A Systematic Scaling Solution Search in Six-Dimensional Inflation

Abstract

We explore the mechanics of inflation within simplified extra-dimensional models involving an inflaton interacting with a Einstein-Maxwell system in two extra dimensions. The models are complicated enough to include a stabilization mechanism for the extra-dimensional radius, but simple enough to solve the full six-dimensional field equations. After performing a consistent truncation, which guarantees our six-dimensional equations are equivalently satisfied by the four dimensional equations of motion, we explore (numerically and analytically) the power-law solutions evident in our initial parameter search. After a comprehensive search for potential power-law scaling solutions in both six and four dimensions, we find two that give rise to interesting inflationary dynamics. They both can generically exist outside of the usual four dimensional effective theory, and yet, we still trust them since our truncation is consistent. One of these is a dynamical 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 transitioning to another solution due to its unstable nature. We present a numerical and analytic discussion of these two solutions. Four of the appendices contain calculations in more explicit detail than are performed in the main text, while a fifth contains a representative Mathematica worksheet and the sixth contains the general results of the systematic sweep for scaling solutions.