# Calculating distance to High Redshift Galaxies based on observations

I am aware that we need to account for the expansion of the universe, changing $$H$$, the relativistic redshift eqn: $$1+z=\sqrt{\frac{c+v}{c-v}}$$ and so on, but can we accurately calculate velocity using just redshift from observations of very high redshift galaxies? I'm thinking redshifts of $$z>7$$. I'd expect the light from these distant objects to be extremely redshifted, faint, and small, near point sources. What kind of assumptions/methods do astronomers use to calculate light-travel distance of these difficult to resolve objects? For instance, how would we know that an object being observed is a QSO at $$z=7.64$$ versus a Galaxy at $$z=7.66$$?

• NB You question isn't clear. Distance can be "calculated" from redshift and a cosmological model. Distance could also be "measured" in various way to help constrain cosmological models, but that would be a very broad question. Dec 31, 2021 at 14:26
• @ProfRob Edited. I wasn't asking if z=7 is nearby, I know it isn't even close, just clarifying what I meant when saying "high redshift" was the most distant and faintest objects. Hopefully, the edits add clarity. Dec 31, 2021 at 14:41

## 1 Answer

The "velocity" of something at high redshift isn't particularly meaningful and is not generally used, since cosmological redshift should not really be thought of as a velocity-related Doppler shift. Indeed this velocity exceeds the speed of light above redshifts above 1.5.

There is a correspondence between recession velocity, distance (you have to think about what sort of distance you want to calculate - comoving distance, light-travel distance etc.) and redshift that depends on the cosmological model and adopted cosmological parameters. The plot below shows how recession velocity depends on redshift for various combinations of cosmological parameters. Details can be found in Davis & Lineweaver (2003).

There are various calculators on the internet you can use to do these calculations - for example this calculator tells me that for a flat universe with $$\Omega_M=0.3$$, then $$z=7$$ corresponds to a light travel distance of 12.79 billion light years or a comoving distance of 28.3 billion light years.

To answer your final question, if the redshift is measured to be 7, then that tells you it isn't a nearby object. Cosmological redshifts of this size are far bigger than any Doppler shift due to actual motion with respect to the Hubble-flow (typically 100-1000 km/s). Thus a redshift of this size would always be dominated by cosmological expansion and we conclude the object is very distant.

Answer to edited final part. QSOs have a different spectrum to a bog-standard galaxy (and are much more luminous). You need a spectrum to get a redshift (certainly one to 3 significant figures). The spectrum tells you it is a QSO. The distance is estimated from the redshift and an assumption about the values of cosmological parameters.