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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, τ↦τ̃=f(τ) , 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(S1) 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.