Abstract
We study the quantum mechanical evolution of the tensor perturbations during inflation with non-linear tensor interactions. We first obtain the Lindblad terms generated by non-linear interactions by tracing out unobservable sub-horizon modes. Then we calculate explicitly the reduced density matrix for the super-horizon modes, and show that the probability of maintaining the unitarity of the squeezed state decreases in time. The decreased probability is transferred to other elements of the reduced density matrix including off-diagonal ones, so the evolution of the reduced density matrix describes the quantum-to-classical transition of the tensor perturbations. This is different from the classicality accomplished by the squeezed state, the suppression of the non-commutative effect, which is originated from the quadratic, linear interaction, and also maintains the unitarity. The quantum-to-classical transition occurs within 5–10 e-folds, faster than the curvature perturbation.
Highlights
On the other hand, the presence of the horizon in de Sitter space-time during inflation enables us to separate the quantum fluctuation modes into those shorter and longer than the horizon scale according to their wavelengths
We study the quantum mechanical evolution of the tensor perturbations during inflation with non-linear tensor interactions
We have considered in the de Sitter (dS) background the time evolution of the squeezed state for tensor perturbations under the influence of non-linear interaction Hamiltonian
Summary
We begin with a general account on the evolution of a system where we have integrated out certain sector. I Ei|ρ(1)(τ )|Ei and ρ(r2ed) (τ ) ≡ i Ei|ρ(2)(τ )|Ei. i Ei|ρ(1)(τ )|Ei and ρ(r2ed) (τ ) ≡ i Ei|ρ(2)(τ )|Ei We can write this term purely in terms of the commutators with free Hamiltonian of the system so the evolution of the reduced density matrix described by this term is unitary. While written purely in terms of the commutator form, one thing we can further note is the existence of the “shift” in the Hamiltonian, Hi(netff1) + Hi(netff2) This means even the unitary part of the evolution is described by a Hamiltonian not identical to the unperturbed free Hamiltonian of the system, because the environment perturbs the free Hamiltonian in such a way that the free Hamiltonian that describes the unitary evolution should be in general shifted.
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