Title: "The covariant Wigner transformation in semiclassical gravity"

We consider the quark-gluon phase of the early universe. Iterating the Dirac equation leads a Klein-Gordon approach, which can be successfully covariantly Wigner-Weyl transformed in curved background spacetime. In the semiclassical approach the mean Einstein equation with first-order quantum corrections can be calculated on phase space. Under high symmetry a singularity-free cosmological solution is found with negative pressure. The quantum effect is out of the scope of the measurement precision of the cosmic microwave background radiation. An extension of the methods to a model for Sgr A* leading chromohydrodynamics for non-abelian symmetry is proposed.