We elaborate on the functional integral describing the stochastic dynamics of a spectator field during inflation, comparing its diagrammatic expansion to that obtained directly from a perturbative solution of the corresponding Langevin equation. We state Feynman rules for computing arbitrary temporal $n$-point functions and perform some illustrative computations for a $\lambda\phi^4$ interaction, paying attention to the role played by a functional Jacobian determinant in the path integral. We also briefly consider the case when the field contributes to the expansion rate, making the noise multiplicative, which introduces additional vertices.

中文翻译：

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随机通胀相关器的费曼规则

我们详细说明了描述膨胀期间观众场随机动力学的函数积分，将其图解扩展与直接从相应朗之万方程的微扰解获得的扩展进行比较。我们陈述了用于计算任意时间 $n$ 点函数的费曼规则，并对 $\lambda\phi^4$ 交互进行了一些说明性的计算，注意函数雅可比行列式在路径积分中的作用。我们还简要考虑了场对扩展率有贡献的情况，使噪声成倍增加，从而引入了额外的顶点。