Reinventing the slide rule for redshifts: the case for logarithmic wavelength shift

Ivan K. Baldry
Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool, L3 5RF, UK

Date: 2018 December 17th

Abstract:

Redshift is not a shift, it is defined as a fractional change in wavelength. Nevertheless, it is a fairly common misconception that $\Delta z \, c$ represents a velocity where $\Delta z$ is the redshift separation between two galaxies. When evaluating large changes in a quantity, it is often more useful to consider logarithmic differences. Defining $\zeta = \ln \lambda_{\rm obs} - \ln \lambda_{\rm em}$ results in a more accurate approximation for line-of-sight velocity and, more importantly, this means that the cosmological and peculiar velocity terms become additive: $\Delta \zeta \, c$ can represent a velocity at any cosmological distance. Logarithmic shift $\zeta$, or equivalently $\ln (1+z)$, should arguably be used for photometric redshift evaluation. For a comparative non-accelerating universe, used in cosmology, comoving distance ($D_{\rm C}$) is proportional to $\zeta$. This means that galaxy population distributions in $\zeta$, rather than $z$, are close to being evenly distributed in $D_{\rm C}$, and they have a more aesthetic spacing when considering galaxy evolution. Some pedagogic notes on these quantities are presented.

Keywords: redshift, wavelength, peculiar velocity, cosmological scalefactor, frame, comoving distance

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