Cosmological evolution of general scalar fields in a brane-world cosmology

We study the cosmology of a general scalar field and barotropic fluid during the early stage of a brane-world where the Friedmann constraint is dominated by the square of the energy density. Assuming both the scalar field and fluid are confined to the brane, we find a range of behavior depending on the form of the potential. Generalizing an approach developed for a standard Friedmann cosmology, in [A. de la Macorra and G. Piccinelli, Phys. Rev. D 61, 123503 (2000)], we show that the potential dependence $V\left(\phi \right)$ can be described through a parameter $\lambda \equiv -\sqrt{2}{m}_5^{3/2}{V}^{\prime }/\left(\sqrt{H}V\right),$ where $m_5$ is the five-dimensional Planck mass, H is the Hubble parameter, and $V^{\prime }\equiv dV/d\phi .$ For the case where $\lambda$ asymptotes to zero, we show that the solution exhibits stable inflationary behavior. On the other hand if it approaches a finite constant, then $V\left(\phi \right)\propto 1/{\phi }^2.$ For $\overrightarrow{\lambda }\infty$ asymptotically, we find examples where it does so both with and without oscillating. In the latter case, the barotropic fluid dominates the scalar field asymptotically. Finally we point out an interesting duality that leads to identical evolution equations in the high-energy ${\rho }^2$ dominated regime and the low-energy $\rho$ dominated regime.