[论文解读] The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities

Nima Arkani–Hamed, Daniel Baumann+2 more|arXiv (Cornell University)|Oct 31, 2018

Cosmology and Gravitation Theories被引用 6

一句话总结

本文发展了宇宙学bootstrap程序，从de Sitter空间中的对称性和奇点推导出暴胀关联函数。通过利用近似共形对称性和解析性约束，该研究通过树图交换精确推导出共形耦合标量的四点函数，其中自旋提升算符和权重转移算符分别对应于有自旋和无质量场，并通过微扰得到暴胀三点头函数，重现了经典结果并分类了所有弱对称性破缺的共形关联函数。

ABSTRACT

Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and particle content. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlators are controlled by an approximate conformal symmetry on the boundary of the spacetime. After classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities completely fixes this correlator. A spin-raising operator relates it to the correlators associated with the exchange of particles with spin, while weight-shifting operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. We reproduce many classic results in the literature and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. The inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta.

研究动机与目标

• 开发一种基于对称性的框架，无需显式时间演化即可计算暴胀关联函数。
• 利用共形对称性和解析性，对de Sitter空间中所有共形耦合标量的四点函数进行分类。
• 通过自旋提升算符和权重转移算符，将形式拓展至有自旋粒子和无质量外部场。
• 通过对de Sitter关联函数进行微扰，获得暴胀三点头函数，并重现如慢滚三谱等已知结果。
• 证明通过外部动量的解析延拓，暴胀关联函数中包含平坦空间散射振幅。

提出的方法

• 系统地利用共形对称性，对de Sitter空间中的接触相互作用和树图交换图进行分类。
• 通过共形对称性导出的微分方程约束，推导出共形耦合标量的四点函数，其交换为大质量标量。
• 实施自旋提升算符，从标量交换解生成有自旋粒子交换的关联函数。
• 应用权重转移算符，将关联函数从Δ=2映射到Δ=3，并进一步映射到无质量外部场。
• 利用慢滚修正，对de Sitter四点函数进行微扰，以获得暴胀三点头函数。
• 通过外部动量的解析延拓，将形式与平坦空间振幅联系起来。

实验结果

研究问题

• RQ1 如何在不显式使用体时间演化的情况下，从对称性和奇点推导出暴胀关联函数？
• RQ2 近似共形对称性对de Sitter四点函数施加了哪些约束？
• RQ3 如何利用自旋提升算符和权重转移算符，从标量交换生成有自旋和无质量粒子的关联函数？
• RQ4 de Sitter关联函数的慢滚修正如何重现暴胀三谱？
• RQ5 通过外部动量的解析延拓，平坦空间散射振幅如何编码于暴胀关联函数中？

主要发现

• de Sitter空间中共形耦合标量的四点函数完全由共形对称性和无物理奇点决定，从而得到树图交换的解析表达式。
• 自旋提升算符将标量交换四点函数映射到涉及任意自旋粒子的关联函数，完全刻画了有自旋粒子交换的暴胀三谱。
• 权重转移算符将Δ=2四点函数与Δ=3情形关联，实现了对无质量外部场关联函数的完整分类。
• 任意自旋交换的暴胀三谱完全由标量交换四点函数的软极限和一系列接触项确定。
• 该形式重现了经典的慢滚三谱，并对弱对称性破缺共形对称性所导致的所有三和四点函数进行了分类。
• 通过外部动量的解析延拓，暴胀关联函数中包含平坦空间散射振幅，粒子产生信号在压缩极限中清晰可见。