标题： 宇宙密度涨落的自相似演化 显示英文标题

标题： Self-Similar Evolution of Cosmological Density Fluctuations

摘要： 在爱因斯坦-德西特宇宙中，尺度自由初始谱$P(k)\propto k^n$的引力演化被认为对于$-3<n<4$是自相似的。 然而，对于$-3<n<-1$，自相似标度的存在尚未得到充分证明。 在这里，我们研究由于长波模式的非线性贡献导致的自相似标度可能被破坏的情况。 对于$n<-1$，傅里叶空间流体方程中的非线性项由于波数$k\to 0$（长波极限）的贡献而发散。 为了评估这种发散可能的动力学效应，使用两种不同的分析方法详细研究了长波贡献的极限。 检查了功率谱的微扰贡献。 结果显示，对于$n<-1$，在所有阶次上都有发散的贡献。 然而，在每一阶次中，主导的发散项恰好相互抵消。 这并不能排除存在较弱但仍然发散的净贡献的可能性。 第二种方法是一种非微扰近似，用于研究长波模式耦合的非线性效应。 得到了傅里叶空间密度相位移的解，该解对于$n<-1$是发散的。 给出了与大尺度整体速度引起的平动运动相关的相位移发散的动力学解释。 我们的分析表明，密度的振幅{\it 不是}受发散项的影响。 因此，两种分析方法都得出结论，物理上有意义的密度扰动增长量的自相似标度得以保持。 显示英文摘要

摘要： The gravitational evolution of scale free initial spectra $P(k)\propto k^n$ in an Einstein-de Sitter universe is widely believed to be self-similar for $-3<n<4$. However, for $-3<n<-1$ the existence of self-similar scaling has not been adequately demonstrated. Here we investigate the possible breaking of self-similar scaling due to the nonlinear contributions of long wave modes. For $n<-1$ the nonlinear terms in the Fourier space fluid equations contain terms that diverge due to contributions from wavenumber $k\to 0$ (the long wave limit). To assess the possible dynamical effects of this divergence the limit of long wave contributions is investigated in detail using two different analytical approaches. Perturbative contributions to the power spectrum are examined. It is shown that for $n<-1$ there are divergent contributions at all orders. However, at every order the leading order divergent terms cancel out exactly. This does not rule out the existence of a weaker but nevertheless divergent net contribution. The second approach consists of a non-perturbative approximation, developed to study the nonlinear effects of long wave mode coupling. A solution for the phase shift of the Fourier space density is obtained which is divergent for $n<-1$. A kinematical interpretation of the divergence of the phase shift, related to the translational motion induced by the large-scale bulk velocity, is given. Our analysis indicates that the amplitude of the density is {\it not} affected by the divergent terms. Thus both analytical approaches lead to the conclusion that the self-similar scaling of physically relevant measures of the growth of density perturbations is preserved.