Cosmological implications of bumblebee vector models

The bumblebee model of spontaneous Lorentz symmetry breaking is explored in a cosmological context, considering a single nonzero time component for the vector field. The relevant dynamic equations for the evolution of the Universe are derived and their properties and physical significance are studied. We conclude that a late-time de Sitter expansion of the Universe can be replicated, and attempt to constrain the parameter of the potential driving the spontaneous symmetry breaking.

Figure 1

Parameter constraints assuming ${\mathbf{x}}_B$ and ${\mathbf{x}}_C$ are unstable (white), only ${\mathbf{x}}_C$ is stable (light grey), and both ${\mathbf{x}}_B$ and ${\mathbf{x}}_C$ are stable (dark grey). The points used in the analysis of the phase-space trajectories are highlighted.

Figure 2

Projection of the phase space onto the ${\mathrm{\Omega }}_M$-$q$ plane for $n=4$ (two darker plots) and $n=6$ (two lighter plots). Fixed points are marked as dots in dark grey ($n=4$) and light grey ($n=6$). For both cases $\xi b^2={10}^{-2}$ and $x_{2i}={10}^{-3}$ (darker), ${10}^{-12}$ (lighter).

Figure 3

Projection of the phase space onto the $x_2$-$q$ plane for $n=4$ (two darker plots) and $n=6$ (two lighter plots). Fixed points are marked as dots in dark grey ($n=4$) and light grey ($n=6$). For both cases $\xi b^2={10}^{-2}$ and $x_{2i}={10}^{-3}$ (darker), ${10}^{-12}$ (lighter).

Figure 4

Projection of the phase space onto the $x_2$-${\mathrm{\Omega }}_M$ plane for $n=4$ (two darker plots) and $n=6$ (two lighter plots). Fixed points are marked as dots in dark grey ($n=4$) and light grey ($n=6$). For both cases $\xi b^2={10}^{-2}$ and $x_{2i}={10}^{-3}$ (darker), ${10}^{-12}$ (lighter).

Figure 5

Projection of the phase space onto the ${\mathrm{\Omega }}_M$-$q$ plane for $n=4$ (two darker plots) and $n=6$ (two lighter plots). Fixed points are marked as dots in dark grey ($n=4$) and light grey ($n=6$). For both cases $\xi b^2={10}^{-2}$ and $x_{2i}={10}^{-3}$ (darker), ${10}^{-12}$ (lighter).

Figure 6

Projection of the phase space onto the $x_2$-$q$ plane for $n=4$ (two darker plots) and $n=6$ (two lighter plots). Fixed points are marked as dots in dark grey ($n=4$) and light grey ($n=6$). For both cases $\xi b^2={10}^{-2}$ and $x_{2i}={10}^{-3}$ (darker), ${10}^{-12}$ (lighter).

Figure 7

Projection of the phase space onto the $x_2$-${\mathrm{\Omega }}_M$ plane for $n=4$ (two darker plots) and $n=6$ (two lighter plots). Fixed points are marked as dots in dark grey ($n=4$) and light grey ($n=6$). For both cases $\xi b^2={10}^{-2}$ and $x_{2i}={10}^{-3}$ (darker), ${10}^{-12}$ (lighter).

Figure 8

Variation of the minimum of $q$ with the initial value of $x_2$ for $n=4$.

Figure 9

Variation of the minimum of $q$ with the initial value of $x_2$ for $n=6$.