  This article presents numerical work on a special case of the cosmological
semiclassical Einstein equation (SCE). The SCE describes the interaction of
relativistic quantum matter by the expected value of the renormalized
stress-energy tensor of a quantum field with classical gravity. Here we
consider a free massless scalar field with general (not necessarily conformal)
coupling to curvature. In a cosmological scenario with flat spatial sections
for special choices of the initial conditions, we observe a separation of the
dynamics of the quantum degrees of freedom from the dynamics of the scale
factor, which extends a classical result by Starobinsky to general coupling.
For this new equation of fourth order for the dynamics of the scale factor, we
study numerical solutions. Typical solutions show a radiation-like Big Bang for
the early universe and de Sitter-like expansion for the late universe. We
discuss a specific solution to the cosmological horizon problem that can be
produced by tuning parameters in the given equation. Although the model
proposed here only contains massless matter, we give a preliminary comparison
of the obtained cosmology with the $\Lambda$CDM standard model of cosmology and
investigate parameter ranges in which the new models to a certain extent
assimilates standard cosmology.
