Proper time reparametrization in cosmology: Mobius symmetry and Kodama charges

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 M\"{o}bius 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 M\"{o}bius 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 M\"{o}bius invariance of cosmology is only the corner of a larger symmetry structure which could be relevant beyond cosmological models.