  The empty space (with no matter fields) is not really empty because of
natural metric fluctuations, quantum and classical. We show that gravitons as
well as classical gravitational waves of super-horizon wavelengths are able to
form the de Sitter state of the empty homogeneous isotropic Universe. This
state is the exact solution to the self-consistent equations of one-loop
quantum gravity for gravitons in the empty FLRW space. It also is the exact
solution to the self-consistent equations of back reaction for classical
gravitational waves. Technically, to get this de Sitter solution in both
quantum and classical cases, it is necessary to make the transition to
imaginary time and then come back to real time which is possible because this
de Sitter state is invariant with respect to Wick rotation. Such a procedure
means that the time was used as a complex variable, and this fact has a deep
but still not understood meaning. De Sitter accelerated expansion of the empty
Universe naturally explains the origin of dark energy and inflation because the
Universe is empty at the start and by the end of its evolution. This theory is
consistent with the existing observational data. The CMB anisotropy of the
order of 10^-5 is produced by fluctuations in the number of gravitons. The
existence of a threshold and unique coincidence of topologically impenetrable
barriers for tunneling takes place for the matter-dominated epoch and De Sitter
State only. These facts provide a solution to the coincidence problem. The
theoretical prediction that the equation-of-state parameter should be w>-1 for
inflation and w<-1 for dark energy is consistent with observational data. To
provide the reader with a complete picture, this paper gather together new and
some published results of the graviton theory of the origin of inflation and
dark energy
