标题： 关于暴胀时空的过去完备性 显示英文标题

标题： On the past-completeness of inflationary spacetimes

摘要： 我们讨论了在无限过去中，膨胀时空是否可以是测地线完备的这个问题。测地线完备性是避免永恒暴胀期间初始奇点的必要条件。通常认为，膨胀足够快（平均哈勃膨胀率 $H_{avg}>0$）的宇宙模型在类空和类时过去方向上必须不完备。这个广为人知的假说依赖于对过去指向的类时或类空测地线上哈勃参数积分的具体限制。如所述，我们表明这一说法是一个开放问题。我们表明， $H_{avg}$ 的计算结果对于给定的时空而言，取决于底层拓扑假设，会得到连续的结果。我们提出了 $H_{avg}$ 的改进定义，并引入了一大群不可数的宇宙学解，它们尽管具有 $H_{avg}>0$，但仍然是测地线完备的。我们讨论了膨胀时空的标准定义以及在物理上合理的标度因子上的量子（半经典）宇宙学问题。 显示英文摘要

摘要： We discuss the question of whether or not inflationary spacetimes can be geodesically complete in the infinite past. Geodesic completeness is a necessary condition for averting an initial singularity during eternal inflation. It is frequently argued that cosmological models which are expanding sufficiently fast (having average Hubble expansion rate $H_{avg}>0$) must be incomplete in null and timelike past directions. This well-known conjecture relies on specific bounds on the integral of the Hubble parameter over a past-directed timelike or null geodesic. As stated, we show this claim is an open issue. We show that the calculation of $H_{avg}$ yields a continuum of results for a given spacetime predicated upon the underlying topological assumptions. We present an improved definition for $H_{avg}$ and introduce an uncountably infinite cohort of cosmological solutions which are geodesically complete despite having $H_{avg}>0$. We discuss a standardized definition for inflationary spacetimes as well as quantum (semi-classical) cosmological concerns over physically reasonable scale factors.