Cosmic string Y-junctions: A comparison between field theoretic and Nambu-Goto dynamics

We explore the formation of cosmic string Y-junctions when strings of two different types collide, which has recently become important since string theory can yield cosmic strings of distinct types. Using a model containing two types of local U(1) string and stable composites, we simulate the collision of two straight strings and investigate whether the dynamics matches that previously obtained using the Nambu-Goto action, which is not strictly valid close to the junction. We find that the Nambu-Goto action performs only moderately well at predicting when the collision results in the formation of a pair of Y-junctions (with a composite string connecting them). However, we find that when they do form, the late-time dynamics matches those of the Nambu-Goto approximation very closely. We also see little radiative emission from the Y-junction system, which suggests that radiative decay due to bridge formation does not appear to be a means via which a cosmological network of such string would rapidly lose energy.

Figure 1

The intersection of two strings to form a pair of Y-junctions, as seen in the Nambu-Goto picture. The initial state of the two strings (top) is with them lying parallel to the $yz$ plane and travelling with velocities $\pm v$ in the $x$ direction, while there is a choice of final state (middle and bottom) depending upon how the two strings connect to each other. In either case, a bridge string links the two $Y$ junctions and the initial strings have kinks travelling along them. We denote the middle case as a cosine linkage, and the bottom case as a sine linkage.

Figure 2

The Nambu-Goto predictions for when cosine bridges can form in cases A, B, and C that we will explore using field theory simulations, with sine bridges also included for case C. Case A has $S=0$ and $R=0.84$ while case B has $S=0.30$ and $R=0.84$. Case C has $S=0.25$ but in the field theory case the value of $R$ depends on the connectivity due to flux cancellation for a cosine bridge (Cc) with $R=0.37$ but not for a sine bridge (Cs), which then has $R=0.90$. Case Cc therefore has $|S|>R^2$. Note that Y-junctions under both Cc and Cs are possible outcomes for a small region of parameter space while for cases A and B there is no overlap between sine bridges (not shown) and the cosine bridges.

Figure 3

Schematic illustrations of the three collision types  A, B, and C considered here, with the difference between the two possible connectivities which exist for type C highlighted: Cc (cosine) and Cs (sine).

Figure 4

Isosurfaces of $|\varphi |=0.5\eta$ and $|\psi |=0.5\nu$ from a type A collision: $(1,0)+(0,1)$. The upper two frames are from $t=-6{\eta }^{-1}$, showing the attraction of the strings towards each other and intersection at negative times rather than the Nambu-Goto value of $t=0$. The lower two frames are from the later time of $t=+30{\eta }^{-1}$, showing the established bridge and Y-junctions. The initial speeds were $v=0.2$ and the strings made an angle $\alpha =20°$ to the $z$ axis.

Figure 5

Energy density slices through a case A simulation, showing the initial burst of radiation as the bridge first forms. These images are heavily saturated in the bridge core, as is required to resolve the small amount of radiation emitted, and are taken from an $\alpha =20°$, $v=0.2$ simulation at $t=21{\eta }^{-1}$. In the uppermost pane, the bridge appears darker than the initial strings because it lies in the $x=0$ plane, while the initial strings are $4.2{\eta }^{-1}$ off this plane, and note that the two panes share different spatial scales.

Figure 6

Incident velocities $v$ and angles $\alpha$ which are seen in simulations to yield Y-junctions when ${\mu }_1={\mu }_2$ (i.e. $S=0$) and $R=0.840$. Here $\mathtt{Y}$ denotes the formation of a bridge with Y-junctions at either side, points that the strings passed through each other, and $\mathtt{X}$ denotes that the strings became locked together to form an X-junction. The CKS prediction is that the region beneath the curve could yield Y-junctions.

Figure 7

Isosurfaces of energy density $T_0^0=0.5{\eta }^4$ from a collision of type A, when $\alpha =40°$ and $v=0.2$, showing the formation of an X-junction. Results are shown for time $t=30{\eta }^{-1}$.

Figure 8

Isosurfaces of energy density $T_0^0=0.5{\eta }^4$ from a type B collision: $(2,0)+(0,1)$, with $\alpha$ and $v$ as in Fig. 4. The snapshot is for $t=24{\eta }^{-1}$. Of the two initial strings, it is the (2, 0) string which appears thicker and notice that the (2, 1) bridge is angled toward this string.

Figure 9

Incident velocities $v$ and angles $\alpha$ which are seen in simulations to yield Y-junctions when $S=0.305$ and $R=0.840$. Here $\mathtt{Y}$ denotes the formation of a bridge with Y-junctions at either side, points denote that the strings passed through each other and $\mathtt{X}$ denotes that the strings became locked together to form an X-junction. The CKS prediction is that the region beneath the curve should yield Y-junctions.

Figure 10

Incident velocities $v$ and angles $\alpha$ which are seen in simulations to yield Y-junctions for case C: $(1,1)+(0,-1)$, and compared to the CKS predictions from Fig. 2. The final state of the system is denoted by a $\mathtt{c}$ when a (1,0) cosine bridge and Y-junction pair formed, an $\mathtt{s}$ when a (1, 2) sine bridge was seen, an X when a X-junction formed, and a dot when the strings passed through each other. Additionally ($\mathtt{c}$) denotes a case when a cosine bridge formed and grew, but that a displacement wave caused a late-time intercommutation between the (1, 1) and (0, $-1$) strings, after which the bridge collapsed.

Figure 11

Isosurfaces of energy density $T_0^0=0.5{\eta }^4$ from the collision of (1, 1) string with a (0, $-1$) string with initial speed $v=0.6$ and $\alpha =20°$, showing a (1, 0) bridge has formed. The snapshot is shown for $t=44{\eta }^{-1}$.

Figure 12

The reconstructed string center-lines for a type Cs collision: $(1,1)+(0,-1)\to (1,0)$, occurring in the region of the $\alpha v$ plane where the Nambu-Goto dynamics suggests that both Cc and Cs are possible: $\alpha =70°$ and $v=0.2$. Results are shown for $t=-24{\eta }^{-1}$ (left), $t=0$ (center) and $t=24{\eta }^{-1}$ (right), showing the $\varphi$ string center line in black (blue online) and the $\psi$ string center line in gray (green online). Center lines are detected using the winding of the scalar field phases around lattice plackettes (see Sec. 6 and appendix). Lengths are shown in lattice units: $\Delta x=0.5{\eta }^{-1}$.

Figure 13

The measured $l_3$ as a function of $t$ measured from a $\Delta x=0.5{\eta }^{-1}$ simulation for a case A collision: $(1,0)+(0,1)$, with $\alpha =20°$ and $v=0.2$. Results shown are derived from the count of sites on $x=y=0$ with $|\varphi |<ϵ\eta$ and $|\psi |<ϵ\nu$ and using $ϵ=0.5$ (crosses, blue online) and $ϵ=0.75$ (triangles, green online), while the CKS prediction is shown by a dashed gray line. Circles (blue online) indicate the results for $ϵ=0.5$ from a shorter simulation but with $\Delta x$ halved from $0.5{\eta }^{-1}$ to $0.25{\eta }^{-1}$, highlighting that the simulated dynamics are not precisely those of the continuum, while the measurements themselves are accurate to within $\Delta x/2$ and the corresponding uncertainties are too small to be shown on this plot.

Figure 14

The measured $d{l}_3/dt$ values from a class A collision, shown as a function of $v$ for fixed $\alpha$, compared to the CKS predictions. Results are shown for $\alpha =10°$ (uppermost), 20° (middle), and 25° (lower). The simulations had $\Delta x=0.5$ and were of sufficient size that signals emitted from the box center at $t=0$ would reach the corners of the $yz$ plane by $t=60{\eta }^{-1}$, except for $(\alpha =20°,v=0.2)$, $(\alpha =20°,v=0.3405)$, and $(\alpha =25°,v=0.31)$ which ran till $t=120$, 90, and $120{\eta }^{-1}$.

Figure 15

The measured physical bridge half-length growth rate $d{l}_3/dt$, bridge speed $u$, and bridge orientation $\theta$ from type B collisions with $\alpha =20°$, compared to CKS predictions. The simulations had $\Delta x=0.5$ and were of sufficient size that signals emitted from the box center at $t=0$ would reach the corners of the $yz$ plane by $t=60{\eta }^{-1}$ (except for $v=0.39$ when this value was $120{\eta }^{-1}$).

Figure 16

The measured physical bridge half-length growth rate $d{l}_3/dt$, bridge speed $u$ (blue), and bridge orientation $\theta$ (red) values from type Cc collisions with $\alpha =60°$, compared to CKS predictions. The simulations at low speed had $\Delta x=0.5$, which was then decreased to accommodate Lorentz contraction, with the simulation size such that signals emitted from the box center at $t=0$ would reach the corners of the $yz$ plane by $t=60{\eta }^{-1}$.

Figure 17

The geometry involved in the tension-based interpretation of Eq. (3). The Y-junction is at position Y and the two kinks are at K1 and K2, while the region upon which the tensions in question act is highlighted in gray. The dashed (blue) lines indicate the consumption of the string 1 due to the motion of Y. (Note that both the unit vector $\mathbf{d}$ along the line YK1 and the orthogonal velocity $\mathbf{w}$ have in general finite components in the $x$ direction; therefore, the projections of them onto the $yz$ plane would not normally be perpendicular, although we shown them as orthogonal for illustrative purposes).