Structure formation and quasi-spherical collapse from initial curvature perturbations with numerical relativity simulations

We use numerical relativity simulations to describe the spacetime evolution during nonlinear structure formation in $\Lambda$CDM cosmology. Fully nonlinear initial conditions are set at an initial redshift $z\approx 300$, based directly on the gauge invariant comoving curvature perturbation $\mathcal{R}_c$ commonly used to model early-universe fluctuations. Assigning a simple 3-D sinusoidal structure to $\mathcal{R}_c$, we then have a lattice of quasi-spherical over-densities representing idealised dark matter halos connected through filaments and surrounded by voids. This structure is implemented in the synchronous-comoving gauge, using a pressureless perfect fluid (dust) description of CDM, and then it is fully evolved with the Einstein Toolkit code. With this, we look into whether the Top-Hat spherical and homogeneous collapse model provides a good description of the collapse of over-densities. We find that the Top-Hat is an excellent approximation for the evolution of peaks, where we observe that the shear is negligible and collapse takes place when the linear density contrast reaches the predicted critical value $\delta^{(1)}_C =1.69$. Additionally, we characterise the outward expansion of the turn-around boundary and show how it depends on the initial distribution of matter, finding that it is faster in denser directions, incorporating more and more matter in the infalling region. Using the EBWeyl code [1] we look at the distribution of the electric and magnetic parts of the Weyl tensor, finding that they are stronger along and around the filaments, respectively. We introduce a method to dynamically classify different regions in Petrov types. With this, we find that the spacetime is of Petrov type I everywhere, as expected, but we can identify the leading order type, finding a transition between different types as non-linearity grows, with production of gravitational waves.