Quantum initial conditions for inflation and canonical invariance

Abstract

We investigate the transformation of initial conditions for primordial curvature perturbations under two types of transformations of the associated action: simultaneous redefinition of time and the field to be quantised, and the addition of surface terms. The latter encompasses all canonical transformations, whilst the time- and field-redefinition is a distinct, non-canonical transformation since the initial and destination systems use different times. Actions related to each other via such transformations yield identical equations of motion and preserve the commutator structure. They further preserve the time-evolution of expectation values of quantum operators unless the vacuum state also changes under the transformation. These properties suggest that it is of interest to investigate vacuum prescriptions that also remain unchanged under canonical 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 under these transformations, whereas those obtained by minimising the local energy density are. 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. We argue that the invariance of the vacuum prescription obtained by minimising the renormalised stress--energy tensor should make it the preferred procedure for setting initial conditions for primordial perturbations. All other procedures reviewed in this work yield ambiguous initial conditions, which is problematic both in theory and practice.