The Large Scale Structure (LSS) of the Universe, that being the large scale distribution of dark and baryonic matter and of the halos and galaxies they form, promises to be a source of unprecedented amounts of information in the near future, with hundreds of millions of galaxies being mapped by various surveys.

Analytic models of LSS begin by treating an ensemble of dark matter particles as a fluid and solving this fluid's equations of motion using a power series in the density field, which is physically interpreted as a perturbative expansion around a linear solution describing idealised, large scale dynamics. Until the past decade, these perturbative methods took the form of what is called Standard Perturbation Theory (SPT) which extends its perturbative description of LSS down to all scales, even beyond those at which the fluid description of dark matter stops being an accurate model. While SPT could not describe the physics of these small scales, it was assumed that small scale effects would not affect the models ability to describe larger scale dynamics. However, this was shown to be incorrect when it was found that SPT was not a convergent perturbation theory and did not accurately describe observations, facts which were attributed to small scale dynamics causing feedback on larger scales which, due to being incompatible with SPT, are referred to as non-perturbative effects. This problem was solved with the development of the Effective Field Theory of Large Scale Structure (EFTofLSS), a variant of SPT which introduces a cutoff, a minimum scale below which the model makes no attempt at describing reality, and a series of terms which correct the large scale calculations for the effects of smaller scale, sub-cutoff physics; these terms are referred to as counterterms.

In this thesis we develop the EFTofLSS, specifically studying the one-loop bispectrum and one-loop trispectrum while testing the commonly used EdS approximation, that being the simplified version of SPT which treates higher order density fields as though we lived in a matter dominated universe for mathematical simplicity. We show that when studying higher order correlators, it is important to use the full $\Lambda$CDM growth factors for the dark matter density fields. We also use the method of perturbation theory on the grid to constrain the EFTofLSS counterterms for the first time, leading to drastic improvements in the precision of our results when compared to previous studies which rely upon more conventional methodologies. We also show that it is important to account for the effects of finite time stepping and the rounding of numbers between time steps in the simulations, such that our model contains corrective terms for simulation imprecisions.

Finally, we discuss how the research presented in this thesis will impact near future LSS surveys and propose a number of possible future projects which could build upon this research.