  We show that on the average, homogeneous and isotropic scalar field and on
the average homogeneous and isotropic ensembles of classical and quantum
gravitational waves generate the de Sitter expansion of the empty (with no
matter) space-time. At the start and by the end of its cosmological evolution
the Universe is empty. The contemporary Universe is about 70% empty, so the
effect of cosmological acceleration should be very noticeable. One can assume
that it manifests itself as dark energy. At the start of the cosmological
evolution, before the first matter was born, the Universe is also empty. The
cosmological acceleration of such an empty space-time can manifests itself as
inflation. To get the de Sitter accelerated expansion of the empty space-time
under influence of scalar fields and classical and quantum gravitational waves,
one needs to make a mandatory Wick rotation, i.e. one needs to make a
transition to the Euclidean space of imaginary time. One can assume that the
very existence of inflation and dark energy could be considered as a possible
observable evidence of the fact that time by its nature could be a complex
value which manifests itself precisely at the start and by the end of the
evolution of the Universe, i.e. in those periods when the Universe is empty (or
nearly empty).
