Constrained correlation functions from the Millennium Simulation

Context. In previous work, we developed a quasi-Gaussian approximation for the likelihood of correlation functions, which, in contrast to the usual Gaussian approach, incorporates fundamental mathematical constraints on correlation functions. The analytical computation of these constraints is only feasible in the case of correlation functions of one-dimensional random fields. Aims. In this work, we aim to obtain corresponding constraints in the case of higher-dimensional random fields and test them in a more realistic context. Methods. We develop numerical methods to compute the constraints on correlation functions which are also applicable for two- and three-dimensional fields. In order to test the accuracy of the numerically obtained constraints, we compare them to the analytical results for the one-dimensional case. Finally, we compute correlation functions from the halo catalog of the Millennium Simulation, check whether they obey the constraints, and examine the performance of the transformation used in the construction of the quasi-Gaussian likelihood. Results. We find that our numerical methods of computing the constraints are robust and that the correlation functions measured from the Millennium Simulation obey them. Despite the fact that the measured correlation functions lie well inside the allowed region of parameter space, i.e. far away from the boundaries of the allowed volume defined by the constraints, we find strong indications that the quasi-Gaussian likelihood yields a substantially more accurate description than the Gaussian one.Comment: 11 pages, 13 figures, updated to match version accepted by A&

Statistical properties of cosmological correlation functions

Correlation functions are an omnipresent tool in astrophysics, and they are routinely used to study phenomena as diverse as the large-scale structure of the Universe, time-dependent pulsar signals, and the cosmic microwave background. In many cases, measured correlation functions are analyzed in the framework of Bayesian statistics, which requires knowledge about the likelihood of the data. In the case of correlation functions, this probability distribution is usually approximated as a multivariate Gaussian, which is not necessarily good approximation -- hence, this work aims at finding a better description. To this end, we exploit fundamental mathematical constraints on correlation functions, which we use to construct a quasi-Gaussian likelihood. We explain how to compute the constraints, in particular for multi-dimensional random fields, where this can only be done numerically, check the quality of the quasi-Gaussian approximation, and compare it to alternative approaches -- most importantly, we test the new-found description of the likelihood in a toy-model Bayesian analysis. Finally, we compute correlation functions from the Millennium Simulation and show that they obey the constraints. By studying statistical properties of the measured correlation functions, we present further indications for the validity of the quasi-Gaussian approach

A quasi-Gaussian approximation for the probability distribution of correlation functions

Context. Whenever correlation functions are used for inference about cosmological parameters in the context of a Bayesian analysis, the likelihood function of correlation functions needs to be known. Usually, it is approximated as a multivariate Gaussian, though this is not necessarily a good approximation. Aims. We show how to calculate a better approximation for the probability distribution of correlation functions, which we call "quasi-Gaussian". Methods. Using the exact univariate PDF as well as constraints on correlation functions previously derived, we transform the correlation functions to an unconstrained variable for which the Gaussian approximation is well justified. From this Gaussian in the transformed space, we obtain the quasi-Gaussian PDF. The two approximations for the probability distributions are compared to the "true" distribution as obtained from simulations. Additionally, we test how the new approximation performs when used as likelihood in a toy-model Bayesian analysis. Results. The quasi-Gaussian PDF agrees very well with the PDF obtained from simulations; in particular, it provides a significantly better description than a straightforward copula approach. In a simple toy-model likelihood analysis, it yields noticeably different results than the Gaussian likelihood, indicating its possible impact on cosmological parameter estimation.Comment: 16 pages, 14 figures, updated to match version accepted by A&