Slow-roll approximation in quantum cosmology

Understand

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 $\Psi_\textsf{NB}(b,\chi)$ for a closed universe to contain a round three-sphere spatial slice of size $b$ covered homogeneously with the scalar field at value $\chi$.

• Our derivation demonstrates a result anticipated by Hartle, Hawking and Hertog in Phys.
• Rev.
• D 77 (2008) 123537 [arXiv:0803.1663], sharpens Vilenkin's result in Phys.