标题： 宇宙微波背景各向异性在紧致双曲宇宙中 I：计算相关函数 显示英文标题

标题： CMB Anisotropy in Compact Hyperbolic Universes I: Computing Correlation Functions

摘要： 宇宙微波背景（CMB）各向异性测量已经使宇宙整体拓扑结构的问题从理论可能性进入了观测的范畴。 宇宙的整体拓扑结构会改变宇宙场的相关性质。 特别是，如果宇宙的大小与CMB最后散射面的距离相当，在最大的可观测尺度上，CMB各向异性模式中会预测出强烈的相关性。 我们详细描述了使用图像正则化方法计算具有非平凡拓扑模型中这类相关函数的完全通用方案，并将其应用于计算复杂的紧致双曲空间。 我们的方法直接在指定半径内求和所有像，理想情况下是空间直径的许多倍，有效地用连续近似处理更远的像，并使用Cesaro重求和来进一步提高结果的准确性。 在所有近似水平下，空间的对称性都在相关函数中得以保留。 这一新技术消除了在这些空间上进行困难的空间本征模分解的必要性。 虽然如果需要，可以通过这种方法获得本征谱，但在给定的近似水平下，相关函数的确定更为准确。 我们通过3维环面的例子证明了该方法非常有效。 我们将它应用于多个紧致双曲（CH）空间的功率谱以及相关函数的评估。 CH空间上CMB各向异性相关性的计算及其由此得出的观测约束将在另一篇配套论文中给出。 显示英文摘要

摘要： CMB anisotropy measurements have brought the issue of global topology of the universe from the realm of theoretical possibility to within the grasp of observations. The global topology of the universe modifies the correlation properties of cosmic fields. In particular, strong correlations are predicted in CMB anisotropy patterns on the largest observable scales if the size of the Universe is comparable to the distance to the CMB last scattering surface. We describe in detail our completely general scheme using a regularized method of images for calculating such correlation functions in models with nontrivial topology, and apply it to the computationally challenging compact hyperbolic spaces. Our procedure directly sums over images within a specified radius, ideally many times the diameter of the space, effectively treats more distant images in a continuous approximation, and uses Cesaro resummation to further sharpen the results. At all levels of approximation the symmetries of the space are preserved in the correlation function. This new technique eliminates the need for the difficult task of spatial eigenmode decomposition on these spaces. Although the eigenspectrum can be obtained by this method if desired, at a given level of approximation the correlation functions are more accurately determined. We use the 3-torus example to demonstrate that the method works very well. We apply it to power spectrum as well as correlation function evaluations in a number of compact hyperbolic (CH) spaces. Application to the computation of CMB anisotropy correlations on CH spaces, and the observational constraints following from them, are given in a companion paper.