Time non-locality in dark matter and LSS

We explore the intriguing phenomenon of time non-locality in the evolution of dark matter and Large Scale Structure (LSS). Recently in [1], it was shown that time non-locality emerges in bias tracer fluctuations, which are SO(3) scalars in real space, at fifth order in the perturbation expansion in dark matter overdensity. We demonstrate that by breaking the symmetry down to SO(2), which is the case whenever line-of-sight effects become important, such as for flux fluctuations in the Lyman α forest, the temporal non-locality appears at the third order in expansion. Additionally, within the framework of EFTofLSS, we demonstrate that time non-locality manifests in the effective stress tensor of dark matter, which is a second rank tensor under SO(3) transformations, again at the third order in dark matter overdensity. Furthermore, we highlight the effectiveness of the standard Π basis [2] in handling time non-local operators.