论文标题
多尺度的方法来解决宇宙学的平均问题
Multiple-Scales Approach to The Averaging Problem in Cosmology
论文作者
论文摘要
宇宙在大尺度上是均匀的和各向同性的，因此，在这些尺度上，它通常被建模为Friedmann-Lema-Roberton-Robertson-Walker（FLRW）时空。爱因斯坦磁场方程的非线性使人们对小规模偏差的平均值和各向同性的平均性产生了关注，这可能对FLRW指标在宇宙中的适用性，即使在大规模上也可能影响。在这里，我提出了一种基于多个奇异扰动理论的多尺度方法，以始终如一地处理小规模的不均匀性。我为大规模时空度量标准获得了领先的有效爱因斯坦方程，其中包含一个反应项。该派生依赖于一系列一致性条件，以确保与大规模时空度量的偏差的增长不会毫无根据。讨论了其满意度的标准，这表明如果物质在小规模上是不相关的，他们确实会感到满足。分析以谐波计进行，并讨论转换为其他仪表。我估计了在NFW Halo的示例中相对于宇宙临界密度的后反应项的大小，并发现其占数百分之几的顺序。在此示例中，后反应项被解释为在小规模上平均为总能量量量的重力势能的能量密度的贡献。
The Universe is homogeneous and isotropic on large scales, so on those scales it is usually modelled as a Friedmann-Lemaître-Robertson-Walker (FLRW) space-time. The non-linearity of the Einstein field equations raises concern over averaging over small-scale deviations form homogeneity and isotropy, with possible implications on the applicability of the FLRW metric to the Universe, even on large scales. Here I present a technique, based on the multiple-scales method of singular perturbation theory, to handle the small-scale inhomogeneities consistently. I obtain a leading order effective Einstein equation for the large-scale space-time metric, which contains a back-reaction term. The derivation relies on a series of consistency conditions, that ensure that the growth of deviations from the large-scale space-time metric do not grow unboundedly; criteria for their satisfiability are discussed, and it is shown that they are indeed satisfied if matter is non-relativistic on small scales. The analysis is performed in harmonic gauge, and conversion to other gauges is discussed. I estimate the magnitude of the back-reaction term relative to the critical density of the Universe in the example of an NFW halo, and find it to be of the order of a few percent. In this example, the back-reaction term is interpreted as a contribution of the energy-density of gravitational potential energy, averaged over the small-scale, to the total energy-momentum tensor.
