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The construction of physical models with local time‐reparametrization invariance is reviewed. Negative‐energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton‐Jacobi theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann‐Lemaître‐Robertson‐Walker type. After a discussion of empty universes, the FLRW theory is extended with homogeneous scalar fields generically described by a σ‐model on some scalar manifold. An explicit gauge‐invariant solution is constructed for the non‐linear O(N)‐models.

Time‐reparametrization invariance and Hamilton‐Jacobi approach to the cosmological σ‐model