Non-linear cosmic structure formation in the mean-field approximation

理解非线性宇宙结构的形成很重要，不仅因为当前和即将到来的数据的分析依赖于它，而且因为这些结构在现实中以及在模拟中揭示了起源不明的普遍性迹象。近年来，基于动力学场理论（KFT），提出了一种新的解析方法来分析处于和非平衡状态的经典粒子系统的非线性结构形成。这种方法通过构造避免了传统方法的壳交叉问题，因此即使在低阶摄动理论下也可以进入相当深入的非线性状态。此外，在KFT范围内发展了一个平均场近似，再现了数值模拟的宇宙密度波动的非线性功率谱，在跨线性和非线性体制的大范围波数中达到百分之百的水平。到目前为止，这个平均场近似是用一种启发式的方法推导出来的。在其成功的激励下，本建议旨在澄清其基础，将其置于严格的推导之上，并将其扩展到更高阶的光谱，如双光谱或三光谱。该工作计划预计发展概率分布，在适当的现实近似下，在经典粒子的演化系综中发现与平均密度场相关的粒子，用这些概率分布平均KFT生成函数中的相互作用项，并推导功率谱、双谱和可能的三谱。它进一步预测了如何在宇宙学背景下将hubard - stratonovich变换引入KFT，以及如何利用这种变换进一步理解和发展宇宙结构形成的平均场近似。这种方法还可以将非线性宇宙结构的研究扩展到非标准的宇宙模型或引力理论。

Understanding the formation of non-linear cosmic structures is important not only because the analysis of current and upcoming data depends on it, but also because these structures in reality as well as in simulations reveal signs of universality whose origin is unclear. Recently, a new analytic approach to non-linear structure formation in classical particle systems in and out of equilibrium has been developed, based on kinetic field theory (KFT). This approach avoids the shell-crossing problem of more conventional approaches by construction and thus allows to enter quite deeply into the non-linear regime even at low orders of perturbation theory. In addition, a mean-field approximation has been developed within KFT that reproduces non-linear power spectra of numerically simulated cosmic density fluctuations at the per-cent level in a wide range of wave numbers in the trans-linear and non-linear regimes. This mean-field approximation has so far been derived in a heuristic way. Motivated by its success, the present proposal aims at clarifying its foundations, placing it on a rigorous derivation, and extending it towards spectra of higher order, such as the bi- or trispectrum. The work programme foresees developing probability distributions for finding particles correlated with the mean density field in evolved ensembles of classical particles out of equilibrium under appropriately realistic approximations, averaging interaction terms in the generating functional of KFT with these probability distributions, and deriving power spectra, bispectra, and possibly trispectra. It further foresees studying how the Hubbard-Stratonovich transformation could be introduced into KFT in a cosmological context, and how this transformation could be used to further understand and develop the mean-field approximation to cosmic structure formation. This approach will also allow to extend studies of non-linear cosmic structures to non-standard cosmological models or theories of gravity.