用球形探针测试高能宇宙线的各向同性
Testing the isotropy of high energy cosmic rays using spherical needlets
摘要：
几十年来，可以从地面观察到的未知来源的高能带电粒子一直是粒子物理学家和天体物理学家的难题。为了区分几种可能的产生方案，天体物理学家试图测试这些宇宙射线到达方向的统计各向同性。在最高能量下，它们应该以良好的准确度指向它们的来源。然而，观测结果是如此罕见，以至于测试球体上的方向数据的这种样本的分布是不平凡的。在本文中，我们选择了一个非参数框架，使弱假设的替代分布，并允许反过来检测各种可能意想不到的形式的各向异性。我们探讨两个特殊的程序。两者都来自拟合的经验分布与小波展开的密度。我们使用[SIAM J. Math. Anal. 38（2006 b）574-594（电子）]，即所谓的针头。扩展在不大于某些${J^{\star}}$的尺度索引处被截断，并计算这些估计值与零密度之间的$L^p$距离。一个测试族（称为多个）是基于针对$J=1,\ldots,{J^{\star}}$的每个选择测试与空值的距离的想法，而所谓的插入方法是基于单个完整的${J^{\star}}$展开，但是具有阈值小波系数。我们描述了这两个程序的实际实施，并将其与文献中的其他方法进行比较。作为各向同性的替代品，我们考虑非常简单的玩具模型和更现实的非各向同性模型的基础上物理启发的模拟。蒙特卡罗研究显示，即使在中等样本量下，对于替代假设的宽样本和参数${J^{\star}}$的不同选择，多重检验也具有良好的性能。在Pierre Auger合作组织公布的69个最具活力的事件中，基于针的程序提出了各向异性的统计证据。使用几个值的方法的参数，我们的程序产生的$p$ -值低于1%，但与不受控制的多重性问题。这种方法的灵活性和修改的可能性，以考虑到大量的各种扩展的问题，使它成为一个有趣的选择，为未来的调查起源的高能宇宙射线。
英文摘要：
For many decades, ultrahigh energy charged particles of unknown origin that can be observed from the ground have been a puzzle for particle physicists and astrophysicists. As an attempt to discriminate among several possible production scenarios, astrophysicists try to test the statistical isotropy of the directions of arrival of these cosmic rays. At the highest energies, they are supposed to point toward their sources with good accuracy. However, the observations are so rare that testing the distribution of such samples of directional data on the sphere is nontrivial. In this paper, we choose a nonparametric framework that makes weak hypotheses on the alternative distributions and allows in turn to detect various and possibly unexpected forms of anisotropy. We explore two particular procedures. Both are derived from fitting the empirical distribution with wavelet expansions of densities. We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574-594 (electronic)], the so-called needlets. The expansions are truncated at scale indices no larger than some ${J^{\star}}$, and the $L^p$ distances between those estimates and the null density are computed. One family of tests (called Multiple) is based on the idea of testing the distance from the null for each choice of $J=1,\ldots,{J^{\star}}$, whereas the so-called PlugIn approach is based on the single full ${J^{\star}}$ expansion, but with thresholded wavelet coefficients. We describe the practical implementation of these two procedures and compare them to other methods in the literature. As alternatives to isotropy, we consider both very simple toy models and more realistic nonisotropic models based on Physics-inspired simulations. The Monte Carlo study shows good performance of the Multiple test, even at moderate sample size, for a wide sample of alternative hypotheses and for different choices of the parameter ${J^{\star}}$. On the 69 most energetic events published by the Pierre Auger Collaboration, the needlet-based procedures suggest statistical evidence for anisotropy. Using several values for the parameters of the methods, our procedures yield $p$-values below 1%, but with uncontrolled multiplicity issues. The flexibility of this method and the possibility to modify it to take into account a large variety of extensions of the problem make it an interesting option for future investigation of the origin of ultrahigh energy cosmic rays.
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