The cosmological constant from conformal transformations: Möbius invariance and Schwarzian action

The homogeneous Friedman–Lemaî tre-Robertson–Walker (FLRW) cosmology of a free scalar field with vanishing cosmological constant was recently shown to be invariant under the one-dimensional conformal group $SL(2,R)$ acting as Möbius transformations on the proper time. Here we generalize this analysis to arbitrary transformations of the proper time, $\tau \mapsto \tilde{\tau }=f\left(\tau \right)$, which are not to be confused with reparameterizations of the time coordinate. First, we show that the FLRW cosmology with a non-vanishing cosmological constant Λ ≠ 0 is also invariant under a $SL(2,R)$ group of conformal transformations. The associated conformal Noether charges form a $sl(2,R)$ Lie algebra which encodes the cosmic evolution. Second, we show that a cosmological constant can be generated from the Λ = 0 case through particular conformal transformations, realizing a compactification or de-compactification of the proper time depending on the sign of Λ. Finally, we propose an extended FLRW cosmological action invariant under the full group $Diff\left(S^1\right)$ of conformal transformations on the proper time, by promoting the cosmological constant to a gauge field for conformal transformations or by modifying the scalar field action to a Schwarzian action. Such a conformally-invariant cosmology leads to a renewed problem of time and to the necessity to re-think inflation in purely time-deparameterized terms. (paper)