Abstract
We show that the simplest FLRW cosmological system consisting in the homo- geneous and isotropic massless Einstein-Scalar system enjoys a hidden conformal symmetry under the 1D conformal group SL(2, ℝ) acting as Mobius transformations in proper time. This invariance is made explicit through the mapping of FLRW cosmology onto conformal mechanics. On the one hand, we identify the corresponding conformal Noether charges, as combinations of the Hamiltonian scalar constraint, the extrinsic curvature and the 3D volume, which form a closed \U0001d530\U0001d529 (2, ℝ) Lie algebra. On the other hand, this approach allows to write FLRW cosmology in terms of a AdS2 phase space and a Schwarzian action. Preserving this conformal structure at the quantum level fixes the ordering ambiguities in the Wheeler-de Witt quantization and allows to formulate FLRW quantum cosmology as a CFT1. We show that the CFT two-points correlator is realized as the overlap of the evolution in proper time of cosmological coherent wave-packets. In particular, the two-points function is built from a vacuum state which, although not conformally invariant, coincides with the cosmological vacuum annihilated by the scalar constraint. These results suggest new perspectives in classical and quantum cosmology, among which the possibility to apply the conformal bootstrap program to quantize cosmological backgrounds.
Highlights
We show that the simplest FLRW cosmological system consisting in the homogeneous and isotropic massless Einstein-Scalar system enjoys a hidden conformal symmetry under the 1D conformal group SL(2, R) acting as Mobius transformations in proper time
Black holes spacetimes suggests that their classical mechanics as described by general relativity is only a statistical description of micro-states associated to more fundamental degrees of freedom of the underlying quantum geometry
Before proceeding to the canonical analysis of the cosmological action,we would like to show how this conformal structure becomes apparent through a simple change of variables. This will provides a direct mapping between FLRW cosmology and to the conformal mechanics developed by de Alfaro, Fubini and Furland [65], which is involved in the study of the AdS2/CFT1 correspondence
Summary
On top of the invariance under 1D diffeomorphism given by time reparametrization, it turns out that the cosmological action admits an unexpected time-dependent conformal symmetry. This invariance is clearer written in proper time. While the Lagrangian is not invariant, the action is modified by a total derivative and the equation of motion are unaffected by the above transformations This shows that the homogeneous and isotropic Einstein-Scalar action enjoys an additional Mobius covariance on top of the time reparametrization freedom of the action.
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