Quintessence in a brane world

We reanalyze a new quintessence scenario in a brane world model, assuming that a quintessence scalar field is confined in our three-dimensional brane world. We study three typical quintessence models: (1) an inverse-power-law potential, (2) an exponential potential, and (3) a kinetic-term quintessence $(k$-essence) model. With an inverse-power-law potential model $\left[V\right(\phi )={\mu }^{\alpha +4\mathrm{}}{\phi }^{-\alpha }],$ we show that in the quadratic dominant stage the density parameter of a scalar field ${\Omega }_{\phi }$ decreases as $a^{-4\left(\alpha -2\right)/(\alpha +2)}$ for $2<\mathrm{}\alpha <6,$ which is followed by the conventional quintessence scenario. This feature provides us wider initial conditions for successful quintessence. In fact, even if the universe is initially scalar-field dominated, it eventually evolves into a radiation dominated era in the ${\rho }^2$-dominant stage. Assuming an equipartition condition, we discuss constraints on parameters, with the result that $\alpha >~4$ is required. This constraint also restricts the value of the five-dimensional Planck mass, e.g., $4\times {10}^{-14}{m}_4\lesssim m_5\lesssim 3\times {10}^{-13}{m}_4$ for $\alpha =5.\mathrm{}$ For an exponential potential model $V={\mu }^4\exp \left(-\lambda \phi {/m}_4\right),$ we may not find a natural and successful quintessence scenario as it is, while for a kinetic-term quintessence, we find a tracking solution even in the ${\rho }^2$-dominant stage, rather than the ${\Omega }_{\phi }$-decreasing solution for an inverse-power-law potential. Then we do find a slight advantage in a brane world. Only the density parameter increases more slowly in the ${\rho }^2$-dominant stage, which provides a wider initial condition for successful quintessence.