Investigating the gauge invariance of quantum initial conditions for inflation

We investigate the transformation of initial conditions for primordial curvature perturbations under two types of transformations of the associated action: field-redefinitions and the addition of surface terms. We consider a subset of these transformations that preserve the form of the equation of motion and the commutator structure of the theory, making them canonical transformations. Actions related to each other via such transformations are physically equivalent (both classically and in the quantum sense), and choosing between the equivalent actions can be thought of as a gauge choice. Under these transformations we find that initial conditions derived via minimising the vacuum expectation value of the Hamiltonian and those obtained using the Danielsson vacuum prescription are not invariant, whereas those obtained by minimising the local energy density are invariant. We derive the range of physically distinct initial conditions obtainable by Hamiltonian diagonalisation, and illustrate their effect on the scalar primordial power spectrum and the Cosmic Microwave Background under the `just enough inflation' model. We also generalise the analogy between the dynamics of a quantum scalar field on a curved, time-dependent spacetime and the gauge-invariant curvature perturbation.