球面拓扑中的引力势
Gravitational potential in spherical topologies
AI总结：
研究球面拓扑宇宙中牛顿引力势的性质，发现非零空间曲率引入额外一阶吸引项，且曲率效应总体主导拓扑效应，并提供了用于结构形成N体模拟的方程。
摘要：
我们研究了不同拓扑下球面宇宙中牛顿引力势的性质。为此，我们使用了 Vigneron [2022, Class. & Quantum Gravity, 39, 155006] 提出的非欧几里得牛顿理论，该理论描述了球面或双曲宇宙中的牛顿引力。我们计算了所有全局均匀规则球面拓扑（即其基本域唯一且为柏拉图立体）中点质量的引力势。我们提供了点质量附近测试位置处引力势的精确解和泰勒展开级数。我们表明，展开的奇数项可解释为源于非零空间标量曲率的存在，而偶数项则与拓扑空间的闭合性质有关。由此带来的一个结果是，与牛顿宇宙学模拟中广泛使用的 3-torus 中的点质量解相比，球面情形均包含一个仅依赖于空间曲率的额外一阶吸引项。拓扑的选择仅在二阶及更高阶影响引力势。对于宇宙学尺度（曲率和拓扑）的典型估计，最强的拓扑效应出现在 Poincaré dodecahedral space 的情况下，但总体而言，曲率效应主导拓扑效应。我们还提供了一组方程，可用于在球面拓扑中执行结构形成的 $N$-body 模拟。
英文摘要：
We study the properties of the Newtonian gravitational potential in a spherical Universe for different topologies. For this, we use the non-Euclidean Newtonian theory developed in Vigneron [2022, Class. & Quantum Gravity, 39, 155006] describing Newtonian gravitation in a spherical or hyperbolic Universe. The potential is calculated for a point mass in all the globally homogeneous regular spherical topologies, i.e. whose fundamental domain is unique and is a platonic solid. We provide the exact solution and the Taylor expansion series of the potential at a test position near the point mass. We show that the odd terms of the expansion can be interpreted as coming from the presence of a non-zero spatial scalar curvature, while the even terms relate to the closed nature of the topological space. A consequence is that, compared to the point mass solution in a 3-torus, widely used in Newtonian cosmological simulations, the spherical cases all feature an additional attractive first order term dependent solely on the spatial curvature. The choice of topology only affects the potential at second order and higher. For typical estimates of cosmological scales (curvature and topology), the strongest topological effect occurs in the case of the Poincaré dodecahedral space, but in general the effect of curvature dominates over topology. We also provide the set of equations that can be used to perform $N$-body simulations of structure formation in spherical topologies.
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