Exact Kantowski–Sachs cosmological models in general theory of relativity

Abstract

We have studied cosmological model generated by perfect fluid coupled with mass less scalar field for Kantowski-Sachs space-time in general theory of relativity. Two different physically viable models of the universe are obtained by using a special law of variation for Hubble's parameter that yields a constant value of deceleration parameter. Some physical consequences of the models have been discussed in case of Zel'dovich fluid.

Dynamical system analysis of Einstein–Skyrme model in a Kantowski–Sachs spacetime

“…Initially we intensify the global behavior of the vector field on the hyperplanes. The flow of the 3D autonomous system (16)(17)(18) carries the behavior of the vector field of 2D autonomous systems. Next we analyze the flow on the hyperplane which is parallel to YZ-plane i.e.…”

“…In this table, the eigenvalues of J(x c , 0, z c ) are 0, x c ,−2x c . One should note that, all the critical points of the system (16)(17)(18) are non-hyperbolic (NH CPs) in nature. The zero eigenvalue of Jacobian matrix makes Hartman-Grobman theorem unbefitting.…”

“…The zero eigenvalue of Jacobian matrix makes Hartman-Grobman theorem unbefitting. Also the Liapunov function is difficult to find for the nonlinear system (16)(17)(18). Center Manifold theory (CMT) extricates from such a situation.…”

“…So we shift the point (x c , 0, z c ) to the origin and find the center manifold at origin. The system of equations (16)(17)(18) in the shifted coordinate system takes the form as follows.…”

“…The vector fields on the 2D autonomous system Let us first consider B(t) is constant. For z is a non zero constant (c) the system(16)(17)(18) turns into an autonomous system as follows,ẋ= k − x 2 y = 0(23)where k = Λ − µ 2c 4 , a constant. For k > 0 there are two critical points at x = ± √ k and corresponding eigenvalues of the Jacobian matrix are ∓2√ k. So x = √k is stable while the critical point at x = − √ k is unstable.…”