During inflation the universe rapidly expands and the observed classical distribution of inhomogeneities originates from the substantially non-classical state. The problem of transition from quantum to classical behavior is also important outside the cosmological context in connection with the study of mesoscopic systems. And this problem can be viewed in the context of the theory of decoherence induced by environment. In short, this phenomenon lies in the fact that the quantum degrees of freedom of the system are entangled with the degrees of freedom of the environment, which leads to the suppression of interference effects and the classicalisation of the system.
In cosmological context, the transition to classical behavior is studied within the Wheeler-DeWitt approach with small inhomogeneous perturbations considered over quantized homogeneous background. The Wheeler-DeWitt equation is solved in the Born-Oppenheimer approximation. Using this approximation, one can represent the wave function as a product of homogeneous and inhomogeneous parts. Inhomogeneous wave function is associated with QFT approach on a clas-sical curved background and homogeneous part provides probability amplitude for different backgrounds. The decoherence process itself is observed by studying the corresponding density matrix.
In this paper we discuss the decoherence of background degrees of freedom due to the loss of information about modes going beyond the cosmological horizon while mantaining the full information about the short wavelength perturbations. We demonstrate the classicalization process of the quantum state of inhomogeneities of the universe by observing reduced density matrix of the short wavelength fluctuations.