Two non-commutative parameters and regular cosmological phase transition in the semi-classical dilaton cosmology

We study cosmological phase transitions from modified equations of motion by introducing two non-commutative parameters in the Poisson brackets, which describes the initial- and future-singularity-free phase transition in the soluble semi-classical dilaton gravity with a non-vanishing cosmological constant. Accelerated expansion and decelerated expansion corresponding to the FRW phase appear alternatively, and then it ends up with the second accelerated expansion. The final stage of the universe approaches the flat spacetime independent of the initial state of the curvature scalar as long as the product of the two non-commutative parameters is less than one. Finally, we show that the initial-singularity-free condition is related to the second accelerated expansion of the universe.