Redshift is not a shift

The definition of redshift is given by

$$z \: = \: \frac{\lambda_{\rm obs} - \lambda_{\rm em}}{\lambda_{\rm em}}$$ (1)

where $\lambda_{\rm obs}$ is the observed wavelength and $\lambda_{\rm em}$ is the emitted or rest-frame wavelength (e.g. eq. 7 of Hubble & Tolman 1935). For low redshifts, it is common to quote $z\,c$ for observed galaxies as a recession velocity in units of $\mathrm{km\,s}^{-1}$. This is related to the approximation

$$z_{\rm pec} \: \simeq \: \frac{v}{c}$$ (2)

where $z_{\rm pec}$ is the redshift (or blueshift) caused by a line-of-sight peculiar velocity ($v$) component. This sometimes leads to the incorrect assumption that the `velocity' due to the cosmological expansion and the peculiar velocity add, or that the redshifts add. Davis & Scrimgeour (2014) show how, that even at modest redshift, the peculiar velocity can be significantly overestimated by naively subtracting the cosmological redshift from the observed redshift.

The correct formula for relating redshift terms, also incorporating the Sun's peculiar motion, can be given by

$$1 + z_{\rm cmb} \: = \: (1 + z_{\rm helio}) (1 + z_{\rm pec,\odot}) \: = \: (1 + z_{\rm cos}) (1 + z_{\rm pec})$$ (3)

where $z_{\rm cmb}$ and $z_{\rm helio}$ are the redshifts of an observed galaxy in the cosmic-microwave-background (CMB) frame and heliocentric frame, respectively, $z_{\rm pec,\odot}$ is the component caused by the motion of our Sun wrt. the CMB frame toward the observed galaxy, $z_{\rm pec}$ is caused by the peculiar velocity of the observed galaxy, and $z_{\rm cos}$ is the cosmological redshift caused by the expansion of the Universe only. This is evident from considering the definition of redshift, i.e., `one plus redshift' has a multiplicative effect on wavelength (Harrison, 1974). Note there is also a term for gravitational redshift and the heliocentric redshift should be determined correctly from the observed redshift.

Taking the difference in redshifts between two galaxies that are at the same distance, we obtain

$$\begin{split} \Delta z \: = \: z_1 - z_2 & = \: (1 + z_{\rm c... ...(1 + z_{\rm cos}) \, \frac{v_1 - v_2}{c} \end{split}\mbox{~~~,}$$ (4)

using the approximation of Eq. 2. So it appears that to estimate the velocity difference requires knowledge of the cosmological redshift, though typically one could just set $\Delta v = \Delta z\,c / (1 + z_1)$, for example, or use one plus the average redshift for the denominator (Danese, 1980). This is a well known consideration when determining the velocity dispersions of galaxy clusters. A related consequence for counting galaxies in cylinders (e.g. Balogh et al. 2004) is that to allow a fixed maximum extent in velocity difference around a galaxy requires increasing the extent in $\Delta z$ with redshift proportional to $1 + z$.

Revisiting the approximation, the peculiar redshift is accurately given by the Doppler shift formula:

$$1 + z_{\rm pec} \: = \: \gamma (1+ \beta_{\rm los})$$ (5)

where $\gamma = (1-\beta^2)^{-1/2}$ is the Lorentz factor and $\beta_{\rm los}$ is the line-of-sight velocity divided by the speed of light. Using Taylor series expansion, we can then simplify to:

$$z_{\rm pec} \: \simeq \: \beta_{\rm los} + \frac{1}{2} \beta^2 + \frac{1}{2} \beta^2 \beta_{\rm los}$$ (6)

This simplifies further to $z_{\rm pec} \simeq \beta_{\rm los}$ after dropping the higher order terms. This is usually sufficiently accurate for use in astrophysics but it is worth bearing in mind that it is an approximation.