Evolution of spherical domain walls in solitonic symmetron models

In this work, inspired by the symmetron model, we analyze the evolution of spherical domain walls by considering specific potentials that ensure symmetry breaking and the occurrence of degenerate vacua that are necessary for the formation of domain walls. By considering a simple analytical model of spherical domain-wall collapse in vacuum, it is shown that this model fits the more accurate numerical results very well until full collapse, after which oscillations and scalar radiation take place. Furthermore, we explore the effect of a central nonrelativistic matter lump on the evolution of a spherical domain wall and show that the central lump can prevent the full collapse and annihilation of the domain-wall bubble, due to the repulsion between the domain wall and matter overdensity within the adopted symmetron-inspired model.

Figure 1

The effective potential of the SG system for $a=b=1$; (a) $\rho =10$ and (b) $\rho =0.01$. See the text for more details.

Figure 2

The effective potential of the DSG system for $a=b=1$ and $ϵ=10$; (a) $\rho =10$ and (b) $\rho =0.01$. See the text for more details.

Figure 3

The effective potential of the ${\phi }^4$ system for $\alpha =\beta =1$; (a) $\rho =10$ and (b) $\rho =0.01$. See the text for more details.

Figure 4

The effective potential of the ${\phi }^6$ system for $\alpha =1$ and $\beta =\sqrt{2}$; (a) $\rho =10$ and (b) $\rho =0.01$. See the text for more details.

Figure 5

The numerical results are shown by the dotted curve depicting the bubble collapse velocity curve in terms of the radius of the bubble for the (a) SG ($a=b=1$), (b) DSG ($a=b=1$ and $ϵ=10$), (c) ${\phi }^4$ ($\alpha =\beta =1$), and (d) ${\phi }^6$ ($\alpha =1$ and $\beta =\sqrt{2}$) systems. The analytical results of Eqs. (21) and (27) for $n=0.3$ are shown in the dashed-dotted and solid curves, respectively.

Figure 6

Evolution of the scalar field (left plot) and the energy density (right plot) of the SG domain wall, where we have considered the parameter values $a=b=1$. See the text for more details.

Figure 7

Evolution of the energy density of the DSG domain wall ($a=b=1$ and $ϵ=10$) from $t=10$ to $t=50$.

Figure 8

Evolution of the energy density of the ${\phi }^4$ domain wall ($\alpha =\beta =1$) from $t=10$ to $t=50$.

Figure 9

Evolution of the scalar field (left plot) and the energy density (right plot) of the ${\phi }^6$ domain wall, for the parameter values $\alpha =1$ and $\beta =\sqrt{2}$. See the text for more details.

Figure 10

Evolution of the scalar field (left plot) and the energy density (right plot) of the SG domain wall with an internal matter density ($\rho ={\rho }_0{e}^{-r/r_0}$) and for the parameter values $a=b=1$. See the text for more details.

Figure 11

Evolution of the scalar field and the energy density of the DSG domain wall with an internal matter density ($\rho ={\rho }_0{e}^{-r/r_0}$), with $ϵ=-0.1$, and for the parameter values $a=b=1$. See the text for more details.

Figure 12

Evolution of the scalar field and the energy density of the DSG domain wall with an internal matter density ($\rho ={\rho }_0{e}^{-r/r_0}$) for $ϵ=10$ and for the parameter values $a=b=1$. See the text for more details.

Figure 13

Evolution of the energy density of the ${\phi }^4$ domain wall ($\alpha =\beta =1$) with an internal matter density ($\rho ={\rho }_0{e}^{-r/r_0}$) from $t=10$ to $t=90$.

Figure 14

Evolution of the scalar field (left plot) and the energy density (right plot) of the ${\phi }^6$ domain wall with an internal matter density, given by $\rho ={\rho }_0{e}^{-r/r_0}$, for the parameter values $\alpha =1$ and $\beta =\sqrt{2}$. See the text for more details.

Figure 15

The plot depicts the result when the gravitational field is switched off (dashed curve) and switched on (dotted curve), using the Newtonian result (39). Note that the Newtonian gravitational force causes the collapse velocity of the bubble to diverge as $R\to 0$. In the absence of the gravitational force, the collapse becomes similar to the analysis depicted in Fig. 5.

Figure 16

Collapse of the domain wall, taking into account the gravitational effects [see Eq. (48)].

Figure 17

Smoothing out the step function (red lines) using the tanh function (54).

Figure 18

The collapse of the domain-wall bubble from rest until it collides with the central mass, in which the symmetron field is highly screened. The parameters chosen are $M_i=10^{48}\mathrm{kg}$, $M_b=1.5\times 10^{43}\mathrm{kg}$,$\alpha =3.33\times 10^{26}\hslash /c^3$, and $\beta =2.36\times {10}^{-22}\sqrt{\hslash }/c^{\frac{3}{2}}$.