Which redshift value is used in the velocity measurement of distance

[Arman777]

Let us say that we have a stellar object so its total velocity is defined as

$$ v_{tot} = v_{pec} + V_{rec}$$

Where

$$V_{rec} = H_0r$$

and $$V(z) = \frac{cz}{1+z}[1+\frac{1}{2}(1-q_0)z - \frac{1}{6}(1-q_0-3q_0^2+j_0)z^2]$$

for small z.So my first question is what is the $z$ value here? Is it the observed redshift or the cosmological redshift?

Also, the relationship between observed and cosmological redshift is given.

$$ 1+z_{obs} = (1 + z_{cos})(1 + z_{earth})((1 + z_{sun})(1 + z_{source})(1 + z_{gravity})$$

If we are using the cosmological redshift then by using above equation we can write,

$$ z_{cos} = \frac{1 + z_{obs}} {(1 + z_{earth})((1 + z_{sun})(1 + z_{source})(1 + z_{gravity})}-1 $$

So is this what we put in (4)?

Edit: For the source you can look here https://arxiv.org/abs/1907.12639 Eqn(16) and (18)

 

[kimbyd]

Generally just the observed redshift is used is my understanding. The redshift imposed by the motion of the Earth relative to the galactic medium is, in most situations, considered to be too small to be relevant. Consider, for example, an object at a redshift of $z=1$. The maximal impact of the Earth's motion on this redshift is about 0.2%, so that if the "actual" redshift is 1, then the measured redshift might be anywhere between $0.998$ and $1.002$.

But that's if only one object is measured. If objects across the sky are measured, the redshifts are effectively averaged, leading to much smaller effects in the final result.

This paper you posted attempts to challenge this accepted understanding, pointing out that even these small effects, if they are systematic, can cause significant issues. They point out, for instance, that large regions of the universe are moving together, so that averaging many objects in those regions will lead to a systematic offset in the redshift. They further claim that even small systematic errors can potentially cause large effects for estimated parameters.

That last claim may be a concern. I definitely know of some situations where small systematic errors can lead to large errors elsewhere. I've only skimmed the paper, so I haven't really evaluated their argument, but on the surface it appears to be incomplete. There are relatively easy ways for teams analyzing large data sets (e.g. galaxy surveys) to do cross-checks that would measure the impact of these kinds of errors. My bet is that as long as they are using spectroscopic redshifts, the systematic errors will tend to remain pretty small. But it's definitely worth further investigation to ensure this is the case.

 

[Arman777]

Generally just the observed redshift is used is my understanding.

I see

They point out, for instance, that large regions of the universe are moving together, so that averaging many objects in those regions will lead to a systematic offset in the redshift.

I did not understand this part.

The changes would be small I guess. Since the current distance ladder error goal is %1 they mentioned this problem..

 

[kimbyd]

I see

I did not understand this part.

The changes would be small I guess. Since the current distance ladder error goal is %1 they mentioned this problem..

Right, so it's worth investigating, but the back-of-the-envelope numbers suggest that errors introduced by these systematic effects should be much less than 1%. This paper you linked argues that it might actually make a difference. My conclusion is that it's definitely worth investigating, but there's a fair chance that this is a red herring.