What does the N-point function hierarchy of the cosmological matter density field really measure ?

Abstract

The cosmological dark matter field is not completely described by its hierarchy of $N$-point functions, a non-perturbative effect with the consequence that only part of the theory can be probed with the hierarchy. We give here an exact characterization of the joint information of the full set of $N$-point correlators of the lognormal field. The lognormal field is the archetypal example of a field where this effect occurs, and, at the same time, one of the few tractable and insightful available models to specify fully the statistical properties of the evolved matter density field beyond the perturbative regime. Nonlinear growth in the Universe in that model is set letting the log-density field probability density functional evolve keeping its Gaussian shape, according to the diffusion equation in Euclidean space. We show that the hierarchy probes a different evolution equation, the diffusion equation defined not in Euclidean space but on the compact torus, with uniformity as the long-term solution. The extraction of the hierarchy of correlators can be recast in the form of a nonlinear transformation applied to the field, 'wrapping', undergoing a sharp transition towards complete disorder in the deeply nonlinear regime, where all memory of the initial conditions is lost.