On super-exponential inflation in a higher-dimensional theory of gravity with higher-derivative terms

Abstract

We consider super-exponential inflation in the early universe, for which Image, with particular reference to the higher-dimensional theory of Shafi and Wetterich, which is discussed in further detail. The Hubble parameter H is given by H2 ≈ (8π/3mp2)V(φ), where the ‘inflaton’ field φ is related to the radius of the internal space, and obeys the equation of motion Image. The spectrum of density perturbations is given by δvarrho/varrho = (M/M0)−s, where s−1 ≈ 3(q + 1); and X ≡ (−∂V/∂φ)/(∂W/∂φ). The parameters q and X are both positive constants, hence the need for two distinct potentials, which can be met in a higher-dimensional theory with higher-derivative terms Image. Some fine-tuning of the parameters Image and/or of the cosmological constant Image is always necessary in order to have super-exponential inflation. It is possible to obtain a spectrum of density perturbations with Image, which helps to give agreement with observations of the cosmic microwave background radiation at very large scales not, vert, similar 1000 Mpc. When Image is proportional to the Euler number density, making the four-dimensional theory free of ghosts, then super-exponential inflation is impossible, but a phase of inflation with Image can still occur.