Quasi-local evolution of cosmic gravitational clustering in a weakly nonlinear regime

We investigate the weakly nonlinear evolution of cosmic gravitational clust ering in phase space by looking at the Zeldovich solution in the discrete w avelet transform (DWT) representation. We show that if the initial perturba tions are Gaussian, the relation between the evolved DWT mode and the initi al perturbations in the weakly nonlinear regime is quasi-local. That is, th e evolved density perturbations are mainly determined by the initial pertur bations localized in the same spatial range. Furthermore, we show that the evolved mode is monotonically related to the initial perturbed mode. Thus, large (small) perturbed modes statistically correspond to the large (small) initial perturbed modes. We test this prediction by using quasi-stellar ob ject Ly alpha absorption samples. The results show that the weakly nonlinea r features for both the transmitted flux and the identified forest lines ar e quasi-localized. The locality and monotonic properties provide a solid ba sis for the DWT scale-by-scale Gaussianization reconstruction algorithm pro posed by L.-L. Feng & L.-Z. Fang for data in the weakly nonlinear regime. W ith the Zeldovich solution, we also find that the major non-Gaussianities c aused by the weakly nonlinear evolution are local scale-scale correlations. Therefore, to have a precise recovery of the initial Gaussian mass field, it is essential to remove the scale-scale correlations.