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

了解非线性宇宙结构的形成是重要的，不仅因为对当前和即将到来的数据的分析依赖于它，还因为这些结构在现实中以及在模拟中揭示了起源尚不清楚的普遍性迹象。最近，基于动力场理论(KFT)发展了一种新的分析方法，用于研究平衡状态下和非平衡状态下经典粒子系统的非线性结构的形成。这种方法通过构造避免了更传统方法的穿壳问题，从而允许即使在低阶摄动理论下也能相当深入地进入非线性区域。此外，在KFT中发展了平均场近似，它在跨线性和非线性区域的大范围波数范围内再现了百分级数值模拟的宇宙密度涨落的非线性功率谱。到目前为止，这种平均场近似是以启发式方式推导出来的。在其成功的激励下，本提案旨在澄清其基础，将其置于严格的推导之上，并将其扩展到更高阶谱，如双谱或三谱。工作计划预计将开发概率分布，用于在适当现实的近似下，在脱离平衡的演化经典粒子系综中找到与平均密度场相关的粒子，用这些概率分布平均KFT生成泛函中的相互作用项，并推导出功率谱、双谱和可能的三谱。它进一步预见了研究如何在宇宙学的背景下将Hubbard-Stratonovich变换引入KFT，以及如何利用这种变换来进一步理解和发展宇宙结构形成的平均场近似。这种方法还将允许将对非线性宇宙结构的研究扩展到非标准宇宙学模型或引力理论。

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.