Fluctuation-dissipation relation for semiclassical cosmology

Using the concept of open systems where the classical geometry is treated as the system and the quantum matter field as the environment, we derive a fluctuation-dissipation theorem for semiclassical cosmology. This theorem, which exists under very general conditions for dissipations in the dynamics of the system, and the noise and fluctuations in the environment, can be traced to the formal mathematical relation between the dissipation and noise kernels of the influence functional depicting the open system, and is ultimately a consequence of the unitarity of the closed system. In particular, for semiclassical gravity, it embodies the back reaction effect of matter fields on the dynamics of spacetime. The back reaction equation derivable from the influence action is in the form of an Einstein-Langevin equation. It contains a dissipative term in the equation of motion for the dynamics of spacetime and a noise term related to the fluctuations of particle creation in the matter field. Using the well-studied model of a quantum scalar field in a Bianchi type-I universe we illustrate how this Langevin equation and the noise term are derived and show how the creation of particles and the dissipation of anisotropy during the expansion of the Universe can be understood as a manifestation of this fluctuation-dissipation relation.