Zipping and unzipping in string networks: Dynamics of Y-junctions

We study, within the Nambu-Goto approximation, the stability of massive string junctions under the influence of the tensions of three strings joining together in a Y-type configuration. The relative angle $\beta$ between the strings at the junction is in general time dependent and its evolution can lead to zipping or unzipping of the three-string configuration. We find that these configurations are stable under deformations of the tension balance condition at the junction. The angle $\beta$ relaxes at its equilibrium value and the junction grows relativistically. We then discuss other potential “unzipping agents” including monopole/string forces for long strings and curvature for loops, and we investigate specific solutions exhibiting decelerated zipping and unzipping of the Y junction. These results provide motivation for incorporating the effects of realistic string interactions in network evolution models with string junctions.

Figure 1

Top: Two strings (denoted by 1, 2) colliding at time $t=0$. Bottom: At time $t=\delta t$, a new segment (denoted by 3) has formed. The junction $J$ is moving to the right and the zipper is growing.

Figure 2

String junction configurations for $v\to 0$ as $t\to 0$ (left) and for $v=\text{const}>0$ (right).

Figure 3

Straight strings collision.

Figure 4

Left: The evolution of ${\dot{s}}_3$ for $\mu =1$, ${\mu }_3=\sqrt{2}\mu$, $m=\sqrt{\mu }$, $\alpha =\pi /6$ and $v=0.4$, shown together with the unperturbed case ${\dot{s}}_3=\text{const}$, given by the CKS solution. Right: Evolution of the angle $\beta (t)$ for the same parameters. The dotted line shows the critical angle for tension and momentum transfer balance.

Figure 5

Schematic representation of the three-string configuration in the general case where the colliding strings have unequal tensions ${\mu }_1\ne {\mu }_2$. Note we only show the $x>0$ part of the configuration.

Figure 6

The evolution of ${\dot{s}}_3$ for ${\mu }_1=1$, ${\mu }_2=0.7$, ${\mu }_3=1.2$, $m=\sqrt{{\mu }_1}$, $\alpha =\pi /6$ and $v=0.4$, shown together with the unperturbed case ${\dot{s}}_3=\text{const}$. given by the CKS solution.

Figure 7

The evolution of $s_3$ including an extra force towards the junction with magnitude $q\epsilon =0.5$ (solid red lines), for $\mu =1$, ${\mu }_3=\sqrt{2}\mu$, $\alpha =\pi /6$, and $v=0$, for $m=\sqrt{\mu }/2$ (top left), $m=\sqrt{\mu }/4$ (top right), and $m=2\sqrt{\mu }$ (bottom), shown together with the unperturbed case given by the CKS solution (dashed black lines).

Figure 8

A snapshot in the evolution of the three-string configuration (99), under the influence of the local string force (96), for $M=12$ and $k=65$. On the left the force is attractive ($q=1$) leading to $\beta <{\beta }_{\text{crit}}$ near the junction, while on the right it is repulsive ($q=-1$) producing $\beta >{\beta }_{\text{crit}}$.

Figure 9

The evolution of $s_3(t)$ under the influence of the local string force (96), for the repulsive case $M=12$ and $k=65$ (solid red line), shown together with the unperturbed case given by the CKS solution (dashed black line). Unzipping happens at $t\sim 0.25$.

Figure 10

Left: Two coplanar loops which are moving in the $z$-direction with opposite velocities collide. Right: After the collision four junctions and eight kinks are formed.

Figure 11

Left panel: The background evolution for F(x) with $F(x=1)=100$, $F^{\prime }(x=1)=0.1$, and $v(x=1)=0.4$ for the radiation era. Right panel: The evolution of junction $B$ for the radiation dominated background, for $a=\pi /9$, ${\mu }_1=2$, ${\mu }_2=1$, ${\mu }_3=2.5$, $v(x=1)=0.4$, $u(x=1)=0.168$, $\theta =0.102$.