From vacuum to dark energy. Exact anisotropic cosmological solution of Petrov type D for a nonlinear scalar field

Abstract

Two exact solutions to Einstein equations, which differ because of its type of initial expansion, are obtained to a nonlinear scalar field with a potential type V = Λ (cid:16) 1 − tanh ( √ 6 φ 2 ) 4 (cid:17) . It is determined that the energy density of solutions is not singular for any time value and for which at the beginning in t = 0, the space-time is a vacuum of Kasner type ( a 1 = a 2 = − 2 a 3 = 2 / 3) for one solution and the flat world for the other. By having studied the temperature, it is established that it is null at the beginning and that once it increases up to a maximum value, it stops increases and asymptotically goes down to zero in respect to time. The Hubble and deceleration parameters were studied, it is showed that the Hubble parameter is indefinite in t = 0 and tends to have a constant value as time increases; then, the deceleration parameter indicates an initial process of a decelerated expansion that continuously changes into an accelerated one as time increases. By the study of the Jacobi stability of the solutions, it is obtained that the solutions are initially unstability but cease to be so in a determined time. The space-time of both solutions transforms into the equivalent of dark energy for FRWL as time increases.