包含色散和高阶累积量的微扰理论：框架与线性理论
Perturbation theory with dispersion and higher cumulants: framework and linear theory
AI总结：
本文提出 Vlasov 微扰理论（VPT），通过纳入轨道交叉产生的第二及更高阶累积量扩展标准微扰理论（SPT），推导了线性解、稳定性条件及标度宇宙中的自洽解，有效解释了小尺度模式的退耦现象。
摘要：
引力聚集的标准微扰理论（SPT）方法基于底层 Vlasov-Poisson 动力学的流体近似，仅考虑相空间分布函数的第零和第一累积量（密度和速度场）。当暗物质粒子轨道交叉时，该假设失效并导致已知问题，例如小尺度模式对更大尺度的异常大反作用损害了预测性。我们通过纳入轨道交叉产生的第二和更高阶累积量来扩展 SPT。对于无碰撞物质，其运动方程完全由 Vlasov-Poisson 系统确定，因此我们将此方法称为 Vlasov 微扰理论（VPT）。偶数累积量发展出背景值，并进入涨落的耦合方程层级。背景值反过来由涨落的功率谱提供源。后者可化为形式上类似于 SPT 的形式，但具有扩展的变量集以及线性与非线性项，我们显式推导了这些项。本文聚焦于线性解，其远比 SPT 丰富，表明穿越由第二累积量设定的色散尺度的模式受到高度抑制。我们从无指数不稳定性的要求推导了偶数累积量背景值的稳定性条件。我们还计算了各种晕模型的平均高阶累积量的预期量级，并表明它们满足稳定性条件。最后，我们推导了标度宇宙中微扰和背景值的自洽解，并研究了累积量展开的收敛性。VPT 框架提供了 SPT 的概念上直接且确定性的扩展，解释了小尺度模式的退耦。
英文摘要：
The standard perturbation theory (SPT) approach to gravitational clustering is based on a fluid approximation of the underlying Vlasov-Poisson dynamics, taking only the zeroth and first cumulant of the phase-space distribution function into account (density and velocity fields). This assumption breaks down when dark matter particle orbits cross and leads to well-known problems, e.g. an anomalously large backreaction of small-scale modes onto larger scales that compromises predictivity. We extend SPT by incorporating second and higher cumulants generated by orbit crossing. For collisionless matter, their equations of motion are completely fixed by the Vlasov-Poisson system, and thus we refer to this approach as Vlasov Perturbation Theory (VPT). Even cumulants develop a background value, and they enter the hierarchy of coupled equations for the fluctuations. The background values are in turn sourced by power spectra of the fluctuations. The latter can be brought into a form that is formally analogous to SPT, but with an extended set of variables and linear as well as non-linear terms, that we derive explicitly. In this paper, we focus on linear solutions, which are far richer than in SPT, showing that modes that cross the dispersion scale set by the second cumulant are highly suppressed. We derive stability conditions on the background values of even cumulants from the requirement that exponential instabilities be absent. We also compute the expected magnitude of averaged higher cumulants for various halo models and show that they satisfy the stability conditions. Finally, we derive self-consistent solutions of perturbations and background values for a scaling universe and study the convergence of the cumulant expansion. The VPT framework provides a conceptually straightforward and deterministic extension of SPT that accounts for the decoupling of small-scale modes.
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