Structure formation beyond shell-crossing: Nonperturbative expansions and late-time attractors

Structure formation in 1+1 dimensions is considered, with emphasis on the effects of shell-crossing. The breakdown of the perturbative expansion beyond shell-crossing is discussed, and it is shown, in a simple example, that the perturbative series can be extended to a transseries including nonperturbative terms. The latter converges to the exact result well beyond the range of validity of perturbation theory. The crucial role of the divergences induced by shell-crossing is discussed. They provide constraints on the structure of the transseries and act as a bridge between the perturbative and the nonperturbative sectors. Then, we show that the dynamics in the deep multistreaming regime is governed by attractors. In the case of simple initial conditions, these attractors coincide with the asymptotic configurations of the adhesion model, but in general they may differ. These results are applied to a cosmological setting, and an algorithm to build the attractor solution starting from the Zel'dovich approximation is developed. Finally, this algorithm is applied to the search of 'haloes' and the results are compared with those obtained from the exact dynamical equations.