Bispectrum covariance in the flat-sky limit
Bispectrum covariance in the flat-sky limit
复制标题
平坦天空极限下的双谱协方差
DOI：
10.1051/0004-6361/200912906
复制
发表时间：
2009
影响因子：
6.5
通讯作者：
Universitat Bonn University College London
中科院分区：
文献类型：
--
作者：
B. Joachimi;X. Shi;P. S. A. F. Astronomie;Universitat Bonn University College London
目标。为了探测高斯水平以上的宇宙学场，可以使用三点统计，所有这些都与双谱有关。因此，测量CMB各向异性，星系集群，和弱引力透镜一样，必须依赖于一个准确的理论背景下的双谱及其噪声特性。如果只考虑天空的一小部分，通常需要在平天极限中进行分析。我们的目标是在一个正式的，详细的推导的双谱协方差在平天近似，专注于一个纯粹的二维傅立叶平面的方法。方法.我们定义了一个无偏估计的双谱，它需要在傅立叶空间中的环重叠的平均值，并计算其全协方差。我们的形式主义的结果相比，平天球谐近似的协方差，宇称变换下的行为，和信息内容。我们介绍了几何解释的平均过程中的估计，从而提供了一个直观的理解。结果与前述工作相反，我们发现傅立叶平面和球谐函数方法的协方差之间的差异的两个因素。我们认为，这种差异可以解释的不同行为方面的奇偶校验。然而，在一个示例性的分析表明，Fisher信息的两种形式主义同意高精度。通过几何解释，我们能够将双谱估计中的归一化与所考虑的三角形配置所包围的区域以及维格纳符号联系起来，这导致了两种方法的协方差的方便近似公式。
Aims. To probe cosmological fields beyond the Gaussian level, three-point statistics can be used, all of which are related to the bispectrum. Hence, measurements of CMB anisotropies, galaxy clustering, and weak gravitational lensing alike have to rely upon an accurate theoretical background concerning the bispectrum and its noise properties. If only small portions of the sky are considered, it is often desirable to perform the analysis in the flat-sky limit. We aim at a formal, detailed derivation of the bispectrum covariance in the flat-sky approximation, focusing on a pure two-dimensional Fourier-plane approach. Methods. We define an unbiased estimator of the bispectrum, which takes the average over the overlap of annuli in Fourier space, and compute its full covariance. The outcome of our formalism is compared to the flat-sky spherical harmonic approximation in terms of the covariance, the behavior under parity transformations, and the information content. We introduce a geometrical interpretation of the averaging process in the estimator, thus providing an intuitive understanding. Results. Contrary to foregoing work, we find a difference by a factor of two between the covariances of the Fourier-plane and the spherical harmonic approach. We argue that this discrepancy can be explained by the differing behavior with respect to parity. However, in an exemplary analysis it is demonstrated that the Fisher information of both formalisms agrees to high accuracy. Via the geometrical interpretation we are able to link the normalization in the bispectrum estimator to the area enclosed by the triangle configuration at consideration as well as to the Wigner symbol, which leads to convenient approximation formulae for the covariances of both approaches.