Observational surveys which probe our universe deeper and deeper into the nonlinear regime of structure formation are becoming increasing accurate. This makes numerical simulations an essential tool for theory to be able to predict phenomena at comparable scales. In the first part of this thesis we study the behaviour of cosmological models involving a scalar field. We are particularly interested in the existence of fixed points of the dynamical system and the behaviour of the system in their vicinity. Upon addition of spatial curvature to the single-scalar field model with an exponential potential, canonical kinetic term, and a matter fluid, we demonstrate the existence of two extra fixed points that are not present in the case without curvature. We also analyse the evolution of the equation-of-state parameter. In the second part, we numerically simulate collisionless particles in the weak field approximation to General Relativity, with large gradients of the fields and relativistic velocities allowed. To reduce the complexity of the problem and enable high resolution simulations, we consider the spherically symmetric case. Comparing numerical solutions to the exact Schwarzschild and Lemaître-Tolman-Bondi solutions, we show that the scheme we use is more accurate than a Newtonian scheme, correctly reproducing the leading-order post-Newtonian behaviour. Furthermore, by introducing angular momentum, configurations corresponding to bound objects are found. In the final part, we simulate the conditions under which one would expect to form ultracompact minihalos, dark matter halos with a steep power-law profile. We show that an isolated object exhibits the profile predicted analytically. Embedding this halo in a perturbed environment we show that its profile becomes progressively more similar to the Navarro-Frenk-White profile with increasing amplitude of perturbations. Next, we boost the power spectrum at a very early redshift during radiation domination on a chosen scale and simulate clustering of dark matter particles at this scale until low redshift. In this scenario halos form earlier, have higher central densities, and are more compact.

In this work we explore three-point statistics applied to the large-scale structure in our Universe. Three-point statistics, such as the bispectrum, encode information not accessible via the standard analysis method-the power spectrum-and thus provide the potential for greatly improving current constraints on cosmological parameters. They also present us with additional challenges, and we focus on two of these arising from a measurement as well as modelling point of view. The first challenge we address is the covariance matrix of the bispectrum, as its precise estimate is required when performing likelihood analyses. Covariance matrices are usually estimated from a set of independent simulations, whose minimum number scales with the dimension of the covariance matrix. Because there are many more possibilities of finding triplets of galaxies than pairs, compared to the power spectrum this approach becomes rather prohibitive. With this motivation in mind, we explore a novel alternative to the bispectrum: the line correlation function (LCF). It specifically targets information in the phases of density modes that are invisible to the power spectrum, making it a potentially more efficient probe than the bispectrum, which measures a combination of amplitudes and phases. We derive the covariance properties and the impact of shot noise for the LCF and compare these theoretical predictions with measurements from N-body simulations. Based on a Fisher analysis we assess the LCF's sensitivity on cosmological parameters, finding that it is particularly suited for constraining galaxy bias parameters and the amplitude of fluctuations. As a next step we contrast the Fisher information of the LCF with the full bispectrum and two other recently proposed alternatives. We show that the LCF is unlikely to achieve a lossless compression of the bispectrum information, whereas a modal decomposition of the bispectrumcan reduce the size of the covariancematrix by at least an order of magnitude. The second challenge we consider in this work concerns the relation between the dark matter field and luminous tracers, such as galaxies. Accurate knowledge of this galaxy bias relation is required in order to reliably interpret the data gathered by galaxy surveys. On the largest scales the dark matter and galaxy densities are linearly related, but a variety of additional terms need to be taken into account when studying clustering on smaller scales. These have been fully included in recent power spectrumanalyses, whereas the bispectrummodel relied on simple prescriptions that were likely extended beyond their realm of validity. In addition, treating power spectrumand bispectrum on different footings means that the two models become inconsistent on small scales. We introduce a new formalism that allows us to elegantly compute the lacking bispectrum contributions from galaxy bias, without running into the renormalization problem. Furthermore, we fit our new model to simulated data by implementing these contributions into a likelihood code. We show that they are crucial in order to obtain results consistent with those fromthe power spectrum, and that the bispectrum retains its capability of significantly reducing uncertainties in measured parameters when combined with the power spectrum.