Kinetic Field Theory: Second-order perturbation theory

尽管宇宙标准模型取得了惊人的成功，但它给我们留下了两个困难而又至关重要的难题，即暗物质和暗能量，它们的起源、组成和在物理学框架中的位置都是谜。大规模的宇宙学调查正在进行，并且即将进行，其主要目标是绘制和量化暗物质的分布，并找出暗能量密度随时间的可能演变。从这些调查中得出的精确的宇宙学推断取决于我们对宇宙结构的后期非线性演化的理解的深度和精度。虽然数值模拟已经返回了大量令人印象深刻和重要的结果，但计算大尺度宇宙结构在后期的高阶统计特性仍然是一项令人生畏的耗时任务。最近，我们发展了一种新的基于非平衡动力学场论（KFT）的非线性宇宙结构形成的解析方法。与更传统的方法相比，这个理论有几个重要的优点。特别是，它作用于相空间中经典粒子的微观自由度。由于相空间中的哈密顿流是双客观和微分同态的，所以我们的方法中不存在众所周知的壳交叉问题。在目前的形式下，该理论没有可调参数。我们在早期的工作中已经证明，即使将一阶摄动理论应用于KFT，也能很好地再现宇宙密度波动的非线性功率谱，在当前宇宙时代的波数为k ~ 10 h/Mpc。我们可以进一步展示KFT的中心数学对象，它的生成函数，如何被完全分解成标准形式。基于这种分解，由此发展的摄动理论的图解方法，以及用于构造这些图的符号计算机代码的第一版，我们在这里提出计算宇宙密度波动的非线性功率谱的二阶摄动项。主要的挑战将是生成泛函分解中出现的通用表达式的可靠和快速的数值积分，以及摄动序列中出现的一环和双环积分的计算。所提出的研究的基本目标是完成和测试我们的微扰图符号计算机代码，为微扰项的数值积分提供代码，并评估、分析和比较这些项对非线性演化的宇宙功率谱的贡献。基于KFT中一阶摄动理论的质量，我们有信心可以解析计算出当前时代达到k ~ 20 ~ 50 h/Mpc波数的精确非线性功率谱。

Astoundingly successful as it is, the cosmological standard model leaves us with two difficult and fundamentally important puzzles, viz. the dark matter and the dark energy, whose origin, composition, and place in the framework of physics are enigmatic. Large cosmological surveys are being and are about to be undertaken with the major goals of mapping and quantifying the dark-matter distribution and finding out about a possible evolution of the dark-energy density with time. Precise cosmological inference from these surveys hinges on the depth and precision of our understanding of the late-time, non-linear evolution of cosmic structures. While numerical simulations have returned a wealth of impressive and important results, calculating higher-order statistical properties of large-scale cosmic structures at late times is still a forbiddingly time-consuming task.Recently, we have developed a novel analytic approach to non-linear cosmic structure formation, based on a non-equilibrium kinetic field theory (KFT). This theory has several important advantages compared to more conventional approaches. In particular, it operates on the microscopic degrees of freedom of classical particles in phase space. Since the Hamiltonian flow in phase space is bijective and diffeomorphic, the notorious shell-crossing problem is absent from our approach. In its present form, the theory is free of adjustable parameters. We have shown in earlier work that even first-order perturbation theory applied to KFT reproduces the non-linear power spectrum of cosmic density fluctuations very well to wave numbers of k ~ 10 h/Mpc at the present cosmic epoch. We could further show how the central mathematical object of KFT, its generating functional, can be fully factorized into terms of a standard form.Based on this factorization, a diagrammatic approach to perturbation theory developed therefrom, and a first version of a symbolic computer code for constructing these diagrams, we propose here to calculate the second-order perturbation terms for the non-linear power spectrum of cosmic density fluctuations. Main challenges will be the reliable and fast numerical integration of the generic expressions appearing in the factorization of the generating functional, and the calculation of the one- and two-loop integrals appearing in the perturbation series. The essential goals of the proposed research are to finalize and test our symbolic computer code for perturbation diagrams, to supply it with a code for numerical integration of perturbation terms, and to evaluate, analyze and compare the contributions of these terms to the non-linearly evolved cosmic power spectrum. Based on the quality of the first-order perturbation theory in KFT, we are confident that we can analytically calculate accurate non-linear power spectra reaching wave numbers of k ~ 20-50 h/Mpc at the present epoch.