Gravitational waves from current-carrying cosmic strings

Benchmark GW spectra emitted by current-carrying cosmic strings. For spectra (A), (C), (E), we assume a cosmic-string tension of $G\mu=0.5\times 10^{-10}$, while for spectra (B) and (E), we assume $G\mu=0.5\times 10^{-17}$. In scenarios (A) and (B), the current flowing on cosmic strings decays around the time of the QCD crossover, while in scenarios (C) and (D), it decays around the time of the electroweak crossover. In scenario (E), the current decays at even earlier times (see text for details). For each scenario, the lower and upper edges of the predicted spectrum respectively correspond to $k=1$ and $k=10^6$ harmonic string modes accounted for in the computation of the GW spectrum. We also show standard (\ie, no current) spectra for string networks with $G\mu=0.5\times 10^{-10}$ and $G\mu=0.5\times 10^{-17}$ (black lines), together with existing constraints and future sensitivities of present and planned GW experiments~\cite{Schmitz:2020syl}. All spectra for $G\mu=0.5\times 10^{-10}$ can explain the PTA signal at nHz frequencies.

Streamlines derived from the autonomous system of ordinary differential equations \eqref{eq:autonomous-alpha}, \eqref{eq:autonomous-v} and \eqref{eq:autonomous-y} for $\tilde{c}=0.23$ and $\nu=1/2$. All the trajectories start from the plane $\alpha = 0.27$ and eventually fall into one of the two different attractors, one at $Y=0$ and another one at $Y\sim 1$.

Two-dimensional slices of \cref{fig:streamtube} around the $Y\sim 1$ attractor.

Two-dimensional slices of \cref{fig:streamtube} around the $Y\sim 1$ attractor.

\emph{Left panel}: Evolution of the parameters $\alpha$, $v_\infty$ and $Y$ when the current is artificially sourced to $1$ at $t=10^{-18}$ s, for $\tilde{c} = 0.23$ and $\nu=1/2$. \emph{Right panel}: Evolution of $\alpha t^{-\delta}$, $v/\alpha$ and $\epsilon/\alpha^2$ that should be approximately time independent for a steady current attractor.

\emph{Left panel}: Evolution of the parameters $\alpha$, $v_\infty$ and $Y$ when the current is artificially sourced to $1$ at $t=10^{-18}$ s, for $\tilde{c} = 0.23$ and $\nu=1/2$. \emph{Right panel}: Evolution of $\alpha t^{-\delta}$, $v/\alpha$ and $\epsilon/\alpha^2$ that should be approximately time independent for a steady current attractor.

$t^4 n(\ell,t)$ for different times $t$ as a function of $x=\ell/t$. \emph{Left panel}: The black dashed line corresponds to the standard scaling. It can be seen that during the current-carrying phase more loops are produced at shorter lengths. In the calculation, we have taken $G\mu=10^{-10}$ and assumed that the transition starts at redshift $z=10^{15}$ and ends at $z=10^{9}$. \emph{Right panel}: $t^4 n(\ell,t)$ after the end of the current-carrying phase is shown. A gap develops in the loop distribution, see text for details.

$t^4 n(\ell,t)$ for different times $t$ as a function of $x=\ell/t$. \emph{Left panel}: The black dashed line corresponds to the standard scaling. It can be seen that during the current-carrying phase more loops are produced at shorter lengths. In the calculation, we have taken $G\mu=10^{-10}$ and assumed that the transition starts at redshift $z=10^{15}$ and ends at $z=10^{9}$. \emph{Right panel}: $t^4 n(\ell,t)$ after the end of the current-carrying phase is shown. A gap develops in the loop distribution, see text for details.

Energy density in small loops, $\rho_L$, gravitational waves, $\rho_\mathrm{GW}$, and long string network, $\rho_\infty$, normalized to the critical density $\rho_c \sim 1/Gt^2$, as a function of time after the onset of the current--carrying phase, $t_\text{ini}$.

GW spectrum as a function of the time at which the current appears, $t_{\mathrm{ini}}$, and disappears, $t_\text{end}$, from the network for a string tension of $G \mu = 5\times 10^{-11}$. The parameter region labeled LVK is disfavored by searches for the stochastic gravitational wave background at ground-based interferometers \cite{KAGRA:2021kbb}. We excluded GW signals with $h^2 \Omega_\mathrm{GW} > 5\times 10^{-9} h^2$ at $20$\,Hz corresponding to the limits set on a flat power spectrum with a log-uniform prior.