Numerical Construction of initial data sets for inhomogeneous cosmological space-times with spatial topology of $\mathbb{T}^3$

In this work, we numerically construct initial data sets for cosmological space-times with a spatial topology of $\mathbb{T}^3$, which are generally inhomogeneous. To do so, we implement a pseudo-spectral approach based on the discrete Fourier transform for numerically solving Einstein's constraint equations in an algebraic-hyperbolic form. We explore the advantages and disadvantages of this method by comparing the numerical solutions with known analytical initial data sets. Additionally, we perform an stability analysis of the system to gain deeper understanding of the problem. Finally, we numerically obtain new families of initial data sets through manipulation of the original system by imposing restrictions in some of the variables.