Understanding the Higgs potential at large field values corresponding to scales in the range above 10 10 GeV is important for questions of vacuum stability, particularly in the early universe where survival of the Higgs vacuum can be an issue. In this paper we show that the Higgs potential can be derived in away which is independent of the choice of conformal frame for the spacetime metric. Questions about vacuum stability can therefore be answered unambiguously. We show that frame independence leads to new relations between the beta functions of the theory and we give improved limits on the allowed values of the Higgs curvature coupling for stability.

We study a Hyperinflation model involving a field doublet on a hyperbolic field-space manifold and an exponential potential, providing a concise treatment of the evolution of the entropic and adiabatic perturbations around the homogeneous hyperbolic attractor solution. We find that the adiabatic spectral index narrows down the admissible values of the potential's slope to a very small region, severely restricting the state space of the allowed background solutions.

We elaborate on the functional integral describing 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 λϕ4 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.

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This final chapter provides details of worked solutions to the various problems set by the lecturers during the course of the school; some of these problems appear within the chapters of this book. This chapter also contains further exercises that were added after the school are not solved here; these are left as a challenge for the enterprising reader. Problems run the range of topics covered. These problems and solutions are associated with topics that include the introduction to EFT, renormalization theory, nuclear and atomic physics, Nambu–Goldstone modes, inflation, and large-scale structure, and how each topic relates to EFTs.