Qualitative analysis of a quasi-magnetic universe

Abstract

We investigate the cosmological dynamics induced by nonlinear electrodynamics in a homogeneous and isotropic universe, focusing on the role of primordial electromagnetic fields with random spatial orientations. Building upon a generalization of the Tolman-Ehrenfest averaging procedure, we derive a modified energy-momentum tensor consistent with the spacetime symmetries, incorporating the influence of the dual invariant G and its statistical contributions. A specific nonlinear electrodynamics model with quadratic corrections to Maxwell's Lagrangian is considered, giving rise to what we define as a quasi-magnetic universe, interpolating between purely magnetic and statistically null field configurations. We analyze the resulting cosmological dynamics through qualitative methods. By casting the equations into autonomous dynamical systems, we identify the equilibrium points, determine their stability, and study the behavior of solutions under various spatial curvatures. Our findings reveal the existence of bouncing and cyclic solutions, regions where energy conditions are violated, and scenarios of accelerated expansion. Special attention is given to two limiting cases, both of which exhibit qualitatively distinct phase portraits and energy-condition behavior. This work provides a comprehensive framework for understanding the influence of nonlinear electromagnetic fields in the early universe and opens avenues for exploring their observational consequences

Figures (9)

• Figure 1: Phase Portrait of Case I. The vertical dot-dashed lines $B_1$ (black), $B_2$ (blue), and $B_3$ (red) are the separatrices. The equilibrium points are $B_{P_1}$ (node) and $B_{P_3}$ (saddle). $B_{P_2}$ is absent. The solid black line indicates the possible flat universes, separating the diagram into disjoint regions in terms of curvature. For this phase diagram, we choose $\tilde{\alpha}=1$, $\sigma=1/2$, without loss of generality, and $w=-1$.
• Figure 2: Phase Portrait of Case II. The vertical dot-dashed lines caption follows the previous case. The equilibrium points are now the $B_{P_1}$ (node), $B_{P_2}$ (center), and $B_{P_3}$ (saddle). Again, the flat universe solutions represented by the solid black line separate the diagram into disjoint regions in terms of curvature. For this phase diagram, we choose $\tilde{\alpha}=1$, $\sigma=1/2$, and $w=-1/2$.
• Figure 3: Phase Portrait of Case III. The vertical dot-dashed lines caption follows the previous case. The equilibrium points are the $B_{P_1}$ (node), $B_{P_2}$ (saddle (see the zoomed region on the left panel)), and $B_{P_3}$ ("center"). Again, the flat universe solutions represented by the solid black line separate the diagram into disjoint regions in terms of curvature, but now there is a branch on the right of $B_3$. For this phase diagram, we choose $\tilde{\alpha}=1$, $\sigma=1/2$, and $w=-1/8$.
• Figure 4: Phase Portrait of Case IV. The vertical dot-dashed lines caption follows the previous case. The equilibrium points are the $B_{P_1}$ (node), $B_{P_2}$ (center), and $B_{P_3}$ ("center"). The zoomed region also shows the appearance of two saddle points (blue) along $B_2$ whose separatrices delineate the attraction basin of $B_{P_3}$. Flat universe solutions separate the diagram again into disjoint regions, and the branch on the right of $B_3$ remains. For this phase diagram, we choose $\tilde{\alpha}=1$, $\sigma=1/2$, and $w=-1/24$.
• Figure 5: Phase Portrait of Case V. The vertical dot-dashed lines caption follows the previous case. The equilibrium points are $B_{P_1}$ (node), $B_{P_2}$ (center), and $B_{P_3}$ (saddle). Flat universe solutions continue separating the diagram into disjoint regions, but now closed orbits are impossible on the right side of $B_3$. For this phase diagram, we choose $\tilde{\alpha}=1$, $\sigma=1/2$, and $w=1/8$.