弱场极限下Klein-Gordon-Einstein方程的流体力学表示：I。一般形式和扰动分析
Hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit: I. General formalism and perturbations analysis
摘要：
利用推广的Madelung变换，我们导出了弱场极限下Klein-Gordon-Einstein方程的流体动力学表示。本文考虑具有$λ|φ|^4$势的复自相互作用标量场。我们研究了流体表象中均匀背景的演化，导出了描述静态宇宙和膨胀宇宙中小扰动演化的线性化方程。我们比较的结果与简化的模型，其中的引力位是手工引入的克莱因-戈登方程，并假定满足（广义）泊松方程。我们研究的扰动在物质时代的演变，使用我们的形式主义的非相对论性的限制。在冷暗物质模型中，波长低于金斯长度的扰动随时间振荡，而波长高于金斯长度的扰动随标度因子线性增长。在标量场模型中扰动的增长要比冷暗物质模型中快得多。当微扰的波长接近宇宙视界（哈勃长度）时，相对论性的处理是强制性的。在这种情况下，我们发现相对论效应减弱甚至阻止微扰的增长。本文揭示了一般的形式主义，并提供了简单的情况下的例子。我们的形式主义的其他应用将被认为是在配套文件。
英文摘要：
Using a generalization of the Madelung transformation, we derive the hydrodynamic representation of the Klein-Gordon-Einstein equations in the weak field limit. We consider a complex self-interacting scalar field with a $λ|φ|^4$ potential. We study the evolution of the homogeneous background in the fluid representation and derive the linearized equations describing the evolution of small perturbations in a static and in an expanding universe. We compare the results with simplified models in which the gravitational potential is introduced by hand in the Klein-Gordon equation, and assumed to satisfy a (generalized) Poisson equation. We study the evolution of the perturbations in the matter era using the nonrelativistic limit of our formalism. Perturbations whose wavelength is below the Jeans length oscillate in time while pertubations whose wavelength is above the Jeans length grow linearly with the scale factor as in the cold dark matter model. The growth of perturbations in the scalar field model is substantially faster than in the cold dark matter model. When the wavelength of the pertubations approaches the cosmological horizon (Hubble length), a relativistic treatment is mandatory. In that case, we find that relativistic effects attenuate or even prevent the growth of pertubations. This paper exposes the general formalism and provides illustrations in simple cases. Other applications of our formalism will be considered in companion papers.
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