We introduce the theory of non-linear cosmological perturbations using the correspondence limit of the Schrödinger equation .
The resulting formalism is equivalent to using the collisionless Boltzman ( or Vlasov ) equations which remain valid during the whole evolution , even after shell crossing .
Other formulations of perturbation theory explicitly break down at shell crossing , e.g .
Eulerean perturbation theory , which describes gravitational collapse in the fluid limit .
This paper lays the groundwork by introducing the new formalism , calculating the perturbation theory kernels which form the basis of all subsequent calculations .
We also establish the connection with conventional perturbation theories , by showing that third order tree level results , such as bispectrum , skewness , cumulant correlators , three-point function are exactly reproduced in the appropriate expansion of our results .
We explicitly show that cumulants up to N = 5 predicted by Eulerian perturbation theory for the dark matter field \delta are exactly recovered in the corresponding limit .
A logarithmic mapping of the field naturally arises in the Schrödinger context , which means that tree level perturbation theory translates into ( possibly incomplete ) loop corrections for the conventional perturbation theory .
We show that the first loop correction for the variance is \sigma ^ { 2 } = \sigma _ { L } ^ { 2 } + ( -1.14 + n ) \sigma _ { L } ^ { 4 } for a field with spectral index n .
This yields 1.86 and 0.86 for n = -3 , -2 respectively , and to be compared with the exact loop order corrections 1.82 , and 0.88 .
Thus our tree-level theory recovers the dominant part of first order loop corrections of the conventional theory , while including ( partial ) loop corrections to infinite order in terms of \delta .