Initial data for general relativistic SPH with Centroidal Voronoi Tessellations

In this work we present an alternative method to obtain a distribution of particles over an hyper surface, such that they obey a rest-mass density distribution $\rho(x^i)$. We use density profiles that can be written as $\rho(x^1,x^2,x^3)=\rho(x^1) \rho(x^2) \rho(x^3)$ in order to be able to use them as a probability density functions. We can find the relation between the chart $x^j$ and a uniform random variable $\bar{x}^j \in (0,1)$, say $F(x^j)=\bar{x}^j$. Using the inverse of this function we relate a set of $N$ arbitrary number of points inside a cube with coordinates $\{ x^j =F^{-1}(\bar{x}^j)\}$ giving the position in order to get the density distribution $\rho(x^j)$. We get some noise due to the random distribution and we can notice that each time we relax the configuration on the cube we also get a better distribution of the desired physical configuration described with $\rho(x^j)$. This relaxation of the position of the particles in the cube has been performed a Lloyd's algorithm in 3D and we have used {\it Voro++} library in order to get the Voronoi tessellations.