Quantum Cosmology and Dark Energy Model of Born-Infeld Type Scalar Field

In this paper, we consider a quantum model of gravitation interacting with a Born-Infeld(B-I) type scalar field $\varphi$. The corresponding Wheeler-Dewitt equation can be solved analytically for both very large and small cosmological scale factor. In the condition that small cosmological scale factor tend to limit, the wave function of the universe can be obtained by applying the methods developed by Vilenkin, Hartle and Hawking. Both Vilenkin’s and Hartle-Hawking’s wave function predicts nonzero cosmological constant. The Vilenkin’s wave function predicts a universe with a cosmological constant as large as possible, while the Hartle-Hawking’s wave function predicts a universe with positive cosmological constant, which equals to $\frac{1}{\lambda }$. It is different from Coleman’s result that cosmological constant is zero, and also different from Hawking’s prediction of zero cosmological constant in quantum cosmology with linear scalar field. We suggest that dark energy in the universe might result from the B-I type scalar field with potential and the universe can undergo a phase of accelerating expansion. The equation of state parameter lies in the range of $-1<$w$<-1\mathrm{/}3$. When the potential $V(\varphi )=\frac{1}{\lambda }$, our Lagrangian describes the Chaplygin gas. In order to give a explanation to the observational results of state parameter w$<-1$, we also investigate the phantom model that posses negative kinetic energy. We find that weak and strong energy conditions are violated for phantom B-I type scalar field. At last, we study a specific potential with the form $V_0(1+\frac{\varphi }{{\varphi }_0})e^{-(\frac{\varphi }{{\varphi }_0})}$ in phantom B-I scalar field in detail. The attractor property of the system is shown by numerical analysis.