We propose a model of cosmological evolution of the early and late Universe which is consistent with observational data and naturally explains the origin of inflation and dark energy. We show that the de Sitter accelerated expansion of the FLRW space with no matter fields (hereinafter, empty space) is its natural state, and the model does not require either a scalar field or cosmological constant or any other hypotheses. This is due to the fact that the de Sitter state is an exact solution of the rigorous mathematically consistent equations of one-loop quantum gravity for the empty FLRW space that are finite off the mass shell. Space without matter fields is not empty, as it always has the natural quantum fluctuations of the metric, i.e. gravitons. Therefore, the empty (in this sense) space is filled with gravitons, which have the backreaction effect on its evolution over time forming a self-consistent de Sitter instanton leading to the exponentially accelerated expansion of the Universe. At the start and the end of cosmological evolution, the Universe is assumed to be empty, which explains the origin of inflation and dark energy. This scenario leads to the prediction that the signs of the parameter 1+w should be opposite in both cases, and this fact is consistent with observations. The fluctuations of the number of gravitons lead to fluctuations of their energy density which in turn leads to the observed CMB temperature anisotropy of the order of 10^-5 and CMB polarization. In the frame of this scenario, it is not a hypothetical scalar field that generates inflation and relic gravitational waves but on the contrary, the gravitational waves (gravitons) generate dark energy, inflation, CMB anisotropy and polarization.
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