As we observe a remote galaxy, we see it with a redshift. The most distant galaxy discovered to date is GN-z11 visible with the redshift of $z=11.09$. For simplicity, let's assume no gravitational redshift.
In Special Relativity, the Doppler effect has two components, the Doppler component $1+\beta$ and the time dilation component, which is simply $\gamma$. The combined relativistic effect is $z+1=(1+\beta)\gamma$.
In case of the expanding universe, the Doppler effect would seem to have similar components, the Doppler component due to the galaxies recession speed and the time dilation component due to the space expansion. Some argue that there is no time dilation in this case, based on the grounds of comoving time. However, this argument holds neither logically, because the relative observed time is different from the cosmological time, nor practically, because without the time dilation component the maximum observed redshit would be $z=1$ for $\beta\approx 1$ near the particle horizon.
Could someone please clarify if there is a relative time dilation in the expanding universe? Do we observe time of remote galaxies moving slower? Otherwise, if there is no such a time dilation, then what additional factors make the Doppler effect redshift so significant for distant objects?
Answer
Here is a derivation of the cosmological redshift (redshift due to cosmological expansion of space): https://en.wikipedia.org/wiki/Redshift#Expansion_of_space
All the $\beta$ and $\gamma$ terms are coming from Lorentz transformations between inertial frames in the framework of Special Relativity. However, an expanding universe cannot be described by Special Relativity, you need the tools of General Relativity, more specifically, one of GR's solutions used in cosmology, the FLRW metric. If you have no knowledge of GR, I'd recommend you the book "Physical foundations of cosmology" by V. Mukhanov.
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