The discretized closed Friedmann–Lemaître–Robertson–Walker (FLRW) universe with positive cosmological constant is investigated by Regge calculus. According to the Collins–Williams formalism, a hyperspherical Cauchy surface is replaced with regular 4-polytopes. Numerical solutions to the Regge equations approximate well to the continuum solution during the era of small edge length. Unlike the expanding polyhedral universe in three dimensions, the 4-polytopal universes repeat expansions and contractions. To go beyond the approximation using regular 4-polytopes we introduce pseudo-regular 4-polytopes by averaging the dihedral angles of the tessellated regular 600-cell. The degree of precision of the tessellation is called the frequency. Regge equations for the pseudo-regular 4-polytope have simple and unique expressions for any frequency. In the infinite frequency limit, the pseudo-regular 4-polytope model approaches the continuum FLRW universe.

中文翻译：

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雷格微积分中的振荡 4-多面体宇宙

通过雷格微积分研究了具有正宇宙学常数的离散封闭弗里德曼-勒梅特-罗伯逊-沃克（FLRW）宇宙。根据柯林斯-威廉姆斯形式主义，超球面柯西曲面被规则的 4 多面体取代。Regge 方程的数值解很好地近似于小边长时代的连续统解。与在三个维度上不断膨胀的多面体宇宙不同，4 多面体宇宙重复膨胀和收缩。为了超越使用规则 4 多面体的近似值，我们通过平均镶嵌规则 600 单元的二面角来引入伪规则 4 多面体。镶嵌的精确程度称为频率。伪正则 4 多面体的 Regge 方程对任何频率都有简单而独特的表达式。