Shape of cosmic string loops

Complicated cosmic string loops will fragment until they reach simple, nonintersecting (“stable”) configurations. Through extensive numerical study we characterize these attractor loop shapes including their length, velocity, kink, and cusp distributions. We find that an initial loop containing $M$ harmonic modes will, on average, split into $3M$ stable loops. These stable loops are approximately described by the degenerate kinky loop, which is planar and rectangular, independently of the number of modes on the initial loop. This is confirmed by an analytic construction of a stable family of perturbed degenerate kinky loops. The average stable loop is also found to have a 40% chance of containing a cusp. We examine the properties of stable loops of different lengths and find only slight variation. Finally we develop a new analytic scheme to explicitly solve the string constraint equations.

Figure 1

When two strings intersect, they reconnect in an intercommutation event with the production of four kinks (discontinuities in the tangent to the string), two on each string.

Figure 2

A sample initial loop with 10 harmonic modes. The darker, thicker segments of the loop are nearer the viewer.

Figure 3

Length of loops versus generation. By generation 10 the distribution is nearly identical to the stable loop length distribution. As fragmentation occurs the length of loops decreases. The distribution of the predominantly small length stable loops can be seen in Fig. 4.

Figure 4

Length distribution of stable loops shown on a logarithmic scale to better see the small length behavior. The resolution limit of the simulations is seen by the small peaks at $L\sim 2/N$. The length distribution is otherwise independent of resolution, $N$. Comparison to the SP (solid, gray line) and CA (dashed, gray line) shows good general agreement.

Figure 5

Average loop length multiplied by the number of harmonics in the parent loop versus generation. This weighted length asymptotes to $\sim 0.33$ for all initial loops.

Figure 6

The number of kinks on stable loops. Each loop must have at least 2 kinks. Most of the loops have between 2 and 5 kinks independent of resolution and the number of harmonic modes.

Figure 7

The velocity distribution of stable loops for various resolutions and numbers of modes. The distribution is universal and sharply peaked at $v\sim 1$. Also shown are the SP (solid, gray line) and CA (dashed, gray line) for comparison. The resolution of SP did not allow them to produce high velocity loops. Our results refine the peak at high velocity found by CA.

Figure 8

The average of the eigenvalues of the $\mathit{P}$ and $\mathit{Q}$ “moment of inertia” tensors (6) versus loop generation. Since both $\mathit{P}$ and $\mathit{Q}$ have the same properties, both are included in the average to increase the statistics. We notice that the smaller eigenvalue approaches zero and the largest eigenvalue is much larger than the middle eigenvalue showing that the $\mathit{P}$ and $\mathit{Q}$ curves are localized on the KT sphere. This plot is for $N=10000$, $M=10$. The asymptotic behavior is independent of resolution and the number of modes.

Figure 9

The kink sharpness distribution [see Eq. (7)]. The distribution is independent of resolution and the number of modes on the initial string. We see that the distribution is sharply peaked at 1, that is, where the $\mathit{P}$ before and after the kink are antiparallel.

Figure 10

The distribution of $⟨(\mathit{P}·\mathit{Q}{)}^2⟩$ showing that the left and right movers become more orthogonal to each other with more fragmentation.

Figure 11

The average number of cusps on each stable loop. On average we see that there is approximately a 40% chance of a stable loop to contain a cusp. This result is independent of resolution and the number of modes on the initial loop.

Figure 12

The velocity of stable loops versus the length of the loop. The $x$ error bars show the width of the length bins employed, and the $y$ error bars show the 95 percentile ranges in each length bin. The square symbol shows the average in the bin. Short loops have center of mass velocity $\sim 1/\sqrt{2}$, which is the root-mean-squared velocity of the string in the initial loop. Longer loops tend to have a lower velocity consistent with the fact that the initial loop is at rest.

Figure 13

Similar to Fig. 12 now for the number of kinks on stable loops versus the length of the loop. Longer loops tend to have a larger number of kinks though the variation is slight. The loop with the maximum number of kinks has 8 kinks instead of the canonical 4.

Figure 14

Similar to Fig. 12 now for the average number of cusps on stable loops versus the length of the loop. Longer loops typically contain more cusps on average. The loop with the maximum number of cusps has only 3 cusps.

Figure 15

Similar to Fig. 12 now for the eigenvalues of the moment of inertia tensors of the stable loops versus the length. Longer loops are also planar and only slightly less so than smaller loops.

Figure 16

Snapshots during the evolution of the degenerate kinky loop. Note that the loop periodically collapses to a double line.

Figure 17

The $\mathit{p}$ and $\mathit{q}$ curves for a perturbed degenerate kinky loop (10) plotted on the KT sphere. The $\mathit{p}$ curve is represented by the two points at the poles. The $\mathit{q}$ curve is represented by the arcs along the equator.

Figure 18

The $\mathit{P}$ and $-\mathit{Q}$ curves plotted on the KT sphere for a stable loop from our simulations. This loop contains three kinks and one cusp. The two curves are represented by the dots. The $\mathit{Q}$ curve is the dots along the equator. The $\mathit{P}$ curve is the dots perpendicular to the equator (running vertically on the left of the sphere) and includes the lone antipodal point (on the right of the sphere). Note that the points on the sphere are not weighted equally and the centroids of both $\mathit{P}$ and $\mathit{Q}$ curves are at the center of the sphere.

Figure 19

Geometric splitting of a loop. Shown in this figure are the ${\sigma }_{\pm }$ grids for various loops. The diagonal lines are lines of constant ${\sigma }_{-}$ (up and to the right) and constant ${\sigma }_{+}$ (up and to the left). Time runs upward and $\sigma$ rightward. The left and right edges of the grids are identified to describe a loop of string. The initial loop is shown on the top with the intersection occurring at points $a$ and $b$. The dashed lines show the constant values of ${\sigma }_{\pm }$ at which the intersection points occur and so at which kinks are formed. The daughter loops are formed by cutting the grid and identifying the new edges. In the figure the gray shaded region is cut from the grid and becomes one loop whereas the remaining unshaded pieces are connected. The new loops are shown below the initial loop. The past light cone of the intersection point has been removed since it no longer contains useful information. The two intersecting diamonds are replaced by the three shaded diamonds in the figure.