# Exact (1 + 3 + 6)-Dimensional Cosmological-Type Solutions in Gravitational Model with Yang–Mills Field, Gauss–Bonnet Term and Λ Term

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## Abstract

**:**

## 1. Introduction

## 2. The 10-Dimensional Model

#### 2.1. The Action and Equations of Motion

#### 2.2. Cosmological Ansatz

## 3. Cosmological Solutions

**Graphical analysis.**The graphical representation $\Lambda \left|\alpha \right|$ upon $x=h/H$ (in this cosmological case) is presented in Figure 2. In drawing this figure, we use the relationship

**Stability.**Using the results of Refs. [22,24], we obtain that the cosmological solutions under consideration obeying $x=h/H\ne {x}_{i}$, $i=a,c,d$, where ${x}_{a}=1$, ${x}_{c}=-\frac{1}{5}$, ${x}_{d}=-\frac{1}{2}$, are stable if (i) $x>{x}_{d}=-1/2$ and unstable if (ii) $x<{x}_{d}=-1/2$. (For isotropic cosmological solutions with $H=h$, see Refs. [17,22] for generic $\Lambda $ and [14,15] for $\Lambda =0$).

**Remark.**It should be noted that here as in the Ref. [22] that we are dealing with a restricted stability problem. We do not consider the general setup for perturbations $\delta {g}_{MN}(t,x)$ and $\delta {A}_{M}(t,x)$ but only consider the cosmological perturbations of scale factors $\delta {a}_{3}\left(t\right)$, $\delta {a}_{6}\left(t\right)$ in the framework of our ansatz (6), () with fixed ${g}^{\left(10\right)}$ and ${\omega}^{\left(6\right)}$. An analogous remark should be addressed to our analysis of static solutions in the next section.

**Zero variation of G.**The cosmological solution with $x=0$, or $h=0$, takes place if $\alpha <0$ and

## 4. Static Analogs of Cosmological Solutions

**Graphical analysis.**The graphical representation of $\Lambda \left|\alpha \right|$ upon $x=h/H$ in the static case is presented at Figure 3. Here, we use the relationship

**Stability.**Using the results of Ref. [26], we can analyse the stability of static solutions under consideration obeying $x=h/H\ne {x}_{i}$, $i=a,c,d$, where ${x}_{a}=1$, ${x}_{c}=-\frac{1}{5}$, ${x}_{d}=-\frac{1}{2}$.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The graphical representation of moduli function $\lambda =f\left(x\right)$ given by relationship (43).

**Figure 2.**The dependence of $\Lambda \left|\alpha \right|$ upon $x=h/H$ in cosmological case. The central branch corresponds to $\alpha >0$. The left and right branches correspond to $\alpha <0$.

**Figure 3.**The dependence of $\Lambda \left|\alpha \right|$ upon $x=h/H$ in static case. The central branch corresponds to $\alpha <0$. The left and right branches correspond to $\alpha >0$.

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Ivashchuk, V.D.; Ernazarov, K.K.; Kobtsev, A.A.
Exact (1 + 3 + 6)-Dimensional Cosmological-Type Solutions in Gravitational Model with Yang–Mills Field, Gauss–Bonnet Term and Λ Term. *Symmetry* **2023**, *15*, 783.
https://doi.org/10.3390/sym15040783

**AMA Style**

Ivashchuk VD, Ernazarov KK, Kobtsev AA.
Exact (1 + 3 + 6)-Dimensional Cosmological-Type Solutions in Gravitational Model with Yang–Mills Field, Gauss–Bonnet Term and Λ Term. *Symmetry*. 2023; 15(4):783.
https://doi.org/10.3390/sym15040783

**Chicago/Turabian Style**

Ivashchuk, V. D., K. K. Ernazarov, and A. A. Kobtsev.
2023. "Exact (1 + 3 + 6)-Dimensional Cosmological-Type Solutions in Gravitational Model with Yang–Mills Field, Gauss–Bonnet Term and Λ Term" *Symmetry* 15, no. 4: 783.
https://doi.org/10.3390/sym15040783