Inflaton and dark matter in a random environment

Abstract

We consider a Lagrangian of interacting scalar fields. We divide the Lagrangian into two parts. The first part is to describe either the dark matter (DM) or the inflaton (IN) depending on the choice of the self-interaction. The second part constitutes an environment of an infinite number of scalar fields interacting linearly with the first part. We approximate the environment by a white noise obtaining a Langevin equation. We show that the resulting Fokker-Planck equation has solutions determining a relation between the diffusion constant, the cosmological constant and the temperature. As a consequence of the Langevin approximation the energy-momentum tensor of the dark matter and the inflaton is not conserved. The compensating energy-momentum tensor is interpreted as the dark energy (DE). We insert the total energy-momentum in Einstein equations. We show that under special initial conditions Einstein equations have a solution with a constant ratio of DM/DE and IN/DE densities.