Inflationary dynamics in the braneworld scenarios

We analyze the attractor behaviour of the inflation field in braneworld scenarios using the Hamilton-Jacobi formalism, where the Friedmann equation has the form of H²=ρ+ε√2ρ₀ρ or H²=ρ+ερ²/2σ, with ε=±1. We find that in all models the linear homogeneous perturbation can decay exponentially as the scalar field rolls down its potential. However, in the case of a -ρ² correction to the standard cosmology with ρ < σ, the existence of an attractor solution requires (σ-ρ)/φ² > 1. Our results show that the perturbation decays more quickly in models with positive-energy correction than in the standard cosmology, which is opposite to the case of negative-energy correction. Thus, the positive-energy modification rather than the negative one can assist the inflation and widen the range of initial conditions.