CMBR功率谱中的峰值。I.相关实空间特征的数学分析
Peaks in the CMBR power spectrum. I. Mathematical analysis of the associated real space features
摘要：
我们的研究目的是了解在宇宙微波背景辐射各向异性中观测到的调制和阻尼正弦峰在实空间中的数学起源。我们使用傅里叶变换的理论，连接本地化功能的两点关联函数在实空间中的功率谱的振荡。我们还说明了分析和蒙特卡罗模拟的角度关联函数的分布填充磁盘具有固定或可变半径能够产生振荡的功率谱。虽然功率谱以多个峰和振荡的形式示出了重复的信息，但是角度关联函数提供了更紧凑的表示，其将多个峰的所有信息浓缩成局部实空间特征。我们已经看到，当角关联函数的给定导数在给定角距离处不连续时，功率谱中的振荡就会出现。这些类型的不连续性不需要在无限小的角距离范围内是突然的，但也可以是平滑的，并且可以通过简单地将天线温度的过度分布在天空上的固定或可变半径的填充盘中来产生，只要存在非零的最小半径和/或最大半径受到约束。
英文摘要：
The purpose of our study is to understand the mathematical origin in real space of modulated and damped sinusoidal peaks observed in cosmic microwave background radiation anisotropies. We use the theory of the Fourier transform to connect localized features of the two-point correlation function in real space to oscillations in the power spectrum. We also illustrate analytically and by means of Monte Carlo simulations the angular correlation function for distributions of filled disks with fixed or variable radii capable of generating oscillations in the power spectrum. While the power spectrum shows repeated information in the form of multiple peaks and oscillations, the angular correlation function offers a more compact presentation that condenses all the information of the multiple peaks into a localized real space feature. We have seen that oscillations in the power spectrum arise when there is a discontinuity in a given derivative of the angular correlation function at a given angular distance. These kinds of discontinuities do not need to be abrupt in an infinitesimal range of angular distances but may also be smooth, and can be generated by simply distributing excesses of antenna temperature in filled disks of fixed or variable radii on the sky, provided that there is a non-null minimum radius and/or the maximum radius is constrained.
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