Cosmic Topology and Software Development

在一个足够小的多连通宇宙中，人们预期宇宙微波背景（CMB）中存在异常微弱的大尺度波动，这正是WMAP卫星所观测到的，置信度约为1/300至1/500。 然而，随着CMB中异常现象的发现，神秘感加深了。具体来说，最宽的涨落（e11 = 2和e11 = 3模式）在宇宙学偶极子的方向上排列。更奇怪的是，射电天文学家和光学天文学家分别观察到指向同一方向的无法解释的各向异性。 虽然现在下结论还为时过早，但暗物质在候选名单上名列前茅。 无论最终的解释是什么，如果大尺度的宇宙微波背景辐射波动比看上去的要弱，这将加强宇宙有限的观点。 目前的项目继续发展有限宇宙候选者的本征模的数学，以便更好地与观测进行比较，同时试图理解数据中的异常现象，并衡量它们对拓扑结论的影响。 该项目的教育部分正在开发和扩展一套几何和拓扑软件，涵盖从专门的研究软件（SnapPea）到广泛使用的中学生拓扑游戏的范围。关于宇宙形状的推测可以追溯到古代，但对宇宙拓扑的严肃数学研究始于1854年，当时黎曼提出了超球面作为宇宙学模型。拓扑学在世纪后期蓬勃发展，导致了各种可能的三维空间。1900年，天文学家卡尔·史瓦西（Karl Schwarzschild）将这些模型作为空间拓扑的候选模型提交给德国天文学会，但由于当时观测数据有限，他只能得出宇宙必须比我们的银河系大得多的结论。 事实上，直到20世纪90年代，COBE和WMAP卫星在宇宙微波背景辐射中发现了有限宇宙的迹象，宇宙拓扑学作为一门观测科学才取得了一些进展。然而，过去的十年是动荡的十年，双曲模型的紧张工作让位于平面和球形模型，以适应1998年暗能量的发现。 最近，这个情节有了新的转折，宇宙微波背景的特征-以及一些光学和无线电异常-都对准了室女座的方向。 这表明了一些新的物理效应--也许是暗物质--这是以前没有考虑过的。本项目的研究部分将几何和拓扑知识应用于正在进行的理解我们宇宙的形状和性质的努力，而教育部分则使这些惊人的美丽想法能够被初中，高中和大学学生所接受。

In a sufficiently small multiconnected universe one expects unusually weak broad-scale fluctuations in the cosmic microwave background (CMB), and that is exactly what the WMAP satellite has observed, at a confidence level of roughly 1-in-300 to 1-in-500. However, the mystery has deepened with the discovery of anomalies in the CMB. Specifically, the broadest fluctuations (the ell=2 and ell=3 modes) align in the direction of the cosmological dipole. Stranger still, radio and optical astronomers have independently observed unexplained anisotropies pointing in the same direction. While it would be premature to draw conclusions, dark matter is high on the list of candidates. Whatever the ultimate explanation, if the broad-scale CMB fluctuations turn out to be weaker than they seem, this will strengthen the case for a finite universe. The current project continues to develop the mathematics of the eigenmodes of finite universe candidates for better comparison with observations, while simultaneously trying to make sense of the anomalies in the data and to gauge their impact on topological conclusions. The educational half of the project is developing and expanding a suite of geometry and topology software, spanning the gamut from specialized research software (SnapPea) to widely used topology games for middle school students.Speculation on the shape of the universe goes back to ancient times, but serious mathematical study of cosmic topology began in 1854 when Riemann proposed the hypersphere as a cosmological model. Topology blossomed in the late 19th century, leading to a variety of possible 3-dimensional spaces. The astronomer Karl Schwarzschild presented these models to the German Astronomical Society in 1900 as candidates for the topology of space, but given the limited observational data available at the time, he could conclude only that the universe must be much larger than our Milky Way galaxy. Indeed cosmic topology made little headway as an observational science until the 1990's when the COBE and WMAP satellites found hints of a finite universe in the cosmic microwave background radiation. The past decade has been a turbulent one, though, with intense work on hyperbolic models giving way to flat and spherical models to accommodate the 1998 discovery of dark energy. More recently the plot has taken a new twist, with features of the cosmic microwave background -- as well as some optical and radio anomalies -- all aligning in the direction of Virgo. This suggests some new physical effect -- perhaps dark matter -- which has not been previously accounted for. The research component of the present project applies geometrical and topological knowledge to the ongoing efforts to understand the shape and nature of our universe, while the educational component makes these breathtakingly beautiful ideas accessible to middle school, high school and college students (www.geometrygames.org).