矩族和关联函数族作为对数正态域探针的不完备性
On the incompleteness of the moment and correlation function hierarchy as probes of the lognormal field
摘要：
我们跟踪分析方法和模型参数独立的方式的独立位的Fisher信息的每个时刻的对数正态分布，作为一个现在的标准处方的宇宙物质密度场的分布，因为它离开高斯初始条件。我们表明，当进入政权的大波动，只有一个很小的，急剧衰减的一部分，总信息内容仍然可以通过提取的领域的全部一系列的时刻。这是由于高尾分布的一个已知特性，即它们不能唯一地恢复给定的所有矩的值。这使得在对数正态假设下，宇宙学探测器，如关联函数层次或等效的傅里叶变换，一旦场变得非线性，对于任何感兴趣的参数，都是基本受限的。我们表明，从两点相关性访问的信息的分数衰减到零以下的字段的平方方差的倒数。我们讨论了一个随机场的概率密度函数的一般性质是使关联函数层次的有效或低效，完整或不完整的任何模型参数的探针集。
英文摘要：
We trace with analytical methods and in a model parameter independent manner the independent bits of Fisher information of each of the moments of the lognormal distribution, as a now standard prescription for the distribution of the cosmological matter density field, as it departs from Gaussian initial conditions. We show that, when entering the regime of large fluctuations, only a tiny, dramatically decaying fraction of the total information content remains accessible through the extraction of the full series of moments of the field. This is due to a known peculiarity of highly tailed distributions, that they cannot be uniquely recovered given the values of all of their moments. This renders under this lognormal assumption cosmological probes such as the correlation function hierarchy or equivalently their Fourier transforms fundamentally limited once the field becomes non linear, for any parameter of interest. We show that the fraction of the information accessible from two-point correlations decays to zero following the inverse squared variance of the field. We discuss what general properties of a random field's probability density function are making the correlation function hierarchy an efficient or inefficient, complete or incomplete set of probes of any model parameter.
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