来自 S-Matrix Marathon 的记录：膨胀宇宙中的可观测量
Records from the S-Matrix Marathon: Observables in Expanding Universes
AI总结：
本文回顾了膨胀宇宙中可观测量数学结构的最新进展，重点分析了 Bunch--Davies 波泛函的解析性质与积分表示，并利用宇宙学多面体的组合学解释了红外发散问题。
摘要：
膨胀宇宙中的可观测量对于理解早期宇宙的物理学至关重要。在这些讲座中，我们回顾了理解其数学结构的一些最新进展，并提取了其中编码的物理学。在讨论了膨胀背景最显著的特征及其对定义可观测量的影响之后，我们重点讨论了所谓的 Bunch--Davies 波泛函。我们在一般基础上分析了其解析性质，并在微扰理论中为特定类别的标量玩具模型引入了它的积分表示。我们讨论了与通常的 Feynman 规则相关的图解法以及图上的组合规则，这些规则生成了没有虚假极点的表示。这些组合规则起源于宇宙学多面体的组合学，我们对其定义和主要特征提供了通俗易懂的介绍。最后，宇宙学多面体的组合学决定了特定类别的嵌套面体的组合学，这些嵌套面体编码了宇宙学积分的渐近行为。我们提供了此类结构和行为的一般描述，这对于理解困扰膨胀背景中可观测量的红外发散至关重要。这些笔记基于 2024 年 3 月 11 日至 22 日在 Institute for Advanced Study 举行的 S-Matrix Marathon 研讨会上的一系列讲座。
英文摘要：
Observables in expanding universes are crucial to understand the physics of the early universe. In these lectures, we review some recent progress in understanding their mathematical structure and extract the physics encoded in them. After discussing the most salient features of an expanding background and their consequences for defining an observable, we focus on the so-called Bunch--Davies wavefunctional. We analyze its analytic properties on general grounds and introduce an integral representation for it in perturbation theory for a special class of scalar toy models. We discuss both the diagrammatics associated to the usual Feynman rules and combinatorial rules on the graphs, which generate a representation free of spurious poles. Such combinatorial rules find their origin in the combinatorics of the cosmological polytopes of which we provide a gentle introduction to its definition and its main features. Finally, the combinatorics of the cosmological polytopes turns out to determine the combinatorics of a special class of nestohedra that encode the asymptotic behaviour of the cosmological integrals. We provide a general description of such structures and behaviour, which is of crucial importance to understand the infrared divergences which plague observables in an expanding background. These notes are based on a series of lectures held during the S-Matrix Marathon workshop at the Institute for Advanced Study on 11--22 March 2024.
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