Goldilocks Models of Higher-Dimensional Inflation (including modulus stabilization)

Abstract

We explore the mechanics of inflation in simplified extra-dimensional models involving an inflaton interacting with the Einstein-Maxwell system in two extra dimensions. The models are Goldilocks-like in that they are just complicated enough to include a mechanism to stabilize the extra-dimensional size, yet simple enough to solve the full 6D field equations using basic tools. The solutions are not limited to the effective 4D regime with H << m_KK (the latter referring to the mass splitting of the Kaluza-Klein excitations) because the full 6D Einstein equations are solved. This allows an exploration of inflationary physics in a controlled regime away from the usual 4D lamp-post. The inclusion of modulus stabilization is important as experience with string models teaches that this is usually what makes models fail: stabilization energies dominate the shallow potentials required by slow roll and open up directions to evolve that are steeper than those of the putative inflationary direction. We explore three representative inflationary scenarios within this simple setup. In one the radion is trapped in an inflaton-dependent local minimum whose non-zero energy drives inflation. Inflation ends as this energy relaxes to zero when the inflaton finds its minimum. The others involve power-law solutions during inflation. One is an attractor whose features are relatively insensitive to initial conditions but whose slow-roll parameters cannot be arbitrarily small; the other is not an attractor but can roll much more slowly, until eventually decaying to the attractor. These solutions can satisfy H > m_KK, but when they do standard 4D fluctuation calculations need not apply. When in a 4D regime the solutions predict eta ~ 0 hence r ~ 0.11 when n_s ~ 0.96 and so are ruled out if tensor modes remain unseen. Analysis of general parameters is difficult without a full 6D fluctuation calculation.