Parametrically driven scalar field in an expanding background

We study the existence and dynamic behavior of localized and extended structures in a massive scalar inflaton field φ in 1+1 dimensions in the framework of an expanding universe with constant Hubble parameter. We introduce a parametric forcing, produced by another quantum scalar field ψ, over the effective mass squared around the minimum of the inflaton potential. For this purpose, we study the system in the context of the cubic quintic complex Ginzburg-Landau equation and find the associated amplitude equation to the cosmological scalar field equation, which near the parametric resonance allows us to find the field amplitude. We find homogeneous null solutions, flat-top expanding solitons, and dark soliton patterns. No persistent non-null solutions are found in the absence of parametric forcing, and divergent solutions are obtained when the forcing amplitude is greater than 4/3.