Proper time reparametrization in cosmology: Möbius 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 Friedmann 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-Λ 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.

Dynamical symmetries of homogeneous minisuperspace models

“…In the case of FLRW cosmology with curvature k, we only have a single scale factor a, so the field space is one-dimensional and (3.33) easily applies. This also extends to the case of a non-vanishing cosmological constant, where it reproduces the results of [49]. On the other hand, without any matter content the dynamics itself is trivial.…”

“…This latter is spanned by the Hamiltonian H together with the volume V and the generator C of isotropic dilations on phase space. This was extended in [49] to include spatial curvature and a cosmological constant. In [50], the present authors have studied the Kantowski-Sachs model describing the black hole interior, and found that there the CVH algebra gets extended to an iso(2, 1) algebra of conserved charges encoding the dynamics of the system in an algebraic manner.…”

Emergent cosmology from quantum gravity in the Lorentzian Barrett-Crane tensorial group field theory model

“…However, should the action (4.7) exhibit a hidden conformal symmetry, it could very well be that the connection to FLRW cosmology could already be established at this point. This would follow along the lines of recent results on mapping of FLRW cosmology onto conformal mechanics, see [111][112][113]. 19 Given the similarities of the equation of motion of the condensate field (4.12) to the non-linear Schrödinger or Klein-Gordon equations [114], it might likewise admit solitary wave solutions.…”

Group quantization of the black hole minisuperspace

“…In a recent work [46] it has been unraveled that the black hole interior homogeneous model actually posses a non-trivial and finite dimensional symmetry algebra, isomorphic to the iso(2, 1) Poincaré algebra, that fully encodes the dynamics on the phase space. This is a generalization of what happens in flat FLRW cosmology coupled with a scalar field, in the isotropic case and for Bianchi I model [47][48][49][50][51][52][53][54], and it has very recently extended to (A)dS Schwarzschild solutions [55].…”