The dark Universe in the light of its morphology

在过去的十年中，星系红移调查在测量更大尺度上的星系分布结构方面非常成功。大量正在进行和计划中的光谱调查，如BOSS， eBOSS， DESI， 4Most和Euclid，最终将提供可观测宇宙中很大一部分的三维地图。该项目的目的是开发基于闵可夫斯基泛函的分析工具，并充分利用这张星系位置图中包含的丰富信息。在精确宇宙学的时代，我们将获得的信息的补充对于揭示支配宇宙命运的黑暗成分的物理起源至关重要。我们使用的四个闵可夫斯基泛函唯一地量化了任何扩展体的形状。第一个函数给出物体的体积，第二个函数给出物体的表面积，第三个函数给出物体的积分平均曲率，最后一个函数给出物体的欧拉特性。加上一个将一组点转换成光滑体的公式，这些函数表征了我们在星系分布中看到的复杂形状，比如细丝和空洞。基于两点相关函数的常规分析并不能完全捕捉到星系调查中看到的这种丰富的结构。后者只测量结构增强在给定距离内发现两个星系的可能性的强度。使用闵可夫斯基泛函的主要优点是，除了标准的两点性质外，它们还通过分析关系包含了所有高阶相关性的信息。我们将在闵可夫斯基泛函中使用这些额外的信息来更精确地测试宇宙模型。为了实现这一目标，我们首先分析哪些宇宙学参数可以用我们将开发的方法进行最佳测试。特别地，我们将尝试确定一个标准的尺子来精确测量宇宙距离，从而测量暗能量。我们还将尝试使用闵可夫斯基泛函来改进暗物质和星系聚类差异的分析。在下一步中，我们将测试将宇宙学模型与观测联系起来的常用建模假设。在第三步中，我们将应用第一步中开发的方法和第二步中确定的最佳建模假设来执行宇宙学模型的数值测试。为此，我们将用我们的数字代码分析当前和未来的星系红移调查。最后，闵可夫斯基泛函被积极应用于不同的科学领域，从固态物理学到生物物理学，甚至在医学应用中也有应用。因此，我们将在这个项目中开发的理论方法和有效代码的影响可以超越宇宙学应用。

Over the last decade, galaxy redshift surveys have been extremely successful in measuring the structure of the galaxy distribution on ever larger scales. The large number of ongoing and planned spectroscopic surveys like BOSS, eBOSS, DESI, 4Most and Euclid will finally deliver a three dimensional map of a significant fraction of the observable universe.The aim of this project is to develop analysis tools that are based on Minkowski Functionals and fully exploit the wealth of information contained in this map of galaxy positions. In the era of precision cosmology, the supplement of information that we will access will be crucial to uncover the physical origin of the dark components that govern the fate of the universe.The four Minkowski Functionals we use uniquely quantify the shape of any extended body. The first functional gives the volume of the body, the second its surface area, the third its integrated mean curvature and the last its Euler characteristic. Together with a prescription that converts a set of points into a smooth body, the functionals characterize the complex shapes that we see in the distribution of galaxies, such as filaments and voids.This rich structure seen in galaxy surveys is not entirely captured by the usual analysis, based on the two-point correlation function. This latter only measures how strongly structure enhances the probability of finding two galaxies at a given distance from one another. The key advantage of using Minkowski Functionals is that, in addition to the standard two-point properties, they include information from all higher-order correlations, through an analytic relation. We will use this extra information in Minkowski Functionals to test cosmological models more precisely.To achieve this goal, we first analyse which cosmological parameters we can test the best with the method we will develop. In particular we will try to identify a standard ruler for a precise measurement of cosmic distances, and thus of dark energy. We will also attempt to use Minkowski Functionals to improve the analysis of the difference of clustering of dark matter and galaxies. In a next step, we will test commonly employed modelling assumptions which connect cosmological models to observations. In the third step, we will apply the method developed in the first step with the best modelling assumptions identified in the second step to perform a numerical test of cosmological models. To do so we will analyze current and future galaxy redshift surveys with our numerical codes.Finally, Minkowski Functionals are actively employed in different areas of Science from solid state physics, to biophysics and are even used in medical applications. Consequently, the impact of the theoretical methods and efficient codes we will develop during this project can go beyond cosmological applications.