We explore the deep ultraviolet (that is, short-distance) limit of the power spectrum (PS) and of the correlation function of a cold dark matter dominated Universe. While for large scales the PS can be written as a double series expansion, in powers of the linear PS and of the wavenumber $k$, we show that, in the opposite limit, it can be expressed via an expansion in powers of the form $1/k^{d+2n}$, where $d$ is the number of spatial dimensions, and $n$ is a non negative integer. The coefficients of the terms of the expansion are nonperturbative in the linear PS, and can be interpreted in terms of the probability density function for the displacement field, evaluated around specific configurations of the latter, that we identify. In the case of the Zel'dovich dynamics, these coefficients can be determined analytically, whereas for the exact dynamics they can be treated as fit, or nuisance, parameters. We confirm our findings with numerical simulations and discuss the necessary steps to match our results to those obtained for larger scales and to actual measurements.

中文翻译：

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大规模结构的渐近展开

我们探索了功率谱 (PS) 和冷暗物质主导宇宙的相关函数的深紫外（即短距离）极限。而对于大尺度，PS 可以写为双级数展开式，在线性 PS 和波数 $k$ 的幂中，我们表明，在相反的限制下，它可以通过形式的幂次展开来表示$1/k^{d+2n}$，其中$d$为空间维度数，$n$为非负整数。扩展项的系数在线性 PS 中是非微扰的，可以根据位移场的概率密度函数来解释，围绕我们确定的后者的特定配置进行评估。在泽尔多维奇动力学的情况下，这些系数可以通过分析确定，而对于确切的动力学，它们可以被视为合适的或令人讨厌的参数。我们通过数值模拟证实了我们的发现，并讨论了将我们的结果与更大尺度和实际测量结果相匹配的必要步骤。