We investigate effective equations governing the volume expansion of spatially averaged portions of inhomogeneous cosmologies in spacetimes filled with an arbitrary fluid. This work is a follow-up to previous studies focused on irrotational dust models (Paper I) and irrotational perfect fluids (Paper II) in flow-orthogonal foliations of spacetime. It complements them by considering arbitrary foliations, arbitrary lapse and shift, and by allowing for a tilted fluid flow with vorticity. As for the first studies, the propagation of the spatial averaging domain is chosen to follow the congruence of the fluid, which avoids unphysical dependencies in the averaged system that is obtained. We present two different averaging schemes and corresponding systems of averaged evolution equations providing generalizations of Papers I and II. The first one retains the averaging operator used in several other generalizations found in the literature. We extensively discuss relations to these formalisms and pinpoint limitations, in particular regarding rest mass conservation on the averaging domain. The alternative averaging scheme that we subsequently introduce follows the spirit of Papers I and II and focuses on the fluid flow and the associated $$1+3$$ 1 + 3 threading congruence, used jointly with the $$3+1$$ 3 + 1 foliation that builds the surfaces of averaging. This results in compact averaged equations with a minimal number of cosmological backreaction terms. We highlight that this system becomes especially transparent when applied to a natural class of foliations which have constant fluid proper time slices.

中文翻译：

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广义相对论中非均匀流体的平均性质 III：广义流体宇宙学

我们研究了在充满任意流体的时空中控制非均匀宇宙学的空间平均部分的体积膨胀的有效方程。这项工作是对先前研究的后续研究，这些研究侧重于时空流动正交叶理中的无旋尘埃模型（论文 I）和无旋完美流体（论文 II）。它通过考虑任意叶理、任意流逝和偏移以及允许具有涡度的倾斜流体流动来补充它们。对于第一项研究，选择空间平均域的传播以遵循流体的一致性，这避免了所获得的平均系统中的非物​​理依赖性。我们提出了两种不同的平均方案和相应的平均演化方程系统，提供了论文 I 和 II 的概括。第一个保留了在文献中发现的其他几个概括中使用的平均算子。我们广泛讨论了与这些形式主义的关系并指出了局限性，特别是关于平均域上的静止质量守恒。我们随后介绍的替代平均方案遵循论文 I 和 II 的精神，侧重于流体流动和相关的 $$1+3$$ 1 + 3 线程同余，与 $3+1$$ 3 + 1 联合使用叶理，建立平均的表面。这导致具有最少数量的宇宙学反向反应项的紧凑平均方程。我们强调，当应用于具有恒定流体适当时间切片的自然叶理类别时，该系统变得特别透明。