Dynamics of a Bianchi Type I Model With a Concave Potential

Abstract

In this paper, we study the dynamics of a Bianchi Type I potential in the presence of a concave potential of the form $V(\phi) = V_0 \left[1- \left( \frac{\phi}{\mu}\right)^n\right]$, where $V_0$ is a constant, and $\mu$ is a mass scale. We show that there are two classes of equilibrium points. The first class corresponds to $\phi \neq 0$, $\mu \neq 0$, $n = 0$, and $\dot{\phi} = 0$, which describe expanding and contracting de Sitter universes, for which the shear anisotropy is zero. We show that the expanding de Sitter universe is a local sink of the system, and therefore has associated to it a stable manifold. Thus, orbits will approach this point at late times. In other words, such a model is found to inflate and isotropize at late times as long as $n = 0$. The second class of equilibrium points corresponds to an expanding and contracting anisotropic universe. However, these points are found to emerge only when $n > 1$, $\phi = \dot{\phi} = 0$, which importantly implies that $V = V_0 < 0$ at this point in order to ensure that the square of the shear scalar, $\sigma^2$ is real. Therefore, such equilibrium points correspond to the ekpyrotic cosmological models. Further, we show that for $n = 2$, by Lyapunov's stability theorem, the expanding equilibrium point is asymptotically stable, while for $n > 2$, a two-dimensional stable manifold exists corresponding to the fact that for $n > 2$, such an equilibrium point represents a local sink of the system. Finally, we give a general condition for inflation to occur in this model in terms of the deceleration parameter, and show that the expanding ekpyrotic equilibrium point undergoes the phenomenon of anisotropic inflation if $-3\Lambda/5 < V_0 < 0$, where $\Lambda > 0$ is the cosmological constant.