标题： 闭合宇宙之间的几何距离 显示英文标题

标题： Geometric distances between closed universes

摘要： 宇宙的大尺度结构由弗里德曼方程很好地近似，这些方程由若干能量密度参数化，这些能量密度可以被观测推断出来。一个自然的问题是：如果这些密度取其他值，宇宙会有什么不同？虽然根据解释和数学严谨性，有多种方法可以解决这个问题，但我们尝试通过构建一个“历史空间”来回答，这个空间包含不同的宇宙学模型。在克服与洛伦兹符号和无限体积相关的技术难题后，至少对于拓扑闭合的情况，引入了一个度量。计算了超空间上两点之间的测地线，以表达两个宇宙在纯粹几何意义上的距离。在这种方法中，年龄可以被视为一个自由参数，相对于几何动力学配置空间，这导致了一个更一般的数学构造。还探讨了与泡膨胀和相变的联系。 显示英文摘要

摘要： The large-scale structure of the Universe is well approximated by the Friedmann equations, parametrized by several energy densities which can be observationally inferred. A natural question to ask is: How different would the Universe be if these densities took on other values? While there are many ways this can be approached depending on interpretation and mathematical rigor, we attempt an answer by building a "history space" of different cosmologies. A metric is introduced after overcoming technical hurdles related to Lorentzian signature and infinite volume, at least for topologically closed cases. Geodesics connecting two points on the superspace are computed to express how distant--in a purely geometric sense--two universes are. Age can be treated as a free parameter in such an approach, leading to a more general mathematical construct relative to geometrodynamical configuration space. Connections to bubble inflation and phase transitions are explored.