# Electrodynamics of a Cosmic Dark Fluid

## Abstract

**:**

## 1. Introduction

#### 1.1. Preface

#### 1.2. Ten Schemes of Description of Dark Fluid Coupling to Electromagnetic Field

- (1)
- Minimal coupling of photons to the axionic Dark Matter.
- (2)
- Non-stationary optical activity induced by the axionic Dark Matter.
- (3)
- Gradient-type interactions with the axionic Dark Matter.
- (4)
- Dynamo-optical interactions associated with the axionic Dark Matter.
- (5)
- Striction-type coupling via a scalar Dark Energy.
- (6)
- Piezo-type coupling via a scalar Dark Energy.
- (7)
- Pyro-type coupling via a scalar Dark Energy.
- (8)
- Dynamo-optical interactions associated with Dark Energy.
- (9)
- Non-minimal coupling of photons to the Dark Fluid.
- (10)
- Electromagnetic interactions induced by the Dark Fluid in a plasma with cooperative field.

## 2. General Formalism

#### 2.1. The Action Functional and Decomposition of the Total Lagrangian

#### 2.2. Master Equations for the Electromagnetic Field

#### 2.3. Velocity Four-Vector and Decompositions of the Maxwell and Excitation Tensors

#### 2.3.1. Eigen Four-Vector of the DE Stress-Energy Tensor

#### 2.3.2. Eigen Four-Vector of the DF Stress-Energy Tensor

#### 2.3.3. Unit Dynamic Vector Field

#### 2.4. Irreducible Representations of Basic Quantities in Terms of Velocity Four-Vector ${U}^{i}$

#### 2.4.1. Irreducible Representation of the Tensor ${\nabla}_{i}{U}_{k}$

#### 2.4.2. Irreducible Representation of the Maxwell Tensor ${F}_{ik}$ and its Dual ${F}_{ik}^{*}$

#### 2.4.3. Irreducible Representation of the Induction Tensor

#### 2.4.4. Irreducible Representation of the Tensor of Spontaneous Polarization-Magnetization ${\mathcal{H}}^{ik}$

#### 2.4.5. Irreducible Representation of the Linear Response Tensor ${C}^{ikmn}$

#### 2.4.6. Reduction of Electrodynamic Equations

#### 2.5. Gravity Field Equations

#### 2.5.1. The Structure of Master Equations for the Gravity Field

#### 2.5.2. Stress-Energy Tensor of the Dark Matter

#### 2.6. Master Equation for the Axion Field

## 3. Model 1. Minimal Coupling of Photons to the Axionic Dark Matter

#### 3.1. Basic Quantities and Equations

#### 3.2. Relic Cosmological Axions, Cold Dark Matter and Terrestrial Magnetic and Electric Fields

#### 3.2.1. Axion Magnetostatics

#### 3.2.2. Axionically Induced Longitudinal Magneto-Electric Oscillations

- (1)
- Relic axions produce oscillations of a new type in the resonator “Earth-Ionosphere”. We indicated them as Longitudinal Magneto-Electric Oscillations, since they possess the following specific feature: the axionically coupled electric and magnetic fields are parallel to one another. When the axions are absent and q = 0, there exist only transversal electromagnetic oscillations, usual for the Faraday–Maxwell version of electrodynamics. Longitudinal Magneto-Electric Oscillations can be considered as a dynamic analog of a static axionically induced effect predicted by Wilczek in [72] (axions produce radial electric field in the vicinity of a monopole with radial magnetic field).
- (2)
- New “hybrid” frequencies of oscillations appear in the global resonator “Earth-Ionosphere” due to the axionic Dark Matter influence.
- (3)
- Estimations of the effect for ${\rho}_{(\mathrm{DM})}\simeq 1.25\phantom{\rule{4pt}{0ex}}\mathrm{GeV}\xb7{\mathrm{cm}}^{-3}$ and $\frac{1}{{\mathrm{\Psi}}_{0}}={\rho}_{\mathrm{A}\gamma \gamma}\simeq {10}^{-9}\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{-1}$, give the value ${\nu}_{(\mathrm{Axion})}\simeq {10}^{-5}\phantom{\rule{4pt}{0ex}}\mathrm{Hz}$ for the effective frequency of Longitudinal Magneto-Electric Oscillations in the Earth Magnetosphere.

#### 3.3. Electromagnetic Response on the Action of Gravitational pp-Waves in an Axionic Environment

## 4. Model 2. Non-Stationary Optical Activity Induced by the Axionic Dark Matter

#### 4.1. Extension of the Axion Electrodynamics: Inertia Effects and Field Theory

#### 4.1.1. Susceptibility of Spatially Isotropic Moving Medium

#### 4.1.2. Axionically Induced Spontaneous Magnetization of the Inertia-Type

#### 4.1.3. Axionically Induced Optical Activity of the Inertia-Type

#### 4.2. An Illustration

## 5. Model 3. Gradient-Type Interactions with the Axionic Dark Matter

#### 5.1. Extended Axion Electrodynamics: Taking into Account Terms Quadratic in the Gradient Four-Vector

#### 5.2. First Illustration: A Spatially Homogeneous Anisotropic Cosmological Model

- (i)
- When ${\lambda}_{(31)}+\frac{1}{2}{\lambda}_{(32)}\ge 0$ and ${\lambda}_{(31)}\ge 0$, there is no anomaly in the electric field, and the quantity ${n}^{2}(t)$ is always positive.
- (ii)
- When ${\lambda}_{(31)}+\frac{1}{2}{\lambda}_{(32)}\ge 0$, and ${\lambda}_{(31)}<0$, there is no anomaly in the electric field, but the quantity ${n}^{2}(t)$ can take infinite value at some moment ${t}^{*}$, for which ${\dot{\varphi}}^{2}({t}^{*})=\frac{1}{|{\lambda}_{(31)}|}$. For infinite refraction index, the phase velocity of electromagnetic waves ${V}_{(\mathrm{ph})}=\frac{1}{n}$ and the group velocity ${V}_{(\mathrm{gr})}=\frac{2n}{{n}^{2}+1}$ take zero values, thus, the electromagnetic energy-information transfer stops. During the interval of cosmological time, for which ${\dot{\varphi}}^{2}(t)>\frac{1}{|{\lambda}_{(31)}|}$ the square of refraction index is negative. Such a situation is indicated in [93] as unlighted epoch in the Universe history, since electromagnetic waves can not propagate in the Universe, when n is pure imaginary quantity. Also, one can say, that it can be called a Dark Epoch of the first kind provided by the coupling of photons to the Dark Matter.
- (iii)
- When ${\lambda}_{(31)}+\frac{1}{2}{\lambda}_{(32)}<0$ and ${\lambda}_{(31)}\ge 0$, a dynamic anomaly in the electric field can appear, if the time moment ${t}^{**}$ exists, for which ${\dot{\varphi}}^{2}({t}^{**})=\frac{1}{|{\lambda}_{(31)}+\frac{1}{2}{\lambda}_{(32)}|}$. The quantity ${n}^{2}(t)$ can change the sign at ${t}^{**}$ providing the existence of a Dark Epoch of the second kind. On the boundary of this Epoch ${n}^{2}({t}^{**})$ = 0, ${V}_{(\mathrm{ph})}({t}^{**})$ = ∞, and the group velocity ${V}_{(\mathrm{gr})}({t}^{**})=0$, i.e., the electromagnetic energy transfer stops.
- (iv)
- When ${\lambda}_{(31)}+\frac{1}{2}{\lambda}_{(32)}<0$, ${\lambda}_{(31)}<0$, and $0<\frac{1}{2}{\lambda}_{(32)}<\left|{\lambda}_{(31)}\right|$, again a dynamic anomaly in the electric field can appear, and the quantity ${n}^{2}(t)$ can be negative, when$$\frac{1}{|{\lambda}_{(31)}|}<{\dot{\varphi}}^{2}<\frac{1}{|\frac{1}{2}{\lambda}_{(32)}-|{\lambda}_{(31)}\left|\right|}\phantom{\rule{0.166667em}{0ex}}$$
- (v)
- When ${\lambda}_{(31)}+\frac{1}{2}{\lambda}_{(32)}<0$, ${\lambda}_{(31)}>\frac{1}{2}\left|{\lambda}_{(32)}\right|$, and ${\dot{\varphi}}^{2}>\frac{1}{|\frac{1}{2}{\lambda}_{(32)}+{\lambda}_{(31)}|}$ a dynamic anomaly in the electric field can appear, and the quantity ${n}^{2}(t)$ also can be negative. On the boundary of the corresponding Dark Epoch ${n}^{2}({t}^{**})$ = 0, ${V}_{(\mathrm{ph})}({t}^{**})$ = ∞, ${V}_{(\mathrm{gr})}({t}^{**})$ = 0.
- (vi)
- When ${\lambda}_{(32)}=0$, but ${\lambda}_{(31)}\ne 0$, one obtains that ${n}^{2}=1$, however, $\epsilon \ne 1$ and $\mu \ne 1$. There are no Dark Epochs, nevertheless, the anomaly in the electric field can exist, if ${\lambda}_{(31)}$ is negative and ${\dot{\varphi}}^{2}(t)>\frac{1}{|{\lambda}_{(31)}|}$ for some interval of the cosmological time.