# Redshift and Velocity relation

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 $$$$V(z) = \frac{cz}{1+z}[1+\frac{1}{2}(1-q_0)z - \frac{1}{6}(1-q_0-3q_0^2+j_0)z^2] ~~(4)$$$$

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)