I would suggest that the OP read something like Ned Wright's Cosmology tutorial. A cosmology textbook would be better, but Ned Wright's tutorial would be a good attempt at start.
Specific quotes from part 2 follows, though I'd recommend starting at part 1.
<<link to part 2>>
<<link to part 1>>
Ned Wright said:
Many Distances
With the correct interpretation of the variables, the Hubble law (
v = HD) is true for all values of D, even very large ones which give v > c. But one must be careful in interpreting the distance and velocity. The distance in the Hubble law must be defined so that if A and B are two distant galaxies seen by us in the same direction, and A and B are not too far from each other, then the difference in distances from us, D(A)-D(B), is the distance A would measure to B. But this measurement must be made "now" -- so A must measure the distance to B at the same proper time since the Big Bang as we see now. Thus to determine Dnow for a distant galaxy Z we would find a chain of galaxies ABC...XYZ along the path to Z, with each element of the chain close to its neighbors, and then have each galaxy in the chain measure the distance to the next galaxy at time to since the Big Bang. The distance to Z, D(us to Z), is the sum of all these subintervals:
Dnow = D(us to Z) = D(us to A) + D(A to B) + ... D(X to Y) + D(Y to Z)
And the velocity in the Hubble law is just the change of Dnow per unit time. It is close to
cz for small
redshifts but deviates for large ones. The
space-time diagram below repeats the example from
Part 1 showing how a change in point-of-view from observer A to observer B leaves the linear velocity
vs. distance Hubble law unchanged:
And another brief quote:
Ned Wright said:
Note that the redshift-velocity law is not the special relativistic Doppler shift law
I won't give the details of the redshift formula at this point, but will refer the original poster (OP) to the article I quoted.
So, there are some important things to notice.
There are many notions of distance in cosmology, and they are not equivalent. Precision is needed to avoid confusion the different notions of distance. I do not believe "photometric distance" is a standard term, though it was the one the Original Poster (OP) asked about. It's not clear to me which of the several possibilities of distance in cosmology might be the same as (or close to) the OP's notion of "photometric distance", as I'm not quite sure what he means. Possibly he could mean luminosity distance.
I am not sure if Ned Wright has any discussion of "luminosity distance". Wiki has some discussion of distance measures in cosmolgoy in
<<link>>., though Wiki is not necessarily a reliable source on this topic. Finding a cosmology book would be a much more reliable approach.
The doppler shift law for the "velocity" as used in the Hubble law formula is NOT the formula from special relativity, because neither the distance in Hubble's law nor the velocity in Hubble's law is the same as used in special relativity.
The origin of the difference is that special relativity is a special case of general relativity that applies when space-time is flat. Because the universe is not flat, on the large scales used in cosmololgy, one cannot use special relativity but must use general relativity instead. This is rather similar to the way that if one wants to sail across a pond, one can approximate the pond as being flat, but if one wants to sail across an ocean, one's calculations will have significant errors if one does not account for the curvature of the Earth.
I will also admit that I do not know the correct relationship between Hubble distance and luminosity distance offhand, but if that is the question that the OP would like answered, someone here or on the cosmology forums might be able to give a better answer. But it's not quite clear to me if that is really the question the OP wants to ask.
I think it'd be safest to use the observed redshift factor, z, as the "common denominator" of distances, as that's what's usually measured when one measured the "distance" to a far galaxy.
With this approach, one can find the relationship between redshift factor z and Hubble distance, and also the relationship between redshift factor z and luminosity distance, and come up with a formula that relates Lumionisty distance to Hubble distance.
