Hubble law, cosmological redshift and distance

Question

I have got confused with some cosmology formulas which calculate distance.

The most famous formula is indeed the Hubble law which says (nowadays) velocity is proportional to Hubble's parameter at current time. On the other hand we can find the velocity from redshift.

By Hubble law it is possible to evaluate distance but it's not clear for me that what does this distance mean...! As the photons, which are now observed, are belong to another redshift (larger) and by traveling toward Earth the have become "redder" because of expansion of universe, what does the observed redshift stand for? The actual redshift of the galaxy at observation time (although we can't recieve the photons now), the redshift at emission time (that means we do not calculate current distance of the galaxy) or something else?

Thanks a lot.

Answer 1

I think that the wikipedia page on Hubble's law is reasonably clear. The distance in Hubble's law is the proper distance. This is the separation between two objects measured by observers at the same cosmic time. That is, if you imagine (instantaneously) stretching a load of metre rulers end-to-end from us to a distance galaxy, it is how many metre rules you would need. The velocity in Hubble's law is the rate of change of proper distance with cosmic time.

On large enough scales we find that that the ratio of velocity to proper distance is the Hubble parameter, which itself changes with cosmic time.

$$ v(t) = H(t) D(t)$$

In practice we cannot measure the velocity or proper distance in Hubble's law - we can generally only estimate these quantities for a galaxy as we observe it some time in the past, when it emitted the light we detect. This is why Hubble's law in terms of measured redshifts and estimated distances is only applicable over relatively small (cosmologically speaking) distances and for recession velocities much less than the speed of light.

For more distant galaxies the distance cannot be calculated from a redshift without a cosmological model for how the Hubble parameter changes with cosmic time. This in turn depends on the adopted cosmological parameters. See Evolution of the Hubble parameter

Proper distance is also discussed here.