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

On the relevance of quantum corrections to the matter stress-energy tensor in eternally expanding universes