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

We study a toy model of continuous infinite expansion of space-time with the flat start. We use as the gravitational background a conformally 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 self-interactions. We perform the calculation in arbitrary dimensions with the use of the Schwinger-Keldysh technique. We get a secular growth that is not of a kinetic type. We provide some results for the case of generic interaction <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mfrac><a:mi>λ</a:mi><a:mrow><a:mi>b</a:mi><a:mo stretchy="false">!</a:mo></a:mrow></a:mfrac><a:msup><a:mi>ϕ</a:mi><a:mi>b</a:mi></a:msup></a:math>.