宇宙的尺度对称性
Scale Symmetry in the Universe
摘要：
尺度对称性是物理学的基本对称性，但在宇宙中似乎还没有完全实现。在这里，我们专注于由引力统治的天文尺度，尺度对称性保持并产生了一个真正的尺度不变的物质分布，即它产生了分形几何。非线性Poisson-Boltzmann-Emden方程对分形宇宙质量分布的特征提供了一个合适的解释。这个方程的另一种解释与量子引力理论有关。我们研究了该方程的分形解，并将其与起源于流体湍流理论的随机乘性级联的统计理论联系起来。这样得到的多重分形质量分布类型与宇宙学模拟分析和星系分布观测的结果一致。
英文摘要：
Scale symmetry is a fundamental symmetry of physics that seems however not to be fully realized in the universe. Here, we focus on the astronomical scales ruled by gravity, where scale symmetry holds and gives rise to a truly scale invariant distribution of matter, namely it gives rise to a fractal geometry. A suitable explanation of the features of the fractal cosmic mass distribution is provided by the nonlinear Poisson--Boltzmann--Emden equation. An alternative interpretation of this equation is connected with theories of quantum gravity. We study the fractal solutions of the equation and connect them with the statistical theory of random multiplicative cascades, which originated in the theory of fluid turbulence. The type of multifractal mass distributions so obtained agrees with results from the analysis of cosmological simulations and of observations of the galaxy distribution.
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