等熵宇宙学
Isochronous Cosmologies
摘要：
最近已经证明了制造（非相对论性，哈密顿）多体问题的可能性，其特征在于具有任意指定周期$T$的等熵时间演化，但具有良好的近似性，甚至精确地模仿任何给定的多体问题（在一个大的，物理相关的类中）随时间的$\tilde{T}$，其也可以是任意大的（但当然是$\tilde{T}<T$）。本文的目的和范围是探讨将这一发现扩展到广义相对论背景下的可能性。为了简单起见，我们限制我们的考虑均匀和各向同性度量的情况下，并表明，通过类似于用于非相对论性多体问题的方法，一类均匀和各向同性的爱因斯坦方程的循环解决方案可能会得到。对于这些解，周期的持续时间不依赖于初始条件，所以我们称这些模型为等熵宇宙学。我们给出了这种度规的物理解释，特别是我们证明了它们可以在任意长的时间内任意接近甚至等同于爱因斯坦方程的Friedman-Robertson-Walker解（当然比它们的周期短，也可以任意分配），这样它们就可以在其周期的一部分中再现标准宇宙学$Λ$ -CDM模型的所有令人满意的唯象特征;而这些等熵宇宙学可能是测地线完备的，因此是无奇点的。
英文摘要：
The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period $T$ yet mimic with good approximation, or even exactly, any given many-body problem (within a large, physically relevant, class) over times $\tilde{T}$ which may also be arbitrarily large (but of course such that $\tilde{T}<T$). Purpose and scope of this paper is to explore the possibility to extend this finding to a general relativity context. For simplicity we restrict our consideration to the case of homogeneous and isotropic metrics and show that, via an approach analogous to that used for the nonrelativistic many-body problem, a class of homogeneous and isotropic cyclic solutions of Einstein's equations may be obtained. For these solutions the duration of the cycles does not depend on the initial conditions, so we call these models isochronous cosmologies. We give a physical interpretation of such metrics and in particular we show that they may behave arbitrarily closely, or even identically, to the Friedman-Robertson-Walker solutions of Einstein's equations for an arbitrarily long time (of course shorter than their period, which can also be assigned arbitrarily), so that they may reproduce all the satisfactory phenomenological features of the standard cosmological $Λ$-CDM model in a portion of their cycle; while these isochronous cosmologies may be geodesically complete and therefore singularity-free.
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