来自欧拉和拉格朗日扰动理论的控制变量：应用于双谱
Control variates from Eulerian and Lagrangian perturbation theory: Application to the bispectrum
AI总结：
本文在欧拉和拉格朗日扰动理论中构建了扰动控制变量，并以物质双谱为例，展示了其能显著降低N体模拟的方差，实现单次模拟即可达到亚2%精度。
摘要：
控制变量作为一种强大的技术，最近引起了人们的兴趣，用于减少从昂贵的宇宙学$N$体模拟中测量的汇总统计量的方差。特别令人感兴趣的是可解析计算的控制变量类别，例如最近引入的用于物质和偏置示踪物功率谱的“Zeldovich控制变量”。在这项工作中，我们提出了在欧拉和拉格朗日扰动理论中构建扰动控制变量的方法，并以物质双谱作为案例研究。欧拉控制变量对所有$n$点函数都是解析可处理的，但我们表明它们与$N$体$n$点函数的相关性以与波数平方和成比例的速率衰减，这阻碍了它们的实用性。我们表明，Zeldovich近似虽然具有可解析计算的双谱，但在低$k$下与欧拉对应物的相关性较小。我们引入了一种替代方案——“移位控制变量”——它可以被构造为具有正确的树级$n$点函数，是Zeldovich重求和后的，并且原则上具有可解析处理的双谱。我们发现，将这种移位控制变量应用于$z=0.5$的物质双谱，相当于对所考虑的最低$k$三角形平均超过$10^4$次模拟。使用单个$V=1({ m Gpc}/h)^3$的$N$体模拟，对于从$k_{ m min} = 0.04 h { m Mpc}^{-1}$到$k_{ m \max} = 0.47 h { m Mpc}^{-1}$、$N\approx 1400$个三角形的分箱方案，我们为每个测量的三角形配置获得了亚2%的精度。这项工作促进了精确双谱模拟器的开发——这是一种非常适合基于模拟的建模的宇宙学探针——并为将控制变量扩展到整个$n$点层次结构奠定了理论基础。
英文摘要：
Control variates have seen recent interest as a powerful technique to reduce the variance of summary statistics measured from costly cosmological $N$-body simulations. Of particular interest are the class of control variates which are analytically calculable, such as the recently introduced 'Zeldovich control variates' for the power spectrum of matter and biased tracers. In this work we present the construction of perturbative control variates in Eulerian and Lagrangian perturbation theory, and adopt the matter bispectrum as a case study. Eulerian control variates are analytically tractable for all $n$-point functions, but we show that their correlation with the $N$-body $n$-point function decays at a rate proportional to the sum-of-squared wavenumbers, hampering their utility. We show that the Zeldovich approximation, while possessing an analytically calculable bispectrum, is less correlated at low-$k$ than its Eulerian counterpart. We introduce an alternative -- the 'shifted control variate' -- which can be constructed to have the correct tree-level $n$-point function, is Zeldovich-resummed, and in principle has an analytically tractable bispectrum. We find that applying this shifted control variate to the $z=0.5$ matter bispectrum is equivalent to averaging over $10^4$ simulations for the lowest-$k$ triangles considered. With a single $V=1({\rm Gpc}/h)^3$ $N$-body simulation, for a binning scheme with $N\approx 1400$ triangles from $k_{\rm min} = 0.04 h {\rm Mpc}^{-1}$ to $k_{\rm \max} = 0.47 h {\rm Mpc}^{-1}$, we obtain sub-2% precision for every triangle configuration measured. This work enables the development of accurate bispectrum emulators -- a probe of cosmology well-suited to simulation-based modeling -- and lays the theoretical groundwork to extend control variates for the entire $n$-point hierarchy.
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