On the luminosity distance and the epoch of acceleration
  Standard cosmological models based on general relativity (GR) with dark
energy predict that the Universe underwent a transition from decelerating to
accelerating expansion at a moderate redshift $z_{acc} \sim 0.7$. Clearly, it
is of great interest to directly measure this transition in a model-independent
way, without the assumption that GR is the correct theory of gravity. We
explore to what extent supernova (SN) luminosity distance measurements provide
evidence for such a transition: we show that, contrary to intuition, the
well-known "turnover" in the SN distance residuals $\Delta\mu$ relative to an
empty (Milne) model does not give firm evidence for such a transition within
the redshift range spanned by SN data. The observed turnover in that diagram is
predominantly due to the negative curvature in the Milne model, {\em not} the
deceleration predicted by $\Lambda$CDM and relatives. We show that there are
several advantages in plotting distance residuals against a flat,
non-accelerating model $(w = -1/3)$, and also remapping the $z-$axis to $u =
\ln(1+z)$; we outline a number of useful and intuitive properties of this
presentation. We conclude that there are significant complementarities between
SNe and baryon acoustic oscillations (BAOs): SNe offer high precision at low
redshifts and give good constraints on the net {\em amount} of acceleration
since $z \sim 0.7$, but are weak at constraining $z_{acc}$; while radial BAO
measurements are probably superior for placing direct constraints on $z_{acc}$.
