Non-Abelian cosmic strings in de Sitter and anti–de Sitter space

In this paper we investigate the non-Abelian cosmic string in de Sitter and anti–de Sitter spacetimes. In order to do that we construct the complete set of equations of motion considering the presence of a cosmological constant. By using numerical analysis we provide the behavior of the Higgs and gauge fields and also of the metric tensor for specific values of the physical parameters of the theory. For the de Sitter case, we find the appearance of an horizon. This horizon is consequence of the presence of the cosmological constant, and its position strongly depends on the value of the gravitational coupling. In the anti–de Sitter case, we find that the system does not present horizons. In fact the new feature of this system is related with the behavior of the (00) and $(zz)$ components of the metric tensor. They present a strong increasing behavior for large distance from the string.

Figure 1

Non-Abelian string. Left: Behavior for the Higgs and gauge fields as functions of $x$. Right: Behavior of the metric functions as functions of $x$. In both plots we consider the parameters $\alpha =0.8$, $\gamma =0.61$, $\overline{\mathrm{\Lambda }}=0.0075$, ${\beta }_2=2.0$, ${\beta }_3=1.0$, and $q=1.0$.

Figure 2

Abelian string. Left: Behavior for the Higgs and gauge fields as functions of $x$. Right: Behavior of the metric functions as functions of $x$. In both plots we consider the parameters $\alpha =0.8$, $\gamma =0.61$, $\overline{\mathrm{\Lambda }}=0.0075$.

Figure 3

(a) The behavior of the cosmological horizon, $x_0$, as a function of $\gamma$ considering the parameters $\alpha =0.8,{\beta }_2=2.0,{\beta }_3=1.0,\overline{\mathrm{\Lambda }}=0.0075,q=1.0$. (b) The dashed line represents the behavior of the cosmological horizon, $x_0$, as a function of $\overline{\mathrm{\Lambda }}$ for the non-Abelian string case, considering $\alpha =0.8,\gamma =0.61,{\beta }_2=2.0,{\beta }_3=1.0,q=1.0$. The solid line corresponds to the trivial behavior of the cosmological horizon, $x_o^v$, in the vacuum case.

Figure 4

Non-Abelian string. Left: Behavior of the Higgs and gauge fields in anti–de Sitter space. Right: Metric functions. In both plots we have considered $\alpha =0.8,\gamma =0.6,{\beta }_2=2.0,{\beta }_3=1.0,\overline{\mathrm{\Lambda }}=-0.03$, and $q=1.0$.

Figure 5

Abelian string. Left: Behavior of the Higgs and gauge fields in anti–de Sitter space as functions of $x$. Right: Behavior of the metric fields in anti–de Sitter space considering $\alpha =0.8,\gamma =0.6,\overline{\mathrm{\Lambda }}=-0.03$.

Figure 6

The metric fields as functions of $x$ for different values of the cosmological constant. For solid lines we adopt $\overline{\mathrm{\Lambda }}=-0.010$, and for dashed lines $\overline{\mathrm{\Lambda }}=-0.007$. In both plots we consider $\alpha =0.8,\gamma =0.6,{\beta }_2=2.0,{\beta }_3=1.0$, and $q=1.0$.

Figure 7

The metric fields $L(x)$ and $N(x)$ as functions of $x$. (a) Comparison of the behavior of the metric fields $L$ and $N$ in de Sitter spacetime (dS), considering $\overline{\mathrm{\Lambda }}=0.0075$, with the metric fields in the Minkowski ($M$) and vacuum (vac) configurations, respectively. (b) Comparison of the behaviour of the metric fields $L$ and $N$ in anti–de Sitter spacetime (AdS), considering $\overline{\mathrm{\Lambda }}=-0.03$, with the metric fields in the Minkowski ($M$) and vacuum (vac) configurations, respectively. In both plots we consider $\gamma =0.6,\alpha =0.8,{\beta }_2=2.0,{\beta }_3=1.0$, and $q=1.0$.