arXiv 2201.031182022-01gr-qchep-th

含$Λ$项的EGB模型的稳定三因子空间指数宇宙型解

Stable exponential cosmological type solutions with three factor spaces in EGB model with a $Λ$-term

K. K. Ernazarov, V. D. Ivashchuk

备注：: Semi-review, 23 pages, 1 figure, LaTex. Several typos in the text and a typo in eq. (2.3) are eliminated. A paragraph is added in the end of Introduction. Two references are added and one ref. is omitted

摘要：

我们研究了一个$D$维爱因斯坦-高斯-博内特模型，该模型包含高斯-博内特项、宇宙学项$Λ$以及两个非零常数：$α_1$和$α_2$。在将度规设为对角度规的情况下，我们发现了在变量$u$中具有三个尺度因子的指数依赖性的宇宙学型解，由三个不重合的哈勃参数控制：$H \neq 0$、$h_1$和$h_2$，服从$m H + k_1 h_1 + k_2 h_2 \neq 0$，分别对应于维度$m > 1$、$k_1 > 1$和$k_2 > 1$的因子空间，并且取决于符号参数$\varepsilon = \pm 1$，其中$\varepsilon = 1$对应于宇宙学情况，而$\varepsilon = - 1$ -对应于静态情况）。我们处理两种情况：i）$m < k_1 < k_2$和ii）$1< k_1 = k_2 = k$，$k \neq m$。我们表明，在这两种情况下的解决方案存在，如果$\varepsilon α= \varepsilon α_2 / α_1 > 0$和$αΛ> 0$满足一定的（上限和下限）。这些解被定义为四阶（或更低阶）的多项式主方程的解，这些解可以在根式中求解。在情况ii）中，给出了显式解。在这两种情况下，我们将稳定和不稳定的解分别称为$u \to \pm \infty$。还考虑了$H = 0$的情况。

英文摘要：

We study a $D$-dimensional Einstein-Gauss-Bonnet model which includes the Gauss-Bonnet term, the cosmological term $Λ$ and two non-zero constants: $α_1$ and $α_2$. Under imposing the metric to be diagonal one, we find cosmological type solutions with exponential dependence of three scale factors in a variable $u$, governed by three non-coinciding Hubble-like parameters: $H \neq 0$, $h_1$ and $h_2$, obeying $m H + k_1 h_1 + k_2 h_2 \neq 0$, corresponding to factor spaces of dimensions $m > 1$, $k_1 > 1$ and $k_2 > 1$, respectively, and depending upon sign parameter $\varepsilon = \pm 1$, where $\varepsilon = 1$ corresponds to cosmological case and $\varepsilon = - 1$ - to static one). We deal with two cases: i) $m < k_1 < k_2$ and ii) $1< k_1 = k_2 = k$, $k \neq m$. We show that in both cases the solutions exist if $\varepsilon α= \varepsilon α_2 / α_1 > 0$ and $αΛ> 0$ satisfies certain (upper and lower) bounds. The solutions are defined up to solutions of certain polynomial master equation of order four (or less) which may be solved in radicals. In case ii) explicit solutions are presented. In both cases we single out stable and non-stable solutions as $u \to \pm \infty$. The case $H = 0$ is also considered.

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英文摘要：

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