Structure formation and quasispherical collapse from initial curvature perturbations with numerical relativity simulations

We use numerical relativity simulations to describe the spacetime evolution during nonlinear structure formation in Λ CDM cosmology. Fully nonlinear initial conditions are set at an initial redshift z ≈300 , based directly on the gauge invariant comoving curvature perturbation R_c commonly used to model early-universe fluctuations. Assigning a simple 3-D sinusoidal structure to R_c, we then have a lattice of quasispherical overdensities representing idealized 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 overdensities. 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 δ_C^{(1 )}=1.69 . Additionally, we characterize 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 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 the different regions of the simulation box 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 in each region and at different times. Along the filaments, the leading Petrov type is D, while the center of the overdensities remains conformally flat, type O, in line with the top-hat model. The surrounding region demonstrates a sort of peeling-off in action, with the spacetime transitioning between different Petrov types as nonlinearity grows, with production of gravitational waves.