柏拉图拓扑和CMB涨落：同伦、各向异性和多极选择规则
Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules
摘要：
宇宙微波背景起源于宇宙历史的早期阶段。观测到的低多极贡献的CMB波动激发了搜索的选择规则的基础拓扑结构的3-空间。Everitt（2004）利用面胶合生成了Platonic球面3-流形的所有同伦。我们将胶水生成器转化为同构的甲板变换。甲板变换作用于一个球形的柏拉图三维流形作为prototile和瓷砖的3球的形象。三维球面上的正交函数的完备集由Wigner调和多项式张成。对于一个四面体，两个立方和三个八面体流形，我们构造代数线性组合的维格纳多项式，甲板下不变的变换和域的流形。从同伦的角度证明了多面体面上的边界条件。通过代数方法，我们转到多极展开式。假设随机模型的CMB辐射，我们推导出多极选择规则，根据点对称的流形。
英文摘要：
The Cosmic Microwave Background CMB originates from an early stage in the history of the universe. Observed low multipole contributions of CMB fluctuations have motivated the search for selection rules from the underlying topology of 3-space. Everitt (2004) has generated all homotopies for Platonic spherical 3-manifolds by face gluings. We transform the glue generators into isomorphic deck transformations. The deck transformations act on a spherical Platonic 3-manifold as prototile and tile the 3-sphere by its images. A complete set of orthonormal functions on the 3-sphere is spanned by the Wigner harmonic polynomials. For a tetrahedral, two cubic and three octahedral manifolds we construct algebraically linear combinations of Wigner polynomials, invariant under deck transformations and with domain the manifold. We prove boundary conditions on polyhedral faces from homotopy. By algebraic means we pass to a multipole expansion. Assuming random models of the CMB radiation, we derive multipole selection rules, depending on the point symmetry of the manifold.
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