It is well known that there are few kinds of redshifts (e.g. relativistic, cosmological, gravitational).
And the formulas are
\begin{align}
1+z_\mathrm{rel}&=\sqrt{\frac{1+v/c}{1-v/c}}\,\,\text{(for completely radial direction)}\\
1+z_\mathrm{cos}&=\frac{a_\mathrm{now}}{a_\mathrm{then}}
\end{align} for relativistic and cosmological, respectively.
However, when two effects are mixed, it confuses me which parameter will go into $v$ for relativistic doppler effect.
Since the proper velocity in cosmology $v$ is addition of Hubble flow and peculiar velocity, i.e. $v=v_\mathrm{pec}+Hd$, where the $d$ is proper distance, I don't know what to put into the formula.
Is the peculiar velocity $v_\mathrm{pec}$ is the parameter for $z_\mathrm{rel}$? Or just $v$ ?
1 Answer
Just compose the two, $1+z=(1+z_\mathrm{rel})(1+z_\mathrm{cos})$, where $z_\mathrm{rel}$ is evaluated using the peculiar velocity.
Both of these contributions can be interpreted as kinematic Doppler shifts due to the relative velocity between source and receiver; the $(1+z_\mathrm{rel})$ factor accounts for the peculiar velocity while the $(1+z_\mathrm{cos})$ factor accounts for the Hubble flow velocity. So if you were to include the Hubble flow velocity in calculating $z_\mathrm{rel}$, then you would improperly double-count it.
Alternatively, it would be correct in principle to forget $z_\mathrm{cos}$ and compute the total redshift $z$ as the kinematic Doppler shift $z_\mathrm{rel}$ associated with a total relative velocity $v$ that includes both the peculiar velocity and the Hubble flow (see e.g. this paper). However, that $v$ is not what you have written, i.e., $v\neq v_\mathrm{pec}+Hd$. The quantity $Hd$ is not even a relative velocity in the relativistic sense of corresponding to an angle in spacetime. Additionally, velocities in relativity do not add linearly.