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 matte
r 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.
It is shown that the classical gravitational waves of super-horizon wavelengths are able to form the de Sitter accelerated expansion of the empty (with no matter fields) Universe. The contemporary Universe is about 70% empty and asymptotically is goi
ng to become completely empty, so the effect caused by emptiness should be already very noticeable. It could manifest itself as the dark energy.
It is shown that the de Sitter state is invariant with respect to Wick rotation. In imaginary time, super- horizon quantum and classical metric fluctuations of the empty FLRW space (with no matter fields) form a self-consistent de Sitter state. The i
nvariance with respect to Wick rotation suggests that this de Sitter state is formed also in the empty space of real time. At the start and by the end of its cosmological evolution the Universe is empty, so that the de Sitter expansion of the empty space can be the cause of inflation and dark energy. This scenario is consistent with the existing observational data
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.
Intrinsic properties of the space itself and quantum fluctuations of its geometry are sufficient to provide a mechanism for the acceleration of cosmological expansion (dark energy effect). Applying Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy approa
ch to self-consistent equations of one-loop quantum gravity, we found exact solutions that yield acceleration. The permanent creation and annihilation of virtual gravitons is not in exact balance because of the expansion of the Universe. The excess energy comes from the spontaneous process of graviton creation and is trapped by the background. It provides the macroscopic quantum effect of cosmic acceleration.
We discuss a special class of quantum gravity phenomena that occur on the scale of the Universe as a whole at any stage of its evolution. These phenomena are a direct consequence of the zero rest mass of gravitons, conformal non-invariance of the gra
viton field, and one-loop finiteness of quantum gravity. The effects are due to graviton-ghost condensates arising from the interference of quantum coherent states. Each of coherent states is a state of gravitons and ghosts of a wavelength of the order of the horizon scale and of different occupation numbers. The state vector of the Universe is a coherent superposition of vectors of different occupation numbers. To substantiate the reliability of macroscopic quantum effects, the formalism of one-loop quantum gravity is discussed in detail. The theory is constructed as follows: Faddeev-Popov path integral in Hamilton gauge -> factorization of classical and quantum variables, allowing the existence of a self-consistent system of equations for gravitons, ghosts and macroscopic geometry -> transition to the one-loop approximation. The ghost sector corresponding to the Hamilton gauge ensures of one-loop finiteness of the theory off the mass shell. The Bogolyubov-Born-Green-Kirckwood-Yvon (BBGKY) chain for the spectral function of gravitons renormalized by ghosts is used to build a self-consistent theory of gravitons in the isotropic Universe. We found three exact solutions of the equations, consisting of BBGKY chain and macroscopic Einsteins equations. The solutions describe virtual graviton, ghost, and instanton condensates and are reproduced at the level of exact solutions for field operators and state vectors. Each exact solution corresponds to a certain phase state of graviton-ghost substratum. We establish conditions under which a continuous quantum-gravity phase transitions occur between different phases of the graviton-ghost condensate.
We show that in imaginary time quantum metric fluctuations of empty space form a self-consistent de Sitter gravitational instanton that can be thought of as describing tunneling from nothing into de Sitter space of real time (no cosmological constant
or scalar fields are needed). For the first time, this mechanism is activated to give birth to a flat inflationary Universe. For the second time, it is turned on to complete the cosmological evolution after the energy density of matter drops below the threshold (the energy density of instantons). A cosmological expansion with dark energy takes over after the scale factor exceeds this threshold, which marks the birth of dark energy at a redshift $1+zapprox 1.3$ and provides a possible solution to the coincidence problem. The number of gravitons which tunneled into the Universe must be of the order of $10^{122}$ to create the observed value of the Hubble constant. This number has nothing to do with vacuum energy, which is a possible solution to the old cosmological constant problem. The emptying Universe should possibly complete its evolution by tunneling back to nothing. After that, the entire scenario is repeated, and it can happen endlessly.
The first mathematically consistent exact equations of quantum gravity in the Heisenberg representation and Hamilton gauge are obtained. It is shown that the path integral over the canonical variables in the Hamilton gauge is mathematically equivalen
t to the operator equations of quantum theory of gravity with canonical rules of quantization of the gravitational and ghost fields. In its operator formulation, the theory can be used to calculate the graviton S-matrix as well as to describe the quantum evolution of macroscopic system of gravitons in the non-stationary Universe or in the vicinity of relativistic objects. In the S-matrix case, the standard results are obtained. For problems of the second type, the original Heisenberg equations of quantum gravity are converted to a self-consistent system of equations for the metric of the macroscopic spacetime and Heisenberg operators of quantum fields. It is shown that conditions of the compatibility and internal consistency of this system of equations are performed without restrictions on the amplitude and wavelength of gravitons and ghosts. The status of ghost fields in the various formulations of quantum theory of gravity is discussed.
We show that cosmological acceleration, Dark Energy (DE) effect is a consequence of the zero rest mass, conformal non-invariance of gravitons, and 1-loop finiteness of quantum gravity (QG). The effect is due to graviton-ghost condensates arising from
the interference of quantum coherent states. The theory is constructed as follows: Faddeev-Popov-De Witt gauged path integral -> factorization of classical and quantum variables -> transition to the 1-loop approximation -> choice of ghost sector, satisfying 1-loop finiteness of the theory off the mass shell. The Bogolyubov-Born-Green-Kirckwood-Yvon (BBGKY) chain for the spectral function of gravitons renormalized by ghosts is used to build a theory of gravitons in the isotropic Universe. We found three exact solutions of the equations that describe virtual graviton and ghost condensates as well as condensates of instanton fluctuations. Exact solutions correspond to various condensates with different graviton-ghost compositions. The formalism of the BBGKY chain takes into account the contribution of non-relativistic matter in the formation of a common self-consistent gravitational field. It is shown that the era of non-relativistic matter dominance must be replaced by an era of dominance of graviton-ghost condensate. Pre-asymptotic state of DE is a condensate of virtual gravitons and ghosts with a constant conformal wavelength. The asymptotic state predicted by the theory is a graviton-ghost condensate of constant physical wavelength in the De Sitter space. Such DE phenomenon is presented in the form of the model that interpolates the exact solutions of equations of 1-loop QG. Processing of observational DE data extracted from the Hubble diagram for supernovae SNIa suggests that the graviton-ghost condensate is an adequate variable component of DE.
