Axially Symmetric Cosmological Models with Perfect Fluid and Cosmological Constant

Abstract

In this paper, we will discuss the behavior of some homogeneous but anisotropic models with axial symmetry, filled with a perfect and pressureless fluid (dust) and a non vanishing cosmological constant. For this, we will restrict our study to the the metric forms $$ d{{s}^{2}} = - {{c}^{2}}d{{t}^{2}} + {{a}^{2}}(t)d{{r}^{2}} + {{b}^{2}}(t)\left( {\frac{{d{{v}^{2}}}}{{1 - k{{v}^{2}}}} + {{v}^{2}}d{{\phi }^{2}}} \right), $$ (1) with the two scale factors a(t) and b(t); k is the curvature index of the 2-dimensional surface and can take the values +1,0,-1, giving the following three different metrics: Kantowski-Sachs, Bianchi I, and Bianchi III, respectively. Relevant Einstein equations for the metric (1), whith a perfect fluid and a cosmological term, ⨀, content are then as follows: $$ 2\frac{{\dot{a}}}{a}\frac{{\dot{b}}}{b} + \frac{{{{{\dot{b}}}^{2}}}}{{{{b}^{2}}}} + \frac{{k{{c}^{2}}}}{{{{b}^{2}}}} = 8\pi G\rho + \Lambda {{c}^{2}}, $$ (2) $$ 2\frac{{\ddot{b}}}{b} + \frac{{{{{\dot{b}}}^{2}}}}{{{{b}^{2}}}} + \frac{{k{{c}^{2}}}}{{{{b}^{2}}}} = - 8\pi G\frac{{{{p}^{2}}}}{{{{c}^{2}}}} + \Lambda {{c}^{2}}$$ (3) , where p s the matter density and p is the (isotropic) pressure of the fluid.