1211.0604 (Hyeong-Chan Kim)

Exact solutions in Einstein cosmology with a scalar field    [PDF]

Hyeong-Chan Kim

We present a simple way to obtain exact solutions of Einstein-scalar field equations on spatially flat Friedmann-Robertson-Walker space-times. The scalar equation turns out to be integrable if the Hubble parameter is written as an appropriate function of the scalar field and its velocity. Eventually, the field equations are reduced to find `generating functions' for a given scalar potential. Once a generating function is found as a function of the scalar field, the evolution of the field and the Universe can be easily obtained with a simple integration. As examples, we obtain the solution spectra in the cases of the constant and the exponential potentials, and find exact solutions for various scalar potentials such as the $\lambda \phi^4$, the power law, and the double-well hyperbolic functions. We additionally analyze the stability of the generating equation. We show that the existence of a fixed point of the equation of motion affect on the evolution so that the Universe experiences a long inflation. We additionally show that small change of the scalar potential cannot get rid of the appearance of the long inflation.

View original: http://arxiv.org/abs/1211.0604