A KMS-like state of Hadamard type on Robertson-Walker spacetimes and its time evolution

In this work we define a new state on the Weyl algebra of the free massive scalar Klein-Gordon field on a Robertson-Walker spacetime and prove that it is a Hadamard state. The state is supposed to approximate a thermal equilibrium state on a Robertson-Walker spacetime and we call it an adiabatic KMS state. This opens the possibility to do quantum statistical mechanics on Robertson-Walker spacetimes in the algebraic framework and the analysis of the free Bose gas on Robertson-Walker spacetimes. The state reduces to an adiabatic vacuum state if the temperature is zero and it reduces to the usual KMS state if the scaling factor in the metric of the Robertson-Walker spacetime is constant. In the second part of our work we discuss the time evolution of adiabatic KMS states. The time evolution is described in terms of semigroups. We prove the existence of a propagator on the classical phase space. This defines a time evolution on the one-particle Hilbert space. We use this time evolution to analyze the evolution of the two-point function of the KMS state. The inverse temperature change is proportional to the scale factor in the metric of the Robertson-Walker spacetime, as one expects for a relativistic Bose gas