In minimally coupled scalar field theories with a potential of the slow-roll type, we give a detailed description of the complex O(4)-symmetric solutions to Einstein’s equations on the four-ball which contribute to the no-boundary amplitude Ψ_NB(b, χ) for a closed universe to contain a round three-sphere spatial slice of size b covered homogeneously with the scalar field at value χ. Our derivation demonstrates a result anticipated by in Hartle etal (2008) Phys. Rev. D 77 123537, sharpens result in Vilenkin’s (1988) Phys. Rev. D 37 888 in the classical regime of the minisuperspace and makes use of a complexified slow-roll approximation. Our technique applies to both the Hartle–Hawking and Vilenkin wave functions, which both predict a family of inflationary universes but weight each member exponentially differently in the semiclassical approximation.

在具有慢滚型势能的最小耦合标量场理论中，我们详细描述了爱因斯坦方程在四球上的复杂O (4)-对称解，这有助于无边界振幅 Ψ _NB ( b , χ ) 用于包含大小为b的圆形三球体空间切片的封闭宇宙，该切片被值χ的标量场均匀覆盖。我们的推导证明了 Hartle等人(2008) Phys 中 预期的结果。Rev. D 77 123537，锐化了 Vilenkin (1988) Phys. 的 结果。修订版 D 37888 在迷你超空间的经典范围内，并利用复杂的慢滚近似。我们的技术适用于 Hartle-Hawking 和 Vilenkin 波函数，它们都预测了一系列暴胀宇宙，但在半经典近似中对每个成员的权重呈指数不同。