  We study a toy-model of continuous infinite expansion of space-time with the
flat start. We use as the gravitational background a conformaly flat metric
with an exponentially growing factor in conformal time. We aim to clarify some
properties of quantum fields in such a gravitational background. In particular,
we calculate one-loop corrections to the Keldysh propagator to verify the fact
of secular growth of the occupation number and anomalous quantum average in the
massless scalar field theory with selfinteractions. We perform the calculation
in arbitrary dimensions with the use of the Schwinger-Keldysh technique. We get
a secular growth which is not of a kinetic type. We provide some results for
the case of generic interaction $\frac{\lambda}{b!}\phi^b$.
