The separate-universe approach gives an intuitive way to understand the evolution of cosmological perturbations in the long-wavelength limit. It uses solutions of the spatially-homogeneous equations of motion to model the evolution of the inhomogeneous universe on large scales. We show that the separate-universe approach fails on a finite range of super-Hubble scales at a sudden transition from slow roll to ultra-slow roll during inflation in the very early universe. Such transitions are a feature of inflation models giving a large enhancement in the primordial power spectrum on small scales, necessary to produce primordial black holes after inflation. We show that the separate-universe approach still works in a piece-wise fashion, before and after the transition, but spatial gradients on finite scales require a discontinuity in the homogeneous solution at the transition. We discuss the implications for the δN formalism and stochastic inflation, which employ the separate-universe approximation.

中文翻译：

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分离宇宙方法和暴胀期间的突然转变

分离宇宙方法提供了一种直观的方式来理解长波长极限下宇宙扰动的演化。它使用空间均匀运动方程的解来模拟大尺度非均匀宇宙的演化。我们证明，在极早期宇宙暴胀过程中，在从慢速滚动到超慢速滚动的突然转变时，分离宇宙方法在有限范围的超哈勃尺度上会失败。这种转变是暴胀模型的一个特征，它在小尺度上大大增强了原始功率谱，这是暴胀后产生原始黑洞所必需的。我们证明，在转变之前和之后，分离宇宙方法仍然以分段方式工作，但有限尺度上的空间梯度需要转变时齐次解的不连续性。我们讨论对 δ 的影响氮形式主义和随机膨胀，它们采用分离宇宙近似。