Dark Energy and the Schwarzian Derivative

Abstract: 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.