Gauge versus (hidden) physical symmetries of FLRW cosmologies

Abstract

In generally covariant theories, evolution in coordinate time is a gauge transformation, so that a symmetry made manifest in a gauge-fixed description need not be a symmetry of the physical dynamics. At the same time, deparametrisation removes gauge redundancies but may hide physical symmetries, in particular those that depend on the chosen physical clock. We investigate the relation between gauge and hidden physical symmetries in flat FLRW geometry coupled to an arbitrary number $n$ of free massless scalar fields. We show that conformal Killing vectors of the minisuperspace metric generate conserved charges that are Dirac observables and hence gauge-invariant. Their Poisson algebra realises the maximal conformal algebra $\mathfrak{conf}(n,1)\simeq\mathfrak{so}(n+1,2)$, extending previous single-field results to arbitrary $n$. We then revisit the Eisenhart--Duval lift in a family of gauges and show that the resulting symmetry algebra is gauge dependent. Only in the harmonic gauge does the algebra enlarge to the Schrödinger algebra, which is thus not a physical symmetry. Finally, we show that deparametrisation maps the lifted charges to gauge-invariant Dirac observables, which always realise a subalgebra of $\mathfrak{conf}(n,1)$, recovering it in full in the harmonic gauge. Our results provide a systematic framework for disentangling gauge from physical symmetries in minisuperspace models, recovering charges to which reduced phase-space descriptions are structurally blind, and extending naturally to models with potentials.