This paper focuses on two aspects of the statistics of cosmological observables that are important for the next stages of precision cosmology. First, we note that the theory of reduced angular $N$-point spectra has only been developed in detail up to the trispectrum case and in a fashion that makes it difficult to go beyond. To fill this gap, here we present a constructive approach that provides a systematic description of reduced angular $N$-point spectra and their covariance matrices, for arbitrary $N$. Second, we focus on the common practice in the literature on cosmological observables, which consists in simply discarding a part of the expression, namely, the terms containing fields evaluated at the observer position. We point out that this is not justified beyond linear order in perturbation theory, as these terms contribute to all the multipoles of the corresponding spectra and with a magnitude that is of the same order as the rest of the nonlinear corrections. We consider the possibility that the reason for neglecting these terms is a conceptual discomfort when using ensemble averages, which originates in an apparent tension between the ergodic hypothesis and the privileged position of the observer on the light-cone. We clarify this subtle issue by performing a careful derivation of the relation between the theoretical statistical predictions and the observational estimators for all $N$. We conclude that there is no inconsistency whatsoever in ensemble-averaging fields at and near the observer position, thus clearing the way for consistent and robust high-precision calculations.

中文翻译：

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宇宙观测的一般和一致统计

本文重点关注宇宙学可观测统计数据的两个方面，这对于下一阶段的精确宇宙学很重要。首先，我们注意到减少角度的理论$ñ$点光谱仅在三光谱情况下才详细开发，并且难以超越。为了填补这一空白，在这里我们提出了一种建设性的方法，该方法提供了减少角度的系统描述。$ñ$点光谱及其协方差矩阵，对于任意 $ñ$。其次，我们关注于宇宙学可观察性文献中的常见实践，即仅丢弃一部分表达式，即包含在观察者位置评估的字段的术语。我们指出，在扰动理论中，这超出线性阶数是没有道理的，因为这些项对相应频谱的所有多极都具有贡献，并且其大小与其余非线性校正的阶数相同。我们认为，忽略这些术语的原因可能是使用整体平均时的概念上的不适，这可能是由于遍历假设和观察者在圆锥体上的特权位置之间存在明显的张力。$ñ$。我们得出的结论是，在观察者位置及其附近的集合平均场中没有任何不一致之处，从而为一致，可靠的高精度计算扫清了道路。