Dark Energy and the Schwarzian Derivative
  Theories with a time dependent Newton's constant admit two natural measures
of time : atomic and astronomical. Temporal parametrisation by SL(2, R)
transformations gives rise to an equivalence between theories with different
time dependence's, including the special Case of no time dependence, a fact
noticed by Mestschersky, Vinti and by Lynden-Bell. I point out that theories
with time dependent dark energy densities admit three natural measures of time
: atomic and astronomical and de Sitter related by temporal re-parametrizations
and I extend Mestschersky-Vinti-Lynden-Bell's result to cover this more general
situation. I find a consequent equivalence between theories in which the
density of dark energy is constant in time and in which it varies with time.
Strikingly a time dependent cosmological constant changes by the addition of a
Schwarzian derivative term unless the temporal reparameterization belongs to
SL(2, R). In General Relativity one may introduce a Schwarzian tensor to
investigate how the notion of dark energy changes under changes of conformal
frame. The general theory is illustrated in the case of Friedmann-Lemaitre
metrics.
