  We develop a path-integral formalism to study the formation of large-scale
structures in the universe. Starting from the equations of motion of
hydrodynamics (single-stream approximation) we derive the action which
describes the statistical properties of the density and velocity fields for
Gaussian initial conditions. Then, we present large-N expansions (associated
with a generalization to N fields or with a semi-classical expansion) of the
path-integral defined by this action. This provides a systematic expansion for
two-point functions such as the response function and the usual two-point
correlation. We present the results of two such expansions (and related
variants) at one-loop order for a SCDM and a LCDM cosmology. We find that the
response function exhibits fast oscillations in the non-linear regime with an
amplitude which either follows the linear prediction (for the direct
steepest-descent scheme) or decays (for the 2PI effective action scheme). On
the other hand, the correlation function agrees with the standard one-loop
result in the quasi-linear regime and remains well-behaved in the highly
non-linear regime. This suggests that these large-N expansions could provide a
good framework to study the dynamics of gravitational clustering in the
non-linear regime. Moreover, the use of various expansion schemes allows one to
estimate their range of validity without the need of N-body simulations and
could provide a better accuracy in the weakly non-linear regime.
