Hagedorn behavior of the superstring propating in a cosmological time dependent background

In this work the Liouville-von Neumann (LvN) quantization of the type IIB superstring is carried on in a time dependent plane wave background with a constant self-dual Ramond-Ramond 5-form and a linear dilaton in the lightlike direction. Such an endeavor allows us to define an invariant density matrix and study important issues in real time string thermodynamics. In particular, a transcendental equation that determines the Hagedorn temperature as a function of the thermalization time is derived. It is shown that the Hagedorn temperature increases as the string approaches the singularity.