有限关联系统中的宇宙方差与各态历经性
Cosmic variance and ergodicity in finite systems with correlations
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中文总结 AI 辅助
该研究探讨宇宙学中系综平均与体积平均的各态历经性偏差，推导其衰减规律，量化有限尺度下的偏差值，指出大尺度结构相关场景需考虑该偏差。
中文摘要 AI 辅助
我们研究宇宙学中系综平均与体积平均的差异。已知对于足够弱的长程关联，该差异的均方根（我们称之为各态历经性偏差）在大体积R³的极限下按R^(-3/2)衰减。我们计算了这对高斯随机场功率谱施加的条件，量化了有限R时的偏差，并表明当测量尺度和关联延伸至可观测宇宙的大小时，R→∞极限对宇宙学观测几乎无意义。我们考虑曲率、密度和速度三类扰动，在大尺度下该偏差对三者均重要。对观测上最相关的密度扰动，相对偏差在间隔r=177 Mpc时首次超过100%，且在所有r>560 Mpc时均大于100%，在比较大尺度结构的系综平均与体积平均时需考虑该偏差。宇宙微波背景温度扰动在大角尺度上的偏差也很大，但因分析不涉及体积平均，故与观测无关。
英文摘要
We consider the difference between ensemble and volume average in cosmology. It is known that for sufficiently weak long-range correlations the root mean square of the difference, which we call ergodicity bias, decays like $R^{-3/2}$ in the limit of large volume $R^3$. We calculate the condition this imposes on the power spectrum of a Gaussian random field. We quantify the bias for finite $R$, and show that the $R\to\infty$ limit is of little relevance for cosmological observations when the measured scales and correlations extend to the size of the observable universe. We consider curvature, density, and velocity perturbations. On large scales the bias is important in all three cases. For the density perturbations, which are observationally the most relevant, the relative bias first exceeds 100% at the separation $r=177$ Mpc, and is larger than 100% for all $r>560$ Mpc. It should be taken into account when comparing ensemble and volume averages for large-scale structure. The bias is also large for the cosmic microwave background temperature perturbations on large angular scales, but this is not relevant for observations, as their analysis does not involve volume averaging.
