# Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Axionic Dark Matter Profiles

#### 2.1. The Total Action Functional

#### 2.2. Background State

#### 2.3. Master Equations of the Axion Electrodynamics

#### 2.4. Static Spacetime with Spherical Symmetry

## 3. On the Features of the Exact Solution Describing the Dirac Magnetic Monopole in the Spacetime with Cosmological Constant

#### 3.1. Geometrical Aspects and Definition of the Fold

- 1.
- the profile $N\left(r\right)$ has the central minimum, the barrier on the left of the minimum, and the maximum on the right;
- 2.
- this domain is inside the cosmological horizon, but it is not hidden inside the event horizon, i.e., $N\left(r\right)>0$ in this domain.

#### 3.2. Horizons

#### 3.2.1. Auxiliary Function Indicating the Number of Horizons

#### 3.2.2. The Case $\Lambda {Q}_{\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}^{2}<\frac{1}{4}$

- When $M<{M}_{1}$, the mass line crosses the indicated graph in one point, i.e., there is only one (cosmological) horizon.
- When $M={M}_{1}$, the mass line is the tangent one with respect to the minimum of this graph, thus, there are two horizons: the double event horizon and the simple cosmological one.
- When ${M}_{1}<M<{M}_{2}$, there are three intersection points, thus, there are three horizons: the inner and outer event horizons and the cosmological one.
- When $M={M}_{2}$ there are two horizons: the simple event horizon and the double cosmological one.

#### 3.2.3. The Case $\Lambda {Q}_{\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}^{2}=\frac{1}{4}$

#### 3.2.4. The Case $\Lambda {Q}_{\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}^{2}>\frac{1}{4}$

- 1.
- $\Lambda {Q}_{\mathrm{m}}^{2}<\frac{1}{4}$, $M<{M}_{1}$;
- 2.
- $\Lambda {Q}_{\mathrm{m}}^{2}=\frac{1}{4}$, $M<\frac{1}{3}\sqrt{\frac{2}{\Lambda}}$;
- 3.
- $\Lambda {Q}_{\mathrm{m}}^{2}>\frac{1}{4}$.

#### 3.3. Folds

- When $M>{M}_{\mathrm{c}}$, the horizontal mass line $y=M$ crosses the graph of the function $\tilde{f}\left(r\right)$ twice; this means that there are two extrema: the minimum and maximum.
- When $M={M}_{\mathrm{c}}$, two extrema coincide forming the cubic inflexion point.
- When $M<{M}_{\mathrm{c}}$, the profile $N\left(r\right)$ is monotonic.

#### 3.4. Final Remarks about the Features of the Spacetime Geometry

#### 3.4.1. The Choice of the Appropriate Scale for the Radial Variable

#### 3.4.2. On the Problem of Naked Singularity and Cosmic Censorship Conjecture

## 4. Analysis of Solutions to the Key Equation of the Axion Field

#### 4.1. The Profile of the Axion Field Distribution

#### 4.2. The Profile of the Energy-Density of the Axionic Dark Matter

#### 4.3. Profiles of the Axionically Induced Electric Field

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Typical sketches of the auxiliary function (20), which illustrate the number of horizons depending on the values of the guiding parameter $\Lambda {Q}_{\mathrm{m}}^{2}$ and of the asymptotic mass M. Panel $\left(\mathbf{a}\right)$ illustrates the case $\Lambda {Q}_{\mathrm{m}}^{2}<\frac{1}{4}$; panel $\left(\mathbf{b}\right)$ relates to the case $\Lambda {Q}_{\mathrm{m}}^{2}=\frac{1}{4}$, and panel $\left(\mathbf{c}\right)$ corresponds to $\Lambda {Q}_{\mathrm{m}}^{2}>\frac{1}{4}$.

**Figure 2.**Folds in the profiles of the metric function $N\left(x\right)$ (29). There exists the infinite barrier on the left-hand side, the central minimum, and the maximum on the right-hand side. The fold is situated in the domain with positive $N\left(x\right)$ and is not harbored by the event horizon. The dimensionless guiding parameters $\sqrt{\Lambda {Q}_{\mathrm{m}}^{2}}$ and $\frac{M}{{M}_{\mathrm{c}}}$ are presented near the graphs in the box.

**Figure 3.**Axion field profiles $\varphi \left(x\right)$, as the solutions to the master Equation (31). The guiding parameters of the model: $\sqrt{\Lambda {Q}_{\mathrm{m}}^{2}}$, and $\frac{M}{{M}_{c}}$ are fixed in the box near the graphs; for the simplicity of illustration we put ${\Psi}_{0}=1$ and ${m}_{A}^{2}=0.1$ in the chosen system of units. The vertical line symbolizes the delimiter associated with the boundary of the solid body of the object, and its intersection with the graph $\varphi \left(x\right)$ defines the boundary value $\varphi \left({x}_{0}\right)$. In the far zone, the graph of the function $\varphi \left(x\right)$ tends to the horizontal asymptotic line, which corresponds to ${\varphi}_{\infty}$ given by (32). The profiles of the axion field distribution inherit the fold-like structure of the profiles of the metric function $N\left(x\right)$.

**Figure 4.**Typical profiles of the energy density scalar of the axion field (34). The basic profile has the typical fold-like structure: the minimum, the barrier on the left of the minimum, the maximum on its right-hand side. In the far zone the axion energy density tends to a constant, and the graphs have the horizontal asymptotes. The vertical line relates to the object boundary and defines the corresponding boundary value $W\left({x}_{0}\right)$.

**Figure 5.**Typical profiles of the axionically induced electric field. The profiles have inverted fold-like structure. The electric field changes sign twice; its profile tends to the Coulombian curve in the far zone near the cosmological horizon. The vertical line relates to the boundary of the solid body of the object; the dot relates to the boundary value of the electric field.

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**MDPI and ACS Style**

Balakin, A.; Groshev, D.
Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior. *Symmetry* **2020**, *12*, 455.
https://doi.org/10.3390/sym12030455

**AMA Style**

Balakin A, Groshev D.
Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior. *Symmetry*. 2020; 12(3):455.
https://doi.org/10.3390/sym12030455

**Chicago/Turabian Style**

Balakin, Alexander, and Dmitry Groshev.
2020. "Fingerprints of the Cosmological Constant: Folds in the Profiles of the Axionic Dark Matter Distribution in a Dyon Exterior" *Symmetry* 12, no. 3: 455.
https://doi.org/10.3390/sym12030455