QUANTUM-TO-CLASSICAL TRANSITION OF COSMOLOGICAL PERTURBATIONS FOR NONVACUUM INITIAL STATES

The transition from quantum to semiclassical behaviour and loss of qua ntum coherence for inhomogeneous perturbations generated from a non-va cuum initial state in the early Universe is considered in the Heisenbe rg and the Schrodinger representations, as well as using the Wigner fu nction. We show explicitly that these three approaches lead to the sam e prediction in the limit of large squeezing (i.e. when the squeezing parameter \r(k)\ --> infinity): each two-modes quantum state (k, - k) of these perturbations is equivalent to a classical perturbation that has a stochastic amplitude, obeying a non-Gaussian statistics which de pends on the initial state, and that belongs to the quasi-isotropic mo de (i.e. it possesses a fixed phase). The Wigner function is not every where positive for any finite r(k), hence its interpretation as a clas sical distribution function in phase space is impossible without some coarse graining procedure. However, this does not affect the transitio n to semiclassical behaviour since the Wigner function becomes concent rated near a classical trajectory in phase space when \r(k)\ --> infin ity even without coarse graining. Deviations of the statistics of the perturbations in real space from a Gaussian one lie below the cosmic v ariance level for the N-particles initial states with N = N(\k\) but m ay be observable for other initial states without statistical isotropy or with correlations between different k modes. As a way to look for this effect, it is proposed to measure the kurtosis of the angular flu ctuations of the cosmic microwave background temperature. (C) 1997 Els evier Science B.V.