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

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_obs - ln lambda_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 is proportional to zeta. This means that galaxy population distributions in zeta, rather than z, are close to being evenly distributed in comoving distance, and they have a more aesthetic spacing when considering galaxy evolution. Some pedagogic notes on these quantities are presented.