Hamiltonian formalism for gauge-invariant cosmological perturbations with multiple scalar fields

We generalise Langlois' Hamiltonian treatment of gauge-invariant linear cosmological perturbations to a cosmological setting with multiple scalar fields minimally coupled to gravity. We review the Hamilton-Jacobi-like technique for a Hamiltonian system with first-class constraints. With this technique, elucidating the gauge-invariant quantities of the system is reduced to solving for the generating function of the appropriate canonical transformation. We then apply it to the case with only two scalar fields, showcasing how the presence of more than one scalar field results in a coupled evolution of the different gauge-invariant scalar degrees of freedom. Their coupled evolution may lead to new phenomenology that cannot be simply inferred by superimposing the results of the single-field case. A simple tracking of the scalar degrees of freedom present in a system with an arbitrary number of scalar fields drives the conclusion that the results for the two-field case can be trivially extended to a multi-field scenario. For completeness, we conclude by recasting explicitly the derivation for the tensor modes, which is nevertheless unaffected by the number of scalar fields in the system. The resulting gauge-invariant Hamiltonian formulation provides a solid foundation for a canonical quantisation of perturbations, necessary to explore the quantum nature of the early universe.