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

我们希望发展一种能够考虑宇宙大尺度结构（LSS）非线性形成中的相对论效应的数值格式。这将允许人们超越结构形成的标准知识，并测试新的模型，例如动态暗能量。为了区分不同的暗能量模型，LSS计算通常比简单的运动学测试（如超新星观测）更有希望。然而，暗能量在几乎所有模型中都是一个相对论源，要考虑到它对非线性聚类的影响，就必须超越牛顿的n体模拟。而不是一个成熟的数值GR方案，这将是技术上太具有挑战性，我们的目标是一个微扰处理，考虑到主要的相对论效应。引力势在所有对宇宙学感兴趣的情况下都是很小的，因此在一阶上处理度规扰动就足够了。然而，我们将保持它们的一阶导数，以及暗物质粒子的速度，高达二阶，以及它们的二阶导数（曲率），与能量密度的波动具有相同的阶数，高达所有阶数。用这样的微扰处理，我们将在时空晶格上对爱因斯坦和刘维尔方程的耦合系统进行数值演化。用相空间中的分布函数来描述暗物质粒子。它的动量依赖在每个网格点将离散使用勒让德多项式的截断层次。离散化的选择使得它适合于应用相对论晶格玻尔兹曼格式，对于这种格式已经存在有效的算法。

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.