The Cosmological Constant from Conformal Transformations: Möbius Invariance and Schwarzian Action

Abstract

The homogeneous Friedman-Lema\^ıtre-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 $\text{SL}(2,\mathbb{R})$ acting as M{ö}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 $\text{SL}(2,\mathbb{R})$ 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 $\text{Diff}({\mathcal{S}}^1)$ 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.

arXiv comment16 pages, last version (published in CQG): title changed and a section added on the physical implications of the new conformally-invariant cosmological model

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