# Dark Energy and Inflation from Gravitational Waves

## Abstract

**:**

## 1. Cosmological Acceleration from Classical Gravitational Waves

^{123}gravitons to create the observed Hubble constant. This number has nothing to do with vacuum energy, which is a possible solution to the “old cosmology constant problem”.

#### 1.1. Introduction

#### 1.2. De Sitter Acceleration from Classical Gravitational Waves

#### 1.3. De Sitter State of Empty Space as the Exact Solution to BBGKY Chain

#### 1.4. Classical Gravitational Waves vs. Quantum Gravitons

#### 1.5. Conclusion

## 2. Gravitons in the Universe4

#### 2.1. Introduction

#### 2.2. The Scheme of the Yang-Mills Quantum Theory

#### 2.3. Scheme of Quantum Theory of Gravitation

#### The Einstein Equations in Normal Coordinates Can Be Obtained in the Following Way

The existing quantum theory of gravity is reduced by means of identical transformations to the operator equations in the Heisenberg representation in the Hamiltonian gauge with the canonical rules for quantizing gravitons and ghosts.

#### 2.4. Extrapolation

#### 2.5. The Problem of the Physical Nature of Ghosts

#### 2.6. One-Loop Approximation

#### 2.7. Conclusion

## 3. Cosmological Acceleration from Virtual Gravitons

#### 3.1. Introduction

#### 3.2. De Sitter State from Gravitons

## 4. Consistency with Observational Data

#### 4.1. Dark Energy

#### 4.1.1. Coincidence Problem

#### 4.1.2. The Threshold Problem

#### 4.1.3. ${\mathsf{\Lambda}}_{vacuum}/{\mathsf{\Lambda}}_{observable}\approx {10}^{123}$ Controversy

#### 4.1.4. The Need to Compare Theory with Other Observational Data

#### 4.2. Inflation

#### 4.2.1. CMB Anisotropy from Fluctuations of Number of Gravitons

#### 4.2.2. Spectrum of Metric Fluctuations

## 5. Origin of Acceleration

#### 5.1. Wick Rotation

#### 5.2. Where Does the Energy Come from?

#### 5.3. Gravitational Waves vs. Scalar Field

#### 5.4. Virtual Gravitons vs. Classical Gravitational Waves

#### 5.5. Conclusion

## 6. Cosmological Scenario

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stochastic Nonlinear Gravitational Waves in an Isotropic Universe15

#### Appendix A.1. Definitions of the Background and Fluctuations

- (1)
- In Einstein’s Equation (A10), the sorting of free indices is carried out according to the rule $i\le k$
- (2)
- The objects of the theory are mixed tensors ${X}_{i}{}^{k}$ and ${\psi}_{i}{}^{k}$, in which the sorting of free indices is carried out according to the same rule $i\le k$. These rules provide the same number of independent equations and independent functions as in the standard formulation of Einstein’s equations.
- (3)
- The derivative of the matrix function with respect to the vector parameter is determined before the expansion of the matrix function with respect to the tensor argument, i.e.,$${X}_{i;l}{}^{k}={X}_{i}{}^{m}{\psi}_{m;l}{}^{k}$$

#### Appendix A.2. Stochastic Nonlinear Gravitational Waves over the FLRW Background

#### Appendix A.3. Stochastic Gravitational Waves over the FLRW Background in the Quasi-Linear Approximation

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1 | We already mentioned such a possibility in our work [9]. In this paper, we present a full consideration of the effect. |

2 | Note that these boundary conditions are the same at the potential barrier for the Schrödinger equation. |

3 | Note that taking into account nonlinear terms can only change the numerical factor in this equation, but not its functional form. It is because the energy density in the equations of state (19) and (25) always takes the following form $\rho =C\cdot <{\left|B\right|}^{2}>{H}^{4}$, where ${\left|B\right|}^{2}$ is the action and C is some numerical factor. This is a consequence of dimensionality [8]. As it was first established by Starobinsky [28], quantum corrections to the Einstein equations due to conformal anomalies leads to the appearance of de Sitter state with the equation of state $\rho ={C}_{1}\cdot \hslash {H}^{4}$ instead of a Bing Bang singularity. It was shown by Zeldovich [29] that ${C}_{1}\le 100$ because it is of the order of the number of all elementary particles. It was shown by us [4] that in the case of gravitons, ${C}_{1}\approx {N}_{g}\approx {10}^{123}$ where N _{g} is the number of gravitons in the contemporary Universe. As we show in Section 3.2, the equation of state of gravitons ${\rho}_{g}=\gamma {N}_{g}\hslash {H}^{4}/3{\pi}^{2}$ where $\gamma \sim 1$ follows from simple qualitative considerations, which do not require discussion of nonlinear effects (as is known, quantum gravity cannot be renormalized in higher loops [30]). |

4 | This Section 2 “Gravitons in the Universe” actually should have two co-authors, Grigory Vereshkov and the author. A significant part of this paper is the content of our unpublished joint paper of 2013. Grigory died in 2014 and I am finishing our joint work alone. The problem is that by the end of his life, our views on the issue under discussion differed. And although much of the calculations in this paper belong to him, I do not consider it possible to put his name on this paper because of the differences in our points of view. He tried to show in the above-mentioned paper that the problem of the “ghost materialization” will find its explanation in the future quantum gravity theory. Perhaps this is so. On my part, I gradually came to the conclusion that “the ghost materialization” does not occur at all in the case of instanton solutions obtained in the Euclidean space of imaginary time. In the absence of a theory of quantum gravity, only such solutions have a physical meaning in my opinion. This idea is at the heart of all three parts of this paper. |

5 | So far, for the description of gravitons in the non-stationary Universe, the models used in the literature have no mathematical connection with the theory of the graviton S-matrix at the level of identity transformations (see, e.g., [15,16,17,18,19]). We do not discuss these models, since they are not relevant to the formalism of the existing quantum gravity. We note only that in all the published papers on the theory of gravitons in the non-stationary Universe, the condition of one-loop finiteness of quantum gravity is not satisfied. |

6 | |

7 | In higher orders of perturbation theory, quantum gravity is not renormalizable [30]. Multi-loop calculations can be carried out only in finite super-gravities. |

8 | At our request, L.D. Faddeev read the paper [7], in which the above-mentioned identical transformations are described in sufficient detail. Here is an excerpt from L.D. Faddeev report: “My impression is that it is correct, but not very new. Of course, when the action includes ghosts is written, it will allow canonical interpretation. However, the problem of the formal quantization of Einstein’s theory of gravity should be considered as solved in all possible formalisms”. Our other colleagues (see acknowledgements in [7] shared a similar opinion. In spite of the above, the paper [3] was rejected by the editorial staff of the PRD based on feedback from reviewers who objected to the presence of the ghost sector in the equations of quantum gravity in the Heisenberg representation. Refusing to recognize the results of obvious mathematical transformations two of the four PRD reviewers wrote that in their opinion, the authors of [3] are not familiar with the fundamentals of quantum field theory. The reply to this comment is the work [7], the review by L.D. Faddeev, as well as the text of this paper. |

9 | As we already mentioned in our work [2], “the effect of dark energy is observed in the contemporary Universe which is far from the Planck time. Therefore, quantum origin of it seems counterintuitive. In fact, this is a macroscopic quantum effect similar to superconductivity and superfluidity [3]. Its origin is related to the formation of quantum coherent condensate. Due to the one-loop finiteness of self-consistent theory of gravitons, all observables are formed by the difference between graviton and ghost contributions. This fact can be seen from the definition of N _{k}. The same final differences of contributions may correspond to the totally different graviton and ghost contributions themselves. All quantum states are degenerate with respect to mutually consistent transformations of gravitons and ghost’s occupation numbers, but providing unchanged values of observable quantities. This is a direct consequence of the internal mathematical structure of the self-consistent theory of gravitons, satisfying the one-loop finiteness condition. The tunneling that unites degenerate quantum states into a single quantum state produces a quantum coherent instanton condensate in imaginary time which can be analytically continued to real time. We refer the reader to Section VII of our work [3] for details. In a general form, hypotheses on the possibility of graviton condensate formation in the Universe were proposed in [41,42] (see [3] for more details)”. |

10 | True in the sense that the contribution of the effect of vacuum polarization of fictitious inertia fields is removed from the gravitational effect of gravitons by renormalization of the number of gravitons |

11 | Recall again that despite the widespread belief that the ghost’s contribution should not appear in the final result of the calculations, in fact, this is only true for the asymptotic states as it takes place in the S-matrix theory. There are no asymptotic states in the Universe which as a whole is the region of interactions. This is the reason why we deal with virtual gravitons (strictly speaking, there are no real gravitons in the Universe). Ghosts appear here to compensate the effect of vacuum polarization of fictitious fields of inertia [2]. This is the reason why they appear only as a factor renormalizing occupation numbers of gravitons. |

12 | In modern cosmology the contemporary epoch of the Universe evolution is commonly called “matter dominated”, although in reality it is better to call it “gravitational wave dominated epoch”. |

13 | Аfter the death of one of the co-authors (Grigory Vereshkov), this work was suspended. |

14 | Recall that the Faddeev-Popov ghosts are fictitious particles, and a possibility of their “materialization” in real time affects the subtle questions of the theory of gauged fields in the non-stationary Universe (Section 2). Because of the present lack of a consistent quantum theory of gravity, it is an open question today. The situation is different in imaginary time which is a “main player” in this work. |

15 | The material constituting the main content of the present Appendix A was prepared by the late Grigory Vereshkov. It was to be included in our joint paper [3]. We did not include it in the final version of [3], since [3] was entirely devoted to quantum effects. |

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Marochnik, L. Dark Energy and Inflation from Gravitational Waves. *Universe* **2017**, *3*, 72.
https://doi.org/10.3390/universe3040072

**AMA Style**

Marochnik L. Dark Energy and Inflation from Gravitational Waves. *Universe*. 2017; 3(4):72.
https://doi.org/10.3390/universe3040072

**Chicago/Turabian Style**

Marochnik, Leonid. 2017. "Dark Energy and Inflation from Gravitational Waves" *Universe* 3, no. 4: 72.
https://doi.org/10.3390/universe3040072