Lagrangian perturbations and the matter bispectrum I: fourth-order model
  for non-linear clustering
  We investigate the Lagrangian perturbation theory of a homogeneous and
isotropic universe in the non-relativistic limit, and derive the solutions up
to the fourth order. These solutions are needed for example for the
next-to-leading order correction of the (resummed) Lagrangian matter
bispectrum, which we study in an accompanying paper. We focus on flat
cosmologies with a vanishing cosmological constant, and provide an in-depth
description of two complementary approaches used in the current literature.
Both approaches are solved with two different sets of initial conditions---both
appropriate for modelling the large-scale structure. Afterwards we consider
only the fastest growing mode solution, which is not affected by either of
these choices of initial conditions. Under the reasonable approximation that
the linear density contrast is evaluated at the initial Lagrangian position of
the fluid particle, we obtain the nth-order displacement field in the so-called
initial position limit: the nth order displacement field consists of 3(n-1)
integrals over n linear density contrasts, and obeys self-similarity. Then, we
find exact relations between the series in Lagrangian and Eulerian perturbation
theory, leading to identical predictions for the density contrast and the
peculiar-velocity divergence up to the fourth order.
