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The Cosmological Trajectories Method: Modelling cosmic structure formation in the non-linear regime

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.