A window for cosmic strings

Schematic trajectories of non self-intersecting loops in the $(\tau,\ell)$ phase space (orange). The blue region corresponds to the production of loops, that is the green line is $\gbr t$, and the blue one is $\gamma_\infty t$. These delimit the $\Theta$-function shown in the loop production function of \cref{eq: def polchinski rocha production}. There are three distinct possibilities for the ordering of $\tstar$, $\tau_0$ and $\tbr$, which appear in the boundaries of \cref{eq: final result n}. This depends on whether the loop had length $\ell_0$ before (\textit{top left}), during (\textit{top right}) or after (\textit{bottom}) loop production.

Schematic trajectories of non self-intersecting loops in the $(\tau,\ell)$ phase space (orange). The blue region corresponds to the production of loops, that is the green line is $\gbr t$, and the blue one is $\gamma_\infty t$. These delimit the $\Theta$-function shown in the loop production function of \cref{eq: def polchinski rocha production}. There are three distinct possibilities for the ordering of $\tstar$, $\tau_0$ and $\tbr$, which appear in the boundaries of \cref{eq: final result n}. This depends on whether the loop had length $\ell_0$ before (\textit{top left}), during (\textit{top right}) or after (\textit{bottom}) loop production.

Schematic trajectories of non self-intersecting loops in the $(\tau,\ell)$ phase space (orange). The blue region corresponds to the production of loops, that is the green line is $\gbr t$, and the blue one is $\gamma_\infty t$. These delimit the $\Theta$-function shown in the loop production function of \cref{eq: def polchinski rocha production}. There are three distinct possibilities for the ordering of $\tstar$, $\tau_0$ and $\tbr$, which appear in the boundaries of \cref{eq: final result n}. This depends on whether the loop had length $\ell_0$ before (\textit{top left}), during (\textit{top right}) or after (\textit{bottom}) loop production.

Loop number density for model A with kinks (left panel) and with cusps (right panel) for $G\mu = 10^{-13}$. Solid colored lines show the density for redshifts, from top to bottom $z = 10^6, 10^7, 10^8, 10^9, 10^{10}, 10^{11}, 10^{12}, 10^{13}, 10^{14}$ and $ 10^{15}$. The dashed dark line shows the scaling distribution when one assumes that all the energy goes into gravitational waves.

Loop number density for model A with kinks (left panel) and with cusps (right panel) for $G\mu = 10^{-13}$. Solid colored lines show the density for redshifts, from top to bottom $z = 10^6, 10^7, 10^8, 10^9, 10^{10}, 10^{11}, 10^{12}, 10^{13}, 10^{14}$ and $ 10^{15}$. The dashed dark line shows the scaling distribution when one assumes that all the energy goes into gravitational waves.

Loop number density for model B with kinks (left panel) and with cusps (right panel) for $G\mu = 10^{-13}$. Solid colored lines show the density for redshifts, from top to bottom $z = 10^6, 10^7, 10^8, 10^9, 10^{10}, 10^{11}, 10^{12}, 10^{13}, 10^{14}$ and $ 10^{15}$. The dashed dark line shows the scaling distribution when one assumes that all the energy goes into gravitational waves.

Loop number density for model B with kinks (left panel) and with cusps (right panel) for $G\mu = 10^{-13}$. Solid colored lines show the density for redshifts, from top to bottom $z = 10^6, 10^7, 10^8, 10^9, 10^{10}, 10^{11}, 10^{12}, 10^{13}, 10^{14}$ and $ 10^{15}$. The dashed dark line shows the scaling distribution when one assumes that all the energy goes into gravitational waves.

Stochastic background of gravitational waves for different $G\mu$. Top panel: Model A, lower panel: Model B. Solid line assume that cusps on the loops emit particles ($n = 1/2$). Dashed lines assume that all the emitted energy goes into gravitational waves.

Stochastic background of gravitational waves for different $G\mu$. Top panel: Model A, lower panel: Model B. Solid line assume that cusps on the loops emit particles ($n = 1/2$). Dashed lines assume that all the emitted energy goes into gravitational waves.

Diffuse $\gamma$-ray background in the presence of only cusps (blue) and only kinks (green) for $f_\mathrm{eff} = 1$ (solid lines) and $f_\mathrm{eff} = 10^{-3}$ (dashed lines). Left panel: Model A, right panel: Model B.

Diffuse $\gamma$-ray background in the presence of only cusps (blue) and only kinks (green) for $f_\mathrm{eff} = 1$ (solid lines) and $f_\mathrm{eff} = 10^{-3}$ (dashed lines). Left panel: Model A, right panel: Model B.

Left panel: One of the two cusps for a loop with $\alpha=3/4$ and an artificial width in the rest frame of the loop. The tip of the cusp goes at the speed of light in the direction of $\vb{\dot{X}}_0$. Right panel: a slice of the cusp along the plane parallel to $\vec{X}_0^{\prime\prime}$. The two branches of the cusp are separated by a distance $x_m$. The ellipses of the two branches are tilted with angle $\theta$.

Left panel: One of the two cusps for a loop with $\alpha=3/4$ and an artificial width in the rest frame of the loop. The tip of the cusp goes at the speed of light in the direction of $\vb{\dot{X}}_0$. Right panel: a slice of the cusp along the plane parallel to $\vec{X}_0^{\prime\prime}$. The two branches of the cusp are separated by a distance $x_m$. The ellipses of the two branches are tilted with angle $\theta$.