Newtonian and relativistic cosmologies
Newtonian and relativistic cosmologies
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DOI：
10.1103/physrevd.85.063512
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发表时间：
2011-11
影响因子：
5
通讯作者：
Stephen R. Green;R. Wald
中科院分区：
文献类型：
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作者：
Stephen R. Green;R. Wald
现在正在使用牛顿引力在大于哈勃半径的尺度上进行宇宙学 N 体模拟。众所周知，牛顿引力中的均匀膨胀、均匀的尘埃球满足与相对论 FLRW 宇宙学中出现的相同方程，而且还知道，牛顿宇宙学和相对论尘埃宇宙学之间的对应关系在边缘束缚/空间平坦情况下的线性微扰理论中继续保持。然而，当小尺度上存在显着的非线性动力学行为时，牛顿引力能否为非均匀宇宙学提供良好的全局描述还远非显而易见。我们根据我们最近开发的微扰框架来研究这个问题，该框架允许在小尺度上出现这种非线性。我们提出了一个相对简单的“字典”——在线性化水平上是精确的——将牛顿尘埃宇宙学映射到广义相对论尘埃宇宙学，并且我们使用我们的“排序方案”来确定所得到的度量和物质分布求解爱因斯坦方程的程度。我们发现爱因斯坦方程在小尺度上不能保持“1 阶”，在大尺度上不能保持“$\epsilon$ 阶”。然后，我们找到满足这些阶数的爱因斯坦方程所需的度量和物质分布的额外修正。虽然这些更正本身就很有趣，但我们计算它们的主要目的是，它们的小量应该为原始词典（以及该词典的简化版本）的有效性提供一个标准。我们预计，在现实的牛顿宇宙学中，这些额外的修正将非常小；如果是这样，这将为使用牛顿模拟来描述相对论宇宙学提供强有力的理由，即使是在比哈勃半径更大的尺度上。
Cosmological N-body simulations are now being performed using Newtonian gravity on scales larger than the Hubble radius. It is well known that a uniformly expanding, homogeneous ball of dust in Newtonian gravity satisfies the same equations as arise in relativistic FLRW cosmology, and it also is known that a correspondence between Newtonian and relativistic dust cosmologies continues to hold in linearized perturbation theory in the marginally bound/spatially flat case. Nevertheless, it is far from obvious that Newtonian gravity can provide a good global description of an inhomogeneous cosmology when there is significant nonlinear dynamical behavior at small scales. We investigate this issue in the light of a perturbative framework that we have recently developed, which allows for such nonlinearity at small scales. We propose a relatively straightforward "dictionary"---which is exact at the linearized level---that maps Newtonian dust cosmologies into general relativistic dust cosmologies, and we use our "ordering scheme" to determine the degree to which the resulting metric and matter distribution solve Einstein's equation. We find that Einstein's equation fails to hold at "order 1" at small scales and at "order $\epsilon$" at large scales. We then find the additional corrections to the metric and matter distribution needed to satisfy Einstein's equation to these orders. While these corrections are of some interest in their own right, our main purpose in calculating them is that their smallness should provide a criterion for the validity of the original dictionary (as well as simplified versions of this dictionary). We expect that, in realistic Newtonian cosmologies, these additional corrections will be very small; if so, this should provide strong justification for the use of Newtonian simulations to describe relativistic cosmologies, even on scales larger than the Hubble radius.