Time dependent particle production and particle number in cosmological de Sitter space

Abstract

In this paper we consider the occupation number of induced quasi-particles produced during a time-dependent process using three-different methods: Instantaneous diagonalization, Bogolyubov transformation between two different vacua, and the Unruh-de Witt detector. Here we consider the Hamiltonian for a time-dependent Harmonic oscillator, with time-dependent mass and frequency. We derive the occupation number of the induced quasi-particles using the invariant operator method; in deriving the occupation number we also point out and make the connection between the Functional Schr\"odinger formalism, quantum kinetic equation, and Bogolyubov transformation between two different Fock space basis at equal times and explain the role in which the invariant operator plays. Using the flat FRW chart of de Sitter spacetime, we show that the different methods lead to different results: Instantaneous diagonalization leads to a power law distribution, while the Bogolyubov transformation and Unruh-de Witt detector both lead to thermal distributions. It is shown that the source of the discrepancy between the instantaneous diagonalization and Bogolyubov methods is the fact that there is no notion of well-defined particles in the out vacuum due to a divergent term. In the Bogolyubov method, this divergent term cancels leading to the thermal distribution, while in the instantaneous diagonalization there is no such cancelation. However, to obtain the thermal distribution in the usual Bogolyubov method, one must use the large mass limit. On physical grounds, one should expect that only the modes which have been allowed to sample the horizon would be thermal, thus in the large mass limit these modes are well within the horizon and, even though they do grow, they remain well within the horizon due to the mass. Thus one should not expect a thermal distribution since the modes won't have a chance to thermalize.