Closing remarks and personal comments

In closing, redshift $z$ started out being considered as a `recession velocity' but is now considered as the inverse scalefactor minus unity when assuming $z = z_{cos}$, noting also that $z \sim \zeta$ at $z \ll 1$ and $z \sim a^{-1}$ at $z \gg 1$. Using the logarithmic shift $\zeta$, the cosmological and peculiar velocity terms are additive (Eq. 10). In addition, linear spacing in $\zeta$ corresponds to logarithmic spacing in $a$, which is often a practical and aesthetically desirable feature for plots highlighting cosmological models and galaxy evolution. Astronomers regularly use logarithmic differences, magnitude and dex, so it would be natural to use logarithmic shift for wavelength.

Selected points are given below:

• Use of $z\,c$ for galaxy recession velocities is poor practice especially beyond a couple of thousand km/s.
• Regarding $\zeta \, c$, it is neat that the quadratic term vanishes for pure line-of-sight motion. I appreciate this is a special case for peculiar velocities but it is arguably more appropriate for `recession velocity' out to $\zeta \sim 0.1$.
• For sources at the same distance, $\Delta z \, c$ is not a velocity, $\Delta \zeta \, c$ is a velocity other than for highly relativistic sources.
• Use of $\zeta$, or $\ln (1+z)$, is natural for studies that deal with the combination of cosmological and velocity terms.
• Photometric redshift analysis should arguably use $\zeta$ as standard including presentation and diagnostics. These measurements are effectively analysing shifts in $\ln (\lambda)$.
• A plot of $z_{\rm phot}$ versus $z_{\rm spec}$ is inelegant on two counts: it does not relate to the logarithmic shift nature of the measurements, and the spacing is aesthetically poor.
• The Hubble-Lemaître law $v=H_0 D$ is exact for a non-accelerating universe if we use velocity and distance definitions $v=\zeta \, c$ and $D = D_{\rm C}$ (line-of-sight comoving distance). Thus any deviations from the `law', in this form, reflect accelerating or decelerating expansion.

Comments on the revision history of this paper are given below:

• An earlier iteration of this paper was rejected by MNRAS (with the title “Shouldn't we be using a shift in logarithmic wavelength as standard?”). The anonymous referee noted that it was just an argument for “re-inventing the slide rule”: harsh but fair. I have used this quote in the revised title.
• The same iteration was also rejected as a tutorial by PASP. The referee noted “It isn't exactly a tutorial, ... it is more a plea to established astronomers for a revision of notation. That notation is so deeply embedded in the literature that most working astronomers would not think that the small benefits of changing it would be worth the disruption and confusion that would result”. I would argue that confusion, related to $z$ and velocity, for example, already exists and will continue; I've noticed it many times. While the referee's view will be common, I think there are some uses mentioned in this paper where switching to $\zeta$, or $\ln (1+z)$ to avoid a new symbol, is more readily justified.
• The tone of the MNRAS submitted version was changed somewhat for arXiv v1, along with other minor changes.
• A reference and note on Hubble-Lemaître law were added, following comments from W. Sutherland, for arXiv v2.