The CWKB particle production and classical condensate in de Sitter spacetime

The complex time WKB approximation is an effective tool in studying particle production in curved spacetime. We use it in this work to understand the formation of classical condensate in expanding de Sitter spacetime. The CWKB leads to the emergence of thermal spectrum that depends crucially on horizons (as in de Sitter spacetime) or observer dependent horizons (as in Rindler spacetime). A connection is sought between the horizon and the formation of classical condensate. We concentrate on de Sitter spacetime and study the cosmological perturbation of $k=0$ mode with various values of $m/H_0$. We find that for a minimally coupled free scalar field for $m^2/H_0^2<2$, the one-mode occupation number grows more than unity soon after the physical wavelength of the mode crosses the Hubble radius and soon after diverges as $N(t)\sim O(1)[\lambda_{phys}(t)/{H_0^{-1}}]^{2\sqrt{\nu^2-1/4}}$, where $\nu\equiv (9/4 -m^2/{H_0^2})^{1/2}$. The results substantiates the previous works in this direction. We also find the correct oscillation and behaviour of $N(z)$ at small $z$ from a single expression using CWKB approximation for various values of $m/H_0$. We also discuss decoherence in relation to the formation of classical condensate. We also find that the squeezed state formalism and CWKB method give identical results.