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

我们希望开发一种能够在宇宙大尺度结构(LSS)的非线性形成中考虑相对论效应的数值方案。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计算往往比像超新星观测这样的简单运动学测试更有前途。然而，暗能量在几乎所有模型中都是相对论来源，要考虑其对非线性聚集的影响，人们必须超越牛顿N体模拟。我们的目标是考虑领先的相对论效应的微扰处理，而不是成熟的数值GR方案，这在技术上太具挑战性。在所有宇宙学感兴趣的情况下，引力势仍然很小，因此，它足以处理一阶度规扰动。然而，我们将保持它们的一阶导数，以及暗物质粒子的速度，直到二阶，以及它们的二阶导数(曲率)，与能量密度的涨落相同的阶，直到所有阶。在这种微扰处理下，我们将在时空格子上数值地发展爱因斯坦的S方程和刘维尔的S方程的耦合系统。暗物质粒子将由它们在相空间中的分布函数来描述。它在每个网格点的动量依赖关系将使用截断的勒让德多项式离散。离散化的选择使得它适合于应用相对论格子Boltzmann格式，对于该格式，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.