In this paper, we investigate the entropy of the free gravitational field for a given epoch for some well-known isotropic and anisotropic cosmologies. We use the definition of gravitational entropy proposed by Clifton, Ellis and Tavakol, where the 2-index square root of the 4-index Bel–Robinson tensor is taken to be the energy– momentum tensor for the free gravity. We examine whether in the vicinity of the initial singularity, the ratio of energy density of free gravity to that of matter density goes to zero, validating Penrose conjecture on Weyl curvature. Whenever this is true, the gravitational entropy increases monotonically with time, leading to structure formation. For the models considered by us, we identify the conditions for which the Weyl curvature hypothesis is valid, and the assumptions under which it is validated, or otherwise.

中文翻译：

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宇宙模型引力熵的研究

在本文中，我们研究了一些著名的各向同性和各向异性宇宙学在给定时期的自由引力场的熵。我们使用 Clifton、Ellis 和 Tavakol 提出的引力熵定义，其中 4 指数 Bel-Robinson 张量的 2 指数平方根被视为自由引力的能量-动量张量。我们检查了在初始奇点附近，自由重力能量密度与物质密度能量密度之比是否为零，验证了彭罗斯关于外尔曲率的猜想。只要这是真的，引力熵就会随着时间单调增加，导致结构形成。对于我们考虑的模型，我们确定了外尔曲率假设有效的条件，以及验证它的假设，或其他。