  In the mathematically rigorous analysis of semiclassical Einstein's
equations, the renormalisation of the stress-energy tensor plays a crucial
role. We address such a topic in the case of a scalar field with both arbitrary
mass and coupling with gravity in the hypothesis that the underlying algebraic
quantum state is of Hadamard type. Particularly, if we focus on highly
symmetric solutions of the semiclassical Einstein's equations, the envisaged
method displays a de Sitter type behaviour even without an a priori introduced
cosmological constant. As a further novel result we shall show that these
solutions turn out to be stable.
