# Einstein–Yang–Mills-Aether Theory with Nonlinear Axion Field: Decay of Color Aether and the Axionic Dark Matter Production

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Formalism

#### 2.1. Action Functional and Basic Definitions

#### 2.1.1. Multiplet of Vector Fields

#### 2.1.2. Yang–Mills Tensors

#### 2.1.3. Axion Field

#### 2.1.4. Extended Jacobson’s Constitutive Tensor

#### 2.2. Master Equations of the Model

#### 2.2.1. Equations for the Yang–Mills Fields

#### 2.2.2. Equations for the Vector Fields

#### 2.2.3. Equation for the Axion Field

#### 2.2.4. Equations for the Gravitational Field

#### 2.2.5. Short Summary

## 3. Decay of the Color Aether

#### 3.1. Ansatz about Parallel Fields

#### 3.1.1. Reduced Equations for the Gauge Fields

#### 3.1.2. Reduced Equations for the Axion Field

#### 3.1.3. Reduced Equation for the Vector Field

#### 3.1.4. Modifications in the Equations for the Gravitational Field

#### 3.2. Bianchi-I Spacetime Platform

#### 3.2.1. Solution to the Equation for the Gauge Field

#### 3.2.2. Solution to the Equation for the Vector Field

#### 3.2.3. Key Equation for the Axion Field

#### 3.2.4. Evolutionary Equations for the Gravitational Field

## 4. On the Solutions of the Equation for the Axion Field

#### 4.1. Special Solutions with $\varphi =2\pi n$: Axions Are in the Equilibrium State

#### 4.2. Special Solutions with $\varphi =\pi (2n+1)$: Axions Are in the Unstable State

#### 4.3. Dynamics of the Axion Field Growth

#### 4.3.1. The First Case: The Function $\mathcal{H}$ Has No Zeros and $\mathcal{H}>0$

#### 4.3.2. The Second Case: The Function $\mathcal{H}$ Has No Zeros and $\mathcal{H}<0$

#### 4.3.3. The Third Case: The Function $\mathcal{H}$ Has Zeros and Changes the Sign

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustration of a typical phase portrait of the dynamic system (60) with (61); when the function $\mathcal{H}$ (61) has no zeros, the saddle points harboring the centers converge thus transforming such phase portrait to the one for the physical pendulum. The phase portrait contains two zones of infinite motion, which are located above the upper line formed by the separatrices and below the lower one; the corresponding curves describe the infinite growth of the axion field.

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**MDPI and ACS Style**

Balakin, A.B.; Kiselev, G.B.
Einstein–Yang–Mills-Aether Theory with Nonlinear Axion Field: Decay of Color Aether and the Axionic Dark Matter Production. *Symmetry* **2022**, *14*, 1621.
https://doi.org/10.3390/sym14081621

**AMA Style**

Balakin AB, Kiselev GB.
Einstein–Yang–Mills-Aether Theory with Nonlinear Axion Field: Decay of Color Aether and the Axionic Dark Matter Production. *Symmetry*. 2022; 14(8):1621.
https://doi.org/10.3390/sym14081621

**Chicago/Turabian Style**

Balakin, Alexander B., and Gleb B. Kiselev.
2022. "Einstein–Yang–Mills-Aether Theory with Nonlinear Axion Field: Decay of Color Aether and the Axionic Dark Matter Production" *Symmetry* 14, no. 8: 1621.
https://doi.org/10.3390/sym14081621