具有非零空间曲率的最优Boltzmann族
Optimal Boltzmann hierarchies with non-vanishing spatial curvature
摘要：
在宇宙学扰动理论中，宇宙微波背景的各向异性通常是从玻尔兹曼方程组耦合到微扰的爱因斯坦方程中计算出来的。在这种设置中，一组多极描述温度各向异性，而另外两组，电和磁类型，描述极化各向异性。为了减少极化所需的多极类型的数量，从而加快数值分辨率，在文献中已经提出了一个最佳层次的Einstein-Boltzmann码。然而，最近已经表明，在空间曲率的存在下，在最优层次中采用的方向和轨道本征函数之间的可分性是不正确的。我们调查如何假设的可分性影响的最佳层次，并表明，它介绍了相对误差的顺序$Ω_K$的完整的层次。尽管如此，我们表明，最佳的层次结构仍然给出了非常好的结果，温度和偏振角谱，相对误差比宇宙方差小得多，即使曲率大$|Ω_K|=0.1$。尽管如此，我们发现，张量扰动的偏振角谱显着改变时，使用最佳的层次结构，导致错误，通常是为了$50 |Ω_K| \%$的组件。
英文摘要：
Within cosmological perturbation theory, the cosmic microwave background anisotropies are usually computed from a Boltzmann hierarchy coupled to the perturbed Einstein equations. In this setup, one set of multipoles describes the temperature anisotropies, while two other sets, of electric and magnetic types, describe the polarization anisotropies. In order to reduce the number of multipoles types needed for polarization, and thus to speed up the numerical resolution, an optimal hierarchy has been proposed in the literature for Einstein-Boltzmann codes. However, it has been recently shown that the separability between directional and orbital eigenfunctions employed in the optimal hierarchy is not correct in the presence of spatial curvature. We investigate how the assumption of separability affects the optimal hierarchy, and show that it introduces relative errors of order $Ω_K$ with respect to the full hierarchy. Despite of that, we show that the optimal hierarchy still gives extremely good results for temperature and polarization angular spectra, with relative errors that are much smaller than cosmic variance even for curvatures as large as $|Ω_K|=0.1$. Still, we find that the polarization angular spectra from tensor perturbations are significantly altered when using the optimal hierarchy, leading to errors that are typically of order $50 |Ω_K| \%$ on that component.
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