Numerical simulation of cosmic structure formation including relativistic effects at leading order using a lattice Boltzmann scheme

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We want to develop a numerical scheme which is capable of taking into account relativistic effects in the nonlinear formation of cosmic large scale structures (LSS). This will allow one to go beyond the standard lore of structure formation and to test new models like e.g. dynamical dark energy. In order to distinguish different dark energy models, LSS calculations are very often more promising than simple kinematical tests like the supernova observations. However, dark energy is a relativistic source in almost all models and to take into account its effect on non-linear clustering one has to go beyond Newtonian N-body simulations.Instead of a full-fledged numerical GR scheme which would be technically too challenging, we are aiming for a perturbative treatment which takes into account the leading relativistic effects. The gravitational potential remains small in all situations of cosmological interest, and therefore it will be sufficient to treat metric perturbations at first order. However, we will keep their first derivatives, as well as velocities of dark matter particles, up to second order,and their second derivatives (curvature), which are of the same order as the fluctuations in the energy density, up to all orders. With such a perturbative treatment, we will evolve the coupled system of Einstein´s and Liouville´s equations numerically on a spacetime lattice. Dark matter particles will be described by their distribution function in phase space. Its momentum dependence at each grid point will be discretized using a truncated hierarchy of Legendre polynomials. The discretization is chosen such that it is suitable for applying a relativistic lattice Boltzmann scheme, for which efficient algorithms do exist already.