Covariances of density probability distribution functions. Lessons from hierarchical models

Abstract

Context. Statistical properties of the cosmic density fields are to a large extent encoded in the shape of the one-point density probability distribution functions (PDF) as measured in surveys. In order to successfully exploit such observables, a detailed functional form of the covariance matrix of the one-point PDF is needed. Aims. The objectives are to model the properties of this covariance for general stochastic density fields and for stochastic fields that reproduce the properties expected in cosmology. The accuracy of the proposed forms is evaluated in specific cases. Methods. The study was conducted in a cosmological context and determined whether the density is defined absolutely or relatively to the sample mean density. Leading and subleading contributions were identified within a large class of models, the so-called hierarchical models. They come from either large or short separation contributions. The validity of the proposed forms for the covariance matrix was assessed with the help of a toy model, the minimum tree model, for which a corpus of exact results could be obtained (forms of the one- and two-point PDF, large-scale density-bias functions, and full covariance matrix of the one-point PDF). Results. It is first shown that the covariance matrix elements are directly related to the spatial average of the two-point density PDF within the sample. The dominant contribution to this average is explicitly given for hierarchical models (coming from large scale contribution), which leads to the construction of specific density-bias functions. However, this contribution alone cannot be used to construct an operational likelihood function. Subdominant large-scale effects are found to provide corrective terms, but also a priori lead to limited information on the covariance matrix. Short distance effects are found to be more important but more difficult to derive as they depend more on the details of the model. However, a simple and generic form of these contributions is proposed. Detailed comparisons in the context of the Rayleigh-Levy flight model show that the large-scale effects capture the bulk of the supersample effects and that, by adding the short-distance contributions, a qualitatively correct model of the likelihood function can be obtained.