Scaling from gauge and scalar radiation in Abelian-Higgs string networks

We investigate cosmic string networks in the Abelian Higgs model using data from a campaign of large-scale numerical simulations on lattices of up to $4096^3$ grid points. We observe scaling or self-similarity of the networks over a wide range of scales and estimate the asymptotic values of the mean string separation in horizon length units $\dot{\xi }$ and of the mean square string velocity ${\overline{v}}^2$ in the continuum and large time limits. The scaling occurs because the strings lose energy into classical radiation of the scalar and gauge fields of the Abelian Higgs model. We quantify the energy loss with a dimensionless radiative efficiency parameter and show that it does not vary significantly with lattice spacing or string separation. This implies that the radiative energy loss underlying the scaling behavior is not a lattice artifact, and justifies the extrapolation of measured network properties to large times for computations of cosmological perturbations. We also show that the core growth method, which increases the defect core width with time to extend the dynamic range of simulations, does not introduce significant systematic error. We compare $\dot{\xi }$ and ${\overline{v}}^2$ to values measured in simulations using the Nambu-Goto approximation, finding that the latter underestimate the mean string separation by about 25%, and overestimate ${\overline{v}}^2$ by about 10%. The scaling of the string separation implies that string loops decay by the emission of massive radiation within a Hubble time in field theory simulations, in contrast to the Nambu-Goto scenario which neglects this energy loss mechanism. String loops surviving for only one Hubble time emit much less gravitational radiation than in the Nambu-Goto scenario and are consequently subject to much weaker gravitational wave constraints on their tension.

Figure 1

Slice through the Nielsen-Olesen vortex, showing total energy density $\rho$, in a $64\times 64$ region with $\mathrm{\Delta }x=0.25$. The total energy density is equal to the negative of the Lagrangian density and the negative of the diagonal component orthogonal to the plane, $T_{33}$.

Figure 2

Variables for computing winding of a lattice plaquette.

Figure 3

Energy isosurfaces of a standing wave configuration with initial amplitude parameter $a_x=0.2$, in a box of $N=64$ sites on a side, with lattice spacing $\mathrm{\Delta }x=0.5$. The dark squares show the positions of the plaquettes initialized with flux $2\pi$. The isosurfaces are at energy densities of 0.01, 0.1 and 1 in units where ${\varphi }_0=1$.

Figure 4

Cross section through a $64^3$ lattice containing a string standing wave, showing both the physical string and the Dirac string. The scalar field ${\varphi }_{\mathbf{x}}$ is shown as a black line at each lattice site, representing its amplitude and complex argument, with blue lines joining them proportional to the value of the link variable ${\theta }_{i,\mathbf{x}}$. The modulus of the scalar field is shown as a smoothed color field. Plaquettes with nonzero winding [see Eq. (84)] are at $(x,y)=(34,32)$ and $(x,y)=(28,32)$. The first is the physical string, the second the Dirac string.

Figure 5

Plots of string length estimators against time for a standing wave with initial amplitude parameter $a_x=0.2$, in a box with $L_{\mathrm{b}}=32$, with lattice spacing (from top to bottom) $\mathrm{\Delta }x=0.5$, $\mathrm{\Delta }x=0.25$ and $\mathrm{\Delta }x=0.125$. The length estimators are ${\ell }_{\mathcal{L}}$ [black, Eq. (74)], ${\ell }_{B,\mathcal{L}}$ [red, Eq. (75)], and ${\ell }_{s,\mathcal{L}}$ [blue, Eq. (76)], The length estimators and simulation times are given in units where ${\varphi }_0=1$. The total energy is shown in dashed grey.

Figure 6

Plots of mean square velocity estimators against time for a standing wave in a box of size $L_{\mathrm{b}}=32$, with initial amplitude parameter $a_x=0.2$, and lattice spacing $\mathrm{\Delta }x=0.5$ (top), $\mathrm{\Delta }x=0.25$ (center) and $\mathrm{\Delta }x=0.125$ (bottom). The velocity estimators are ${\overline{v}}_{\mathcal{L}}^2$ [black, Eq. (78)], ${\overline{v}}_{B,\mathcal{L}}^2$ [red, Eq. (79)], ${\overline{v}}_{s,\mathcal{L}}^2$ [blue, Eq. (80)], and ${\overline{v}}_{w,\mathcal{L}}^2$ [green, Eq. (82)]. The velocities estimated from the string position as given from the winding (see Sec. 3e) is shown with open circles. The Nambu-Goto prediction is shown in dashed black. Note that ${\overline{v}}_{\mathcal{L}}^2$ and ${\overline{v}}_{s,\mathcal{L}}^2$ are almost identical. The lengths and times are given in units where ${\varphi }_0=1$.

Figure 7

Plots of ${\xi }_{\mathcal{L}}$ versus time for all network simulations in the radiation (top) and matter (bottom) eras. The error bands mark the $1\text{−}\sigma$ and $2\text{−}\sigma$ variations between realizations. Simulations with $s=0$ are marked in red, $s=1$ in blue. The string separation parameter and simulation times are given in units where ${\varphi }_0=1$.

Figure 8

Plots of $\dot{\xi }$ versus $r_{\mathrm{s}}/\xi$ for all simulations, for radiation (top) and matter era (bottom) simulations. Two time ranges are plotted, detailed in Table 5, one around half way through the simulation (open symbols), and one near the end (filled symbols). Circles in blue are $s=1$, while circles in red have $s=0$. Black symbols are the resolution test runs with lattice spacing 0.5 (circle), 0.25 (triangle) and 0.125 (square). Dashed lines show linear fits to the $s=1$ data and $s=0$ data separately, with the same color code.

Figure 9

Plots of mean square velocity ${\overline{v}}^2$ versus time for all network simulations in the radiation (top) and matter (bottom) eras. Simulations with $s=0$ are marked in red, $s=1$ in blue. Solid lines are the field estimator ${\overline{v}}_{\mathcal{L}}^2$, while velocities from string positions are marked with dashed lines. Simulations with lattice sizes $512^3$, $1024^3$, $2048^3$ and $4096^3$ end at approximate times 150, 300, 600 and 1100 respectively. The error bands mark the $1\text{−}\sigma$ and $2\text{−}\sigma$ variations between realizations. Note that there is velocity data for only one simulation with $s=0$ and $N=4096$ in the matter era. The simulation times are given in units where ${\varphi }_0=1$.

Figure 10

Mean square string velocity from the equation of state estimator ${\overline{v}}_{w,\mathcal{L}}^2$ in radiation (top) and matter (bottom) eras, plotted against lattice spacing $\mathrm{\Delta }x$ in units of the string width $r_{\mathrm{s}}$. The same color and symbol code as Fig. 8 is used.

Figure 11

Radiative efficiency parameter $\tilde{\epsilon }$ (91) against the ratio of lattice spacing to string width for radiation (top) and matter (bottom) eras. The same color and symbol code as Fig. 8 is used.

Figure 12

Plots of the radiative efficiency parameter $\tilde{\epsilon }$ (91) versus $r_{\mathrm{s}}/\xi$, in radiation (top) and matter (bottom) eras. The same color and symbol code as Fig. 8 is used. The mean values of $\tilde{\epsilon }$ are listed in Table 9.