In this talk I will present recent results on the semiclassical Einstein equation in flat cosmological spacetimes driven by a massive conformally coupled scalar field (cf. arXiv:1309.6303). It is possible to give initial conditions at finite time to get a state for the quantum field which has finite expectation values for the stress-energy tensor. This expectation value can be controlled by means of a global estimate on regular cosmological spacetimes. The obtained estimates permit to write a theorem about the existence and uniqueness of the local solutions encompassing both the spacetime metric and the matter field (via the state) simultaneously. It is then possible to extend the local solution up to a point where the scale factor becomes singular or the Hubble function reaches a critical value. I will also discuss how the results of arXiv:1309.6303 can be improved to obtain Hadamard states and not only “quasi”-adiabatic states as solutions.