  We introduce a novel approach, the Cosmological Trajectories Method (CTM), to
model nonlinear structure formation in the Universe by expanding
gravitationally-induced particle trajectories around the Zel'dovich
approximation. A new Beyond Zel'dovich approximation is presented, which
expands the CTM to leading second-order in the gravitational interaction and
allows for post-Born gravitational scattering. In the Beyond Zel'dovich
approximation we derive the exact expression for the matter clustering power
spectrum. This is calculated to leading order and is available in the CTM
MODULE. We compare the Beyond Zel'dovich approximation power spectrum and
correlation function to other methods including 1-loop Standard Perturbation
Theory (SPT), 1-loop Lagrangian Perturbation Theory (LPT) and Convolution
Lagrangian Perturbation Theory (CLPT). We find that the Beyond Zel'dovich
approximation power spectrum performs well, matching simulations to within
$\pm{10}\%$, on mildly non-linear scales, and at redshifts above $z=1$ it
outperforms the Zel'dovich approximation. We also find that the Beyond
Zel'dovich approximation models the BAO peak in the correlation function at
$z=0$ more accurately, to within $\pm{5}\%$ of simulations, than the Zel'dovich
approximation, SPT 1-loop and CLPT.
