标题： 循环宇宙与无限过去 显示英文标题

标题： Cyclic Universe and Infinite Past

摘要： 我们针对无限循环宇宙学的两个关于过去的问题展开讨论。 第一个问题是，在Borde-Guth-Vilenkin（BGV）“不可能性”定理的背景下，它是否可以包含一条过去的无穷长度的零测地线。第二个问题是，考虑到只有极小比例的新生宇宙能够循环，是否存在在无限时间后成功宇宙的足够概率。 我们对这两个问题给出了肯定的回答，随后证明在无限循环中，总宇宙数在任意长的时间内已经是无限的。 显示英文摘要

摘要： We address two questions about the past for infinitely cyclic cosmology. The first is whether it can contain an infinite length null geodesic into the past in view of the Borde-Guth-Vilenkin (BGV) "no-go" theorem, The second is whether, given that a small fraction of spawned universes fail to cycle, there is an adequate probability for a successful universe after an infinite time. We give positive answers to both questions then show that in infinite cyclicity the total number of universes has been infinite for an arbitrarily long time.