Numerical simulations of necklaces in SU(2) gauge-Higgs field theory

We perform the first numerical simulations of necklaces in a non-Abelian gauge theory. Necklaces are composite classical solutions which can be interpreted as monopoles trapped on strings, rather generic structures in a Grand Unified Theory. We generate necklaces from random initial conditions, modeling a phase transition in the early Universe, and study the evolution. For all cases, we find that the necklace system shows scaling behavior similar to that of a network of ordinary cosmic strings. Furthermore, our simulations indicate that comoving distance between the monopoles or semipoles along the string asymptotes to a constant value at late times. This means that, while the monopole-to-string energy density ratio decreases as the inverse of the scale factor, a horizon-size length of string has a large number of monopoles, significantly affecting the dynamics of string loops. We argue that gravitational wave bounds from millisecond pulsar timing on the string tension in the Nambu-Goto scenario are greatly relaxed.

Figure 1

Views of a small $360^3$ simulation for three different parameter choices at $t\approx 240$. The two fields ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ have blue and green shading respectively. At left, a simulation with $m_1^2=0.25$ and $m_2^2=0.025$ (contours shown with $\mathrm{Tr}{\mathrm{\Phi }}_1^2=0.2$ and $\mathrm{Tr}{\mathrm{\Phi }}_2^2=0.04$), giving rise to monopoles (blue) as beads on strings (green). The other two images show a system with $m_1^2=m_2^2=0.25$ (contours shown with $\mathrm{Tr}{\mathrm{\Phi }}^2=0.2$ for both ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$). In the center, $\kappa =2$, showing the semipoles at the boundaries between the two colours. At right, $\kappa =1$, and the theory has a continuous global symmetry, meaning that the boundaries between the colors have no extra energy.

Figure 2

Plot of the network length scale ${\xi }_{\mathrm{n}}$, defined in Eq. (16), with core growth parameter $s=1$ (top) and $s=0$ (bottom). Fits to linear growth are also shown, within the range indicated by the vertical dashed lines. The gradients of the fit are given in Tables 3 and 4.

Figure 3

The ratio of monopole to string energy density (3) in simulations with $s=1$. The legend gives the expansion rate parameter $\nu =d\log a/d\log t$, the mass ratio of the fields $m_2/m_1$, and in the degenerate case the value of the cross-coupling $\kappa$, which is otherwise $\kappa =1$. The mass parameter $m_1^2=0.25$.

Figure 4

The number of monopoles per comoving string length in simulations with $s=1$ (top) and $s=0$ (bottom). The legend gives the expansion rate parameter $\nu =d\log a/d\log t$, the mass ratio of the fields $m_2/m_1$, and in the degenerate case the value of the cross-coupling $\kappa$, which is otherwise $\kappa =1$. The mass parameter $m_1^2=0.25$ ($s=1$) and $m_1^2=0.1$ ($s=0$).

Figure 5

A small $360^3$ box at $t=1080$, simulated at $m_2^2/m_1^2=1$ and $\kappa =2$. The high density of semipoles on one of the strings shows that semipoles can avoid annihilation in some cases.

Figure 6

The difference of the linear monopole density $n$ from its asymptotic value $n_{\infty }$. The parameter $n_{\infty }$ is extracted from a fit of $n$ to a constant to exponential decay [see Eq. (17)]; the fits are shown as dashed lines. Both $n$ and the time are scaled by $m_1$ to make dimensionless quantities. Only those values of $m_2/m_1$ where a reliable fit is possible are shown; for other values, the change in $n$ is too small.

Figure 7

Plot of $\overline{v}$ and ${\overline{v}}_{\mathrm{m}}$, the root mean square string and monopole/semipole velocities, for $s=1$ (top) and $s=0$ (bottom). The time axis is scaled by $d_{\mathrm{BV}}$, defined in Eq. (2).

Figure 8

Comoving core widths of monopoles ($w_{\mathrm{m}}$, solid) and strings ($w_{\mathrm{s}}$, dashed) in the $s=1$ simulations. The core widths are defined as $w_{\mathrm{m}}=(a{m}_1{)}^{-1}$ and $w_{\mathrm{s}}=(a{m}_2{)}^{-1}$, where $a$ is the scale factor.