Evolution of semilocal string networks. II. Velocity estimators

We continue a comprehensive numerical study of semilocal string networks and their cosmological evolution. These can be thought of as hybrid networks comprised of (nontopological) string segments, whose core structure is similar to that of Abelian Higgs vortices, and whose ends have long–range interactions and behavior similar to that of global monopoles. Our study provides further evidence of a linear scaling regime, already reported in previous studies, for the typical length scale and velocity of the network. We introduce a new algorithm to identify the position of the segment cores. This allows us to determine the length and velocity of each individual segment and follow their evolution in time. We study the statistical distribution of segment lengths and velocities for radiation- and matter-dominated evolution in the regime where the strings are stable. Our segment detection algorithm gives higher length values than previous studies based on indirect detection methods. The statistical distribution shows no evidence of (anti)correlation between the speed and the length of the segments.

Figure 1

In the left panel, the string segments in a typical simulation box are shown; the blue points are points with a magnetic field higher than the threshold described in the text. The right panel zooms in on a portion of one segment. The blue dots correspond again to points with high magnetic field, and the red, cyan and black points correspond to points where the windings of the scalar fields ${\varphi }_1$, ${\varphi }_2$ or both (respectively) are nontrivial. Notice how these dots are not nicely aligned in a smooth curve, but jitter around (this is the Manhattan effect). Performing the smoothing over four nearest neighbors on each side of each point yields the black line, which is the smoothed version of the center of the core of the segment. Actually, the black line is also formed of points, but here those points have been joined by a line to show how they interpolate between the lattice points with winding.

Figure 2

In this figure the relation between the two length-measuring approaches is shown. The horizontal axis has the lengths measured with the magnetic flux approach, while the vertical axis has the lengths measured with the winding approach. Every diamond represents a segment in a simulation box for $\beta =0.20$ at time $\tau =256$. It is clear that the proportionality is almost the same for all the segments in the network. The diagonal is shown with a dashed line, emphasizing the bias factor between the two approaches.

Figure 3

In this figure one can see the box after the characterization. The black lines represent the string segments detected using the windings, where each point is smoothed using the four nearest neighbors in each direction. The red dots represent the antimonopoles and the blue ones the monopoles.

Figure 4

In the left panel the windings of both fields are shown, without taking into account the magnetic field: blue for windings of ${\varphi }_1$, green for windings in ${\varphi }_2$ and red for windings in both. As one can see each one of the fields creates a network of closed or infinite strings. On the right panel, we plot the same windings, but also the regions where there is concentration of magnetic field (orange). This figure shows that semilocal strings are actually the regions where both fields wind.

Figure 5

In this figure we show the merging of two string segments, in what our algorithm would report as ultrarelativistic velocities. Initially the separation between both string-ends is $d=30$ and in less than $\mathrm{\Delta }\tau =10$ both segments have merged. This process can be understood as small string segments being formed both close to the string ends, as well as in between both strings (depicted by those single points in between strings).

Figure 6

This figure shows that our simulations reach a scaling regime for all $\beta$. The left panel shows the case of radiation domination, the right panel the case of matter domination.

Figure 7

Scalings in total string length and monopole number for $\beta =0.15$ in radiation era (left) and for $\beta =0.35$ in matter era (right).

Figure 8

This plot shows the distribution of segments with respect to velocity for simulation in radiation domination and $\beta =0.04$ for all seven simulations. Each point represents a segment in the network, where in the x-axis the length of the segment divided by time is shown and its velocity in the y-axis. This is the case where segments flow through the network, i.e., they do interact with any other segment in the next time step.

Figure 9

This figure is similar to Fig. 8, but in this case the segments that are plotted are those that merge with other segments before the next time step.

Figure 10

These histograms show the distribution of the segments during the network evolution for $\beta =0.04$ in the radiation dominated era for all seven simulations, where the segments are binned in 10 bins with uniform width. The top panels show the fractional distribution of the number of segments $n$ with their lengths divided by time, whereas the bottom panels show the analogous distribution for the total lengths divided by time $L_n/\tau$. The colors represent different types of segments, depending on their future behaviour: in blue segments which are flowing, in green segments that are merging and in yellow segments that are collapsing before the next time-step. We write arrows to remark that in those instances, there are a few (one or two) segments in that bin, which are hard to see in the top panels, but can be seen in the bottom ones. Note that in the last time step we have no information whether the segments will flow, merge or collapse, so we just choose to show them as flow segments.

Figure 11

These histograms show the velocity $v$ distribution of the segments (top) and monopoles (bottom) during their evolution, for radiation and $\beta =0.04$. The velocities are binned in 10 bins with uniform width. The color code is analogous to that of Fig. 10: blue corresponds to strings that do not interact with other strings before the next time step (flow), green is for strings which merge with other segments and yellow is for strings that disappear before the next time step because the segment collapses. Note that in the last time step of segment velocities we have no information whether the segments will flow, merge or collapse, so we just choose to show them as flow segments. Note also that unlike in the segment case, where the velocity has been calculated forward and backward, for monopoles we have only calculated the velocity forward, and therefore we do not have information to compute the velocity at the last time step.

Figure 12

The “family tree” of a segment for $\beta =0.04$ in radiation. The number inside the box denotes the length of each segment, and time runs upwards.