The Cosmological Constant from Conformal Transformations: M\"{o}bius Invariance and Schwarzian Action

The homogeneous Friedman-Lema\^\i 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 $\textrm{SL}(2,\mathbb{R})$ acting as M{\"o}bius transformations on the proper time. Here we generalize this analysis to arbitrary transformations of the proper time, $\tau\mapsto \tilde{\tau}=f(\tau)$, which are not to be confused with reparametrizations of the time coordinate. First, we show that the FLRW cosmology with a non-vanishing cosmological constant $\Lambda\ne 0$ is also invariant under a $\textrm{SL}(2,\mathbb{R})$ group of conformal transformations. The associated conformal Noether charges form a $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra which encodes the cosmic evolution. Second, we show that a cosmological constant can be generated from the $\Lambda=0$ case through particular conformal transformations, realizing a compactification or de-compactification of the proper time depending on the sign of $\Lambda$. Finally, we propose an extended FLRW cosmological action invariant under the full group $\textrm{Diff}({\cal S}^1)$ 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.