosmological redshift. This wiki states that the amound of cosmological redshift coming from let’s say, a star, is calculated by

$$z=\frac{H_0 D}{c}$$

Since $H_0D$ simply equals the recession speed $v$ of the star, I’d deduce that the amount of redshift depends solely on the star's recession speed the moment the light left it. This would be an analogue to the Doppler redshift.

However, some sources like this one claims that the longer light has been traveling, the longer it has been exposed to the expansion of the universe making the redshift larger.

This statement makes me deduce that a portion of ligth that already has left the star, can be even more redshifted because of the space expansion during its travel, being it a constant or an accelerated expansion.

If this is true, why does the redshift formula then solely rely on the recession speed?

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I think this website answers your question. In a word, "cosmological redshift due to recession speed or expansion of space", both are correct in local. They appear different because you choose different coordinate.

If you go beyond local, "cosmological redshift is due to expansion of space" may be more appropriate. And Hubble's law should be generalized as redshift-distance relation.

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  • $\begingroup$ Thanks, this helped me quite a bit. Is the expansion of the universe the same as stretching spacetime? Such that if we put a very long ruler in space between earth and a galaxy, it also stretches out making us not notice any distance increase between them? $\endgroup$
    – Phy
    Feb 17, 2021 at 13:08
  • $\begingroup$ @Phy I think (not quite sure) space does not stretch its contents as the universe expands. In fact Baryon acoustic oscillation is usually used as standard ruler (very similar to ruler), which can be used to measure the expansion history. Please check this wiki page. $\endgroup$
    – R.Y. Zhao
    Feb 18, 2021 at 1:48
  • $\begingroup$ @Z.Y. Zhao Thanks. I have one last question. With local do you mean significantly lower than speed $c$? Such that the formula Doppler redshift formula in my opening post would always give the correct redshift in local, regardless of which coordinates you use? $\endgroup$
    – Phy
    Feb 18, 2021 at 10:57
  • $\begingroup$ @Phy Yes, local means $z\ll 1$. $\endgroup$
    – R.Y. Zhao
    Feb 18, 2021 at 12:23
  • $\begingroup$ @Z.Y. Zhao Sorry for coming back with another question after a while. Does this mean that the "redshift due to recession speed" approach is also less accurate than the other if space expansion has accelerated significantly after the concerning light portion has left a star? After all, the "redshift due to recession speed" would deduce that the light portion would not be influenced by any recession speed change after leaving a star. $\endgroup$
    – Phy
    Mar 30, 2021 at 21:05

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