Abstract

We study a \(D\)-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss-Bonnet term and the cosmological constant \(\Lambda\). We find a class of cosmological type solutions with exponential dependence of two scale factors on the variable \(u\) (either cosmological time or a spatial coordinate), governed by two Hubble-like parameters \(H\neq 0\) and \(h\), corresponding to factor spaces of dimensions \(m>2\) and \(l>2\), respectively, and depending on the sign parameter \(\varepsilon=\pm 1\) (\(\varepsilon=1\) corresponds to cosmological solutions and \(\varepsilon=-1\) to static ones). These solutions are governed by a certain master equation \(\Lambda\alpha=\lambda(x)\) and the restriction \(\alpha\varepsilon(x-x_{+})(x-x_{-})<0\) (\(x_{-}<x_{+}<0\)) for the ratio \(h/H=x\), where \(\alpha=\alpha_{2}/\alpha_{1}\) is the ratio of two constants of the model . The master equation is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. Imposing certain restrictions on \(x\), and we prove the stability of the solutions for \(u\to\pm\infty\) in a certain class of cosmological solutions with diagonal metrics.

中文翻译：

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具有二元符号{\ Lambda} $$的Einstein–Gauss–Bonnet模型中具有两个因子空间的指数宇宙型解的稳定性

摘要

我们研究了一个\（D \）维的爱因斯坦-高斯-邦尼特引力模型，其中包括高斯-邦尼特项和宇宙常数\（\ Lambda \）。我们发现了一类宇宙学类型的解，它由两个比例因子对变量\（u \）（宇宙学时间或空间坐标）呈指数依赖性，由两个类似哈勃的参数\（H \ neq 0 \）和\ （h \），分别对应于尺寸\（m> 2 \）和\（l> 2 \）的因数空间，并取决于符号参数\（\ varepsilon = \ pm 1 \）（\（\ varepsilon = 1 \）对应宇宙学解，\（\ varepsilon = -1 \）到静态的）。这些解决方案由某个主方程\（\ Lambda \ alpha = \ lambda（x）\）和约束\（\ alpha \ varepsilon（x-x _ {+}）（x-x _ {-}）<0决定\）（\（x _ {-} <x _ {+} <0 \））的比率\（h / H = x \），其中\（\ alpha = \ alpha_ {2} / \ alpha_ {1} \ ）是模型的两个常数之比。主方程式等效于四阶或三阶多项式方程式，并且可以求解为根。对\（x \）施加某些限制，我们证明了在具有对角线度量的特定类宇宙学解中\（u \ to \ pm \ infty \）的解的稳定性。