Matter and dynamics in closed cosmologies

To systematically analyze the dynamical implications of the matter content in cosmology, we generalize earlier dynamical systems approaches so that perfect fluids with a general barotropic equation of state can be treated. We focus on locally rotationally symmetric Bianchi type IX and Kantowski-Sachs orthogonal perfect fluid models, since such models exhibit a particularly rich dynamical structure and also illustrate typical features of more general cases. For these models, we recast Einstein’s field equations into a regular system on a compact state space, which is the basis for our analysis. We prove that models expand from a singularity and recollapse to a singularity when the perfect fluid satisfies the strong energy condition. When the matter source admits Einstein’s static model, we present a comprehensive dynamical description, which includes the qualitative asymptotic behavior, of models in the neighborhood of the Einstein model; the results refute earlier claims about “homoclinic phenomena and chaos.” We also discuss aspects of the global dynamics of models; in particular, we give criteria for the collapse to a singularity, and we describe when models expand forever to a state of infinite dilution; possible initial and final states are analyzed. Numerical investigations complement the analytical results.

Figure 1

Qualitative shapes of potentials for FRW in the Cases (iii) [small dashed line], (iv) [solid line], and (v) [large dashed line] of Table .

Figure 2

Qualitative shapes of potentials for FRW Cases (i) and (ii) in Table , with associated flows of the FRW dynamical system (17).

Figure 3

Typical phase portraits for $-1/3<w\equiv \mathrm{const}<1$ and $-1\le w\equiv \mathrm{const}<-1/3$.

Figure 4

The FRW plane $\Vert (q_{+},m_1)\Vert =0$ in the space ${}^3\mathbf{X}$. Four null rays, ${\tilde{\nu }}_{q,l}^{\pm }$, converge to the Einstein point E as $\tau \to \pm \infty$.

Figure 5

The light cone $E=V_{\max }$ in ${}^3\mathbf{X}$ divides the state space in three regions that generalize the regions in the FRW picture 2d.

Figure 6

Case $E>V_{\max }$. Intersection of the mass hyperboloids of energy $E$ with a plane $\Vert (q_{+},m_1)\Vert =\mathrm{const}$ in ${}^3\mathbf{X}$ yields two hyperbolas, one in region I and one in region II. The axes are the same as in Fig. 5.

Figure 7

Illustration of the close relationship between the dynamics close to E and FRW dynamics. At the scale of (c) there is no visible difference between this figure and the analogous figure depicting the FRW separatrices in the same projection, except that the orbits do not converge to ${}_{\pm }{}^0\mathrm{F}$ in general, but to the generic limits ${}_{\mp }{}^0\mathrm{T}_{\pm }$, cf. Appendix  and Fig. 10. In this figure the matter content is radiation and a cosmological constant; however, the qualitative behavior is the same for any matter source that admits the Einstein static solution.

Figure 8

Case $E<V_{\max }$. Intersection of the timelike hyperboloids with energy $E$ with different planes $\Vert (q_{+},m_1)\Vert =\mathrm{const}$. We obtain timelike hyperbolas in Case (a), a vertex and a null cone in Case (b), and spacelike hyperbolas in (c). The vertex of the null cone in (b) represents a closed orbit in the center manifold. The global asymptotic behavior of the orbits is governed by the null ray orbits ${\tilde{\nu }}_{l,q}^{\pm }$, cf. Figure 4. The axes in the figures are the same as in Fig. 5.

Figure 9

Example of the effect of the center manifold on dynamics. The source is orthogonal dust and a positive cosmological constant; however, the pictures depict behavior that is typical for all sources that admit a unique Einstein point E.

Figure 10

Illustration of the dynamics far from the Einstein point: the qualitative structure of the flow is still the one of a small neighborhood of E. Clearly there is no “homoclinic chaos.” In this figure the matter content is radiation and a cosmological constant; however, the qualitative behavior is the same for any matter source that admits the Einstein static solution.

Figure 11

The Kantowski-Sachs submanifold. Case (ii).

Figure 12

In $Q_0=0$, for $L=\mathrm{const}>{\tilde{\mathcal{L}}}_E$, the curve $\mathcal{E}=E$ is entirely contained in the region $F<0$ where $Q_0^{\prime }>0$.