Abstract
We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quasi-de Sitter spacetime. Our main interest lies in the question of to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation, where we also compare our results to already existing ones in the literature. We show that its generalization to Fock space has to be chosen appropriately in order to not violate the Shale-Stinespring condition. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation, to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution of the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map.
Highlights
In the framework of linear cosmological perturbation theory the Mukhanov-Sasaki equation plays a central role. It encodes the dynamics of the Mukhanov-Sasaki variable, which is a linearized and gauge invariant quantity that is built from a specific combination of matter and gravitational perturbations such that the resulting expression is gauge invariant up to linear order
This article is structured as follows: In Section 2 we introduce the framework of the extended phase space and rederive the canonical transformation that maps the system of the time-dependent harmonic oscillator to the time-independent one generalizing the approach in Reference [12]
In this work we used the method of the Lewis-Riesenfeld invariant in order to analyze in which sense the dynamical properties of the Mukhanov-Sasaki equation select possible candidates for initial states in the context of inflation
Summary
In the framework of linear cosmological perturbation theory the Mukhanov-Sasaki equation plays a central role. In the framework considered in our work this corresponds to the question whether there exists a canonical transformation that maps the Mukhanov-Sasaki Hamiltonian to the time-independent harmonic oscillator for each moment in time. If we aim at relating the framework of the Lewis-Riesenfeld invariant to the notion of initial states associated with the Mukhanov-Sasaki equation, we need to generalize this approach to the field theory context. This article is structured as follows: In Section 2 we introduce the framework of the extended phase space and rederive the canonical transformation that maps the system of the time-dependent harmonic oscillator to the time-independent one generalizing the approach in Reference [12].
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