Proper time reparametrization in cosmology: Mobius symmetry and Kodama charges

Abstract

It has been noticed that for a large class of cosmological models, the gauge fixing of the time-reparametrization invariance does not completely fix the clock. Instead, the system enjoys a surprising residual Noether symmetry under a Möbius reparametrization of the proper time, which maps gauge-inequivalent solutions to the Friedman equations onto each other. In this work, we provide a unified treatment of this hidden conformal symmetry and its realization in the homogeneous and isotropic sector of the Einstein-Scalar-$\Lambda$ system. We consider the flat Friedmann-Robertson-Walker (FRW) model, the (A)dS cosmology and provide a first treatment of the model with spatial constant curvature. We derive the general condition relating the choice of proper time and the conformal weight of the scale factor, and give a detailed analysis of the conserved Noether charges generating this physical symmetry. Our approach allows us to identify new realizations of this symmetry while recovering previous results in a unified manner. We also present the general mapping onto the conformal particle and discuss the solution-generating nature of the transformations beyond the Möbius symmetry. Finally, we show that, at least in a restricted context, this hidden conformal symmetry is intimately related to the Kodama charges of spherically symmetric gravity. This new connection suggests that the Möbius invariance of cosmology is only the corner of a larger symmetry structure which could be relevant beyond cosmological models.

34 pages, matched the published version in JCAP

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• Emergent Cosmology from Quantum Gravity in the Lorentzian Barrett-Crane Tensorial Group Field Theory Model
• Dynamical symmetries of homogeneous minisuperspace models
• Group quantization of the black hole minisuperspace
• Scale invariance beyond criticality within the mean-field analysis of tensorial field theories
• Conformally related vacuum gravitational waves, and their symmetries
• Quantum Gravity, Hydrodynamics and Emergent Cosmology: A Collection of Perspectives
• On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC
• Dynamical axisymmetric compact objects in General Relativity