We study some aspects of dynamical compactification scenario where stabilization of extra dimensions occurs due to the presence of Gauss–Bonnet term and nonzero spatial curvature. In the framework of the model under consideration, there exists two-stages scenario of evolution of a Universe: in the first stage, the space evolves from a totally anisotropic state to the state with three-dimensional (corresponding to our “real” world) expanding and D-dimensional contracting isotropic subspaces; on the second stage, constant curvature of extra dimensions begins to play role and provide compactification of extra dimensions. It is already known that such a scenario is realizable when constant curvature of extra dimensions is negative. Here we show that a range of coupling constants for which exponential solutions with three-dimensional expanding and D-dimensional contracting isotropic subspaces are stable is located in a zone where compactification solutions with positively curved extra space are unstable, so that two-stage scenario analogous to the one described above is not realizable. Also we study “nearly-Friedmann” regime for the case of arbitrary constant curvature of extra dimensions and describe new parametrization of the general solution for the model under consideration which provide elegant way of describing areas of existence over parameters space.

中文翻译：

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爱因斯坦-高斯-博内引力中宇宙学动力学的某些方面

我们研究了动态紧化场景的某些方面，其中由于高斯-博内项和非零空间曲率的存在而发生额外维度的稳定。在所考虑的模型框架中，存在宇宙演化的两个阶段情景：在第一阶段，空间从完全各向异性状态演变为具有三维状态（对应于我们的“真实”世界）扩大和D-维收缩各向同性子空间；在第二阶段，额外维度的恒定曲率开始发挥作用并提供额外维度的紧凑化。众所周知，当额外维度的恒定曲率为负时，这种情况是可以实现的。在这里，我们展示了一系列耦合常数，其中具有三维扩展的指数解和D维收缩各向同性子空间是稳定的位于具有正弯曲额外空间的紧化解不稳定的区域，因此类似于上述的两阶段场景是不是可实现的。此外，我们研究了额外维度任意恒定曲率情况下的“近弗里德曼”机制，并描述了所考虑模型的通用解的新参数化，这提供了描述参数空间上存在区域的优雅方式。