  In this seven-part paper, we show that gravitational waves (classical and
quantum) produce the accelerated de Sitter expansion at the start and at the
end of the cosmological evolution of the Universe. In these periods, the
Universe contains no matter fields but contains classical and quantum metric
fluctuations, i.e., it is filled with classical and quantum gravitational
waves. In such evolution of the Universe, dominated by gravitational waves, the
de Sitter state is the exact solution to the self-consistent equations for
classical and quantum gravitational waves and background geometry for the empty
space-time with FLRW metric. In both classical and quantum cases, this solution
is of the instanton origin since it is obtained in the Euclidean space of
imaginary time with the subsequent analytic continuation to real time. The
cosmological acceleration from gravitational waves provides a transparent
physical explanation to the coincidence, threshold and 'old cosmological
constant' paradoxes of dark energy avoiding recourse to the anthropic
principle. The cosmological acceleration from virtual gravitons at the start of
the Universe evolution produces inflation, which is consistent with the
observational data on CMB anisotropy. Section 1 is devoted to cosmological
acceleration from classical gravitational waves. Section 2 is devoted to the
theory of virtual gravitons in the Universe. Section 3 is devoted to
cosmological acceleration from virtual gravitons. Section 4 discusses the
consistency of the theory with observational data on dark energy and inflation.
The discussion of mechanism of acceleration and cosmological scenario are
contained in Sections 5 and 6. Appendix contains the theory of stochastic
nonlinear gravitational waves of arbitrary wavelength and amplitude in an
isotropic Universe.
