Constrained correlation functions from the Millennium Simulation

TL;DR

This paper develops numerical methods to compute constraints on correlation functions in higher-dimensional random fields, tests them against analytical results in 1D, and applies them to Millennium Simulation data, improving likelihood modeling.

Contribution

It introduces numerical techniques for higher-dimensional correlation function constraints and demonstrates their effectiveness on simulation data, enhancing likelihood approximation methods.

Findings

Correlation functions from Millennium Simulation obey the constraints.

Numerical methods for constraints are robust and accurate.

Quasi-Gaussian likelihood outperforms Gaussian likelihood in modeling.

Abstract

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.