Gravitational Wave Background from Metastable Current-Carrying Cosmic Strings

Author(s)

Rybak, Ivan, Maji, Rinku, Shafi, Qaisar

Abstract

We study the stochastic gravitational wave background (SGWB) generated by networks of current-carrying metastable cosmic strings (MSCSs). Extending previous studies, we include the possibility of vorton formation phenomenologically. We construct an analytical template that provides a fast and accurate approximation to the spectrum in the frequency range probed by pulsar timing arrays (PTAs). We also show how the parameters of cosmic string networks determine the low-frequency slope. For the frequency range relevant for LIGO-Virgo-KAGRA (LVK), we derive an analytical expression that constrains the scale-invariant part of the SGWB. Finally, we confront the model with current constraints from PTAs and the LVK collaboration. We identify regions of the parameter space that can produce a signal consistent with PTA observations but are in tension with LVK constraints, suggesting that additional mechanisms to suppress the high-frequency spectrum are required for a consistent interpretation of the PTA signal

Figures

Scaling values of the characteristic length parameter $\zeta$ and the RMS velocity $v$ as functions of the charge amplitude $Y$, obtained from the stationary solutions of Eq.~\eqref{AlgebrEq}.
Caption Scaling values of the characteristic length parameter $\zeta$ and the RMS velocity $v$ as functions of the charge amplitude $Y$, obtained from the stationary solutions of Eq.~\eqref{AlgebrEq}.
SGWB produced by MSCSs without current, illustrating the impact of varying the model parameters. In particular, the figure shows the effect of varying the loop size, string tension, and decay parameter $\kappa_d$, demonstrating how these parameters modify the low-frequency spectral slope of the SGWB, with spectral index $q=4/3$, $\mathcal{F}=0.1$ and $\alpha_T \equiv \zeta \alpha$
Caption SGWB produced by MSCSs without current, illustrating the impact of varying the model parameters. In particular, the figure shows the effect of varying the loop size, string tension, and decay parameter $\kappa_d$, demonstrating how these parameters modify the low-frequency spectral slope of the SGWB, with spectral index $q=4/3$, $\mathcal{F}=0.1$ and $\alpha_T \equiv \zeta \alpha$
SGWB spectra from metastable current-carrying cosmic strings for $\sqrt{\kappa_d}=8$, $\mathcal{F}=0.1$, and $\alpha_T=0.1$, where $\alpha_T \equiv \zeta \alpha$ is a commonly used parameterization \cite{Blanco-Pillado:2013qja}. The spectra are shown for different values of the string tension, charge amplitude $Y$, vector coupling $e^2$, and $P_{\rm vort}$. The dashed lines indicate the power laws $f^3$ and $f^2$. Compared with the standard MSCSs, current-carrying strings exhibit both a shift in the characteristic frequency and a change in the spectral amplitude. For the parameters considered here, vorton formation predominantly affects the high-frequency part of the spectrum, while leaving the low-frequency behavior nearly unchanged. Following \cite{Rybak:2024our}, we take the spectral index to be $q=4/3$ for $Y=0$ and $q=5/3$ for $Y\neq 0$.
Caption SGWB spectra from metastable current-carrying cosmic strings for $\sqrt{\kappa_d}=8$, $\mathcal{F}=0.1$, and $\alpha_T=0.1$, where $\alpha_T \equiv \zeta \alpha$ is a commonly used parameterization \cite{Blanco-Pillado:2013qja}. The spectra are shown for different values of the string tension, charge amplitude $Y$, vector coupling $e^2$, and $P_{\rm vort}$. The dashed lines indicate the power laws $f^3$ and $f^2$. Compared with the standard MSCSs, current-carrying strings exhibit both a shift in the characteristic frequency and a change in the spectral amplitude. For the parameters considered here, vorton formation predominantly affects the high-frequency part of the spectrum, while leaving the low-frequency behavior nearly unchanged. Following \cite{Rybak:2024our}, we take the spectral index to be $q=4/3$ for $Y=0$ and $q=5/3$ for $Y\neq 0$.
Comparison between the analytical approximation and the full numerical calculation of the SGWB produced by metastable current-carrying cosmic strings. The fundamental mode is shown for $Y=0.5$, $\sqrt{\kappa_d}=8$, $\alpha_T=10^{-3}$, $\mathcal{F}=0.1$, and $e^2=0$, for two different values of $G\mu$. The two characteristic regimes with slopes $f^2$ and $f^3$ are visible, with the transition between them determined by Eq.~\eqref{FrChar}.
Caption Comparison between the analytical approximation and the full numerical calculation of the SGWB produced by metastable current-carrying cosmic strings. The fundamental mode is shown for $Y=0.5$, $\sqrt{\kappa_d}=8$, $\alpha_T=10^{-3}$, $\mathcal{F}=0.1$, and $e^2=0$, for two different values of $G\mu$. The two characteristic regimes with slopes $f^2$ and $f^3$ are visible, with the transition between them determined by Eq.~\eqref{FrChar}.
\textit{Upper panel:} Comparison between the analytical approximation and the full numerical calculation of the SGWB produced by current-carrying MSCSs. The numerical calculation includes $10^5$ terms, while the analytical approximation includes $10^3$ terms, using the following cosmic string parameters: $Y=0.3$, $\sqrt{\kappa_d}=8$, $G \mu = 10^{-7}$, $\mathcal{F}=0.1$, and $e^2=0$. The approximation reproduces the characteristic $f^2$ and $f^3$ slopes, together with the changes in the relativistic degrees of freedom that create small bumps. \\ \textit{Lower panel:} Comparison between the full numerical spectrum and the analytical approximation with different numbers of terms in the summation for standard cosmic strings with cusps, corresponding to $q=4/3$, $G\mu=10^{-7}$, $\sqrt{\kappa_d}=8$, $\mathcal{F}=0.1$ and $\alpha_T = 0.1$. As the number of terms increases, the analytical approximation converges toward the full numerical result. The convergence is slower for cusps ($q=4/3$) than for kinks ($q=5/3$), requiring more terms to achieve the same level of accuracy.
Caption \textit{Upper panel:} Comparison between the analytical approximation and the full numerical calculation of the SGWB produced by current-carrying MSCSs. The numerical calculation includes $10^5$ terms, while the analytical approximation includes $10^3$ terms, using the following cosmic string parameters: $Y=0.3$, $\sqrt{\kappa_d}=8$, $G \mu = 10^{-7}$, $\mathcal{F}=0.1$, and $e^2=0$. The approximation reproduces the characteristic $f^2$ and $f^3$ slopes, together with the changes in the relativistic degrees of freedom that create small bumps. \\ \textit{Lower panel:} Comparison between the full numerical spectrum and the analytical approximation with different numbers of terms in the summation for standard cosmic strings with cusps, corresponding to $q=4/3$, $G\mu=10^{-7}$, $\sqrt{\kappa_d}=8$, $\mathcal{F}=0.1$ and $\alpha_T = 0.1$. As the number of terms increases, the analytical approximation converges toward the full numerical result. The convergence is slower for cusps ($q=4/3$) than for kinks ($q=5/3$), requiring more terms to achieve the same level of accuracy.
\textit{Upper panel:} Comparison between the analytical approximation and the full numerical calculation of the SGWB produced by current-carrying MSCSs. The numerical calculation includes $10^5$ terms, while the analytical approximation includes $10^3$ terms, using the following cosmic string parameters: $Y=0.3$, $\sqrt{\kappa_d}=8$, $G \mu = 10^{-7}$, $\mathcal{F}=0.1$, and $e^2=0$. The approximation reproduces the characteristic $f^2$ and $f^3$ slopes, together with the changes in the relativistic degrees of freedom that create small bumps. \\ \textit{Lower panel:} Comparison between the full numerical spectrum and the analytical approximation with different numbers of terms in the summation for standard cosmic strings with cusps, corresponding to $q=4/3$, $G\mu=10^{-7}$, $\sqrt{\kappa_d}=8$, $\mathcal{F}=0.1$ and $\alpha_T = 0.1$. As the number of terms increases, the analytical approximation converges toward the full numerical result. The convergence is slower for cusps ($q=4/3$) than for kinks ($q=5/3$), requiring more terms to achieve the same level of accuracy.
Caption \textit{Upper panel:} Comparison between the analytical approximation and the full numerical calculation of the SGWB produced by current-carrying MSCSs. The numerical calculation includes $10^5$ terms, while the analytical approximation includes $10^3$ terms, using the following cosmic string parameters: $Y=0.3$, $\sqrt{\kappa_d}=8$, $G \mu = 10^{-7}$, $\mathcal{F}=0.1$, and $e^2=0$. The approximation reproduces the characteristic $f^2$ and $f^3$ slopes, together with the changes in the relativistic degrees of freedom that create small bumps. \\ \textit{Lower panel:} Comparison between the full numerical spectrum and the analytical approximation with different numbers of terms in the summation for standard cosmic strings with cusps, corresponding to $q=4/3$, $G\mu=10^{-7}$, $\sqrt{\kappa_d}=8$, $\mathcal{F}=0.1$ and $\alpha_T = 0.1$. As the number of terms increases, the analytical approximation converges toward the full numerical result. The convergence is slower for cusps ($q=4/3$) than for kinks ($q=5/3$), requiring more terms to achieve the same level of accuracy.
Results of the analysis with \texttt{PTArcade} for standard MSCSs, with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$ and $\log(G\mu) \in \left[-10,-4\right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_std_Constr}.
Caption Results of the analysis with \texttt{PTArcade} for standard MSCSs, with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$ and $\log(G\mu) \in \left[-10,-4\right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_std_Constr}.
Constraints imposed by the LVK collaboration \cite{LIGOScientific:2025bgj} on cosmic string models producing a scale-invariant SGWB at $f=25 \,{\rm Hz}$, shown for different values of the current amplitude $Y$ and the vorton parameter $P_{\rm vort}$.
Caption Constraints imposed by the LVK collaboration \cite{LIGOScientific:2025bgj} on cosmic string models producing a scale-invariant SGWB at $f=25 \,{\rm Hz}$, shown for different values of the current amplitude $Y$ and the vorton parameter $P_{\rm vort}$.
Results of the analysis with \texttt{PTArcade} for current-carrying MSCSs, with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $Y \in \left[ 0, 0.99 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Caption Results of the analysis with \texttt{PTArcade} for current-carrying MSCSs, with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $Y \in \left[ 0, 0.99 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Results of the analysis with \texttt{PTArcade} for current-carrying MSCSs with vortons $P_{\rm vort}=1$, with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $Y \in \left[ 0, 0.99 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Caption Results of the analysis with \texttt{PTArcade} for current-carrying MSCSs with vortons $P_{\rm vort}=1$, with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $Y \in \left[ 0, 0.99 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Spectra for the best-fit cases of standard and current-carrying MSCSs with and without vortons. The standard MSCS spectrum is represented by the solid line for $\log (G\mu) = -5.873$ and $\sqrt{\kappa_d} = 7.943$. The spectrum of current-carrying MSCSs without vortons is shown by the dash-dotted line for $\log (G\mu) = -6.230$, $\sqrt{\kappa_d} = 8.089$, and $Y=0.543$. The spectrum of current-carrying MSCSs with vortons is shown by the dashed line for $\log (G\mu) = -5.860$, $\sqrt{\kappa_d} = 8.037$, $Y=0.472$, and $P_{\rm vort}=1$. We also show the power-law integrated sensitivity curves \cite{Thrane:2013oya, Schmitz:2020syl} for several proposed experiments, namely SKA \cite{Janssen:2014dka}, LISA \cite{Bartolo:2016ami}, DECIGO \cite{Sato_2017}, BBO \cite{Crowder:2005nr, Corbin:2005ny}, HLVK \cite{KAGRA:2013rdx}, ET \cite{Mentasti:2020yyd}, and CE \cite{Regimbau:2016ike}, as well as bounds from big bang nucleosynthesis (BBN) \cite{Maggiore:1999vm}, LVK Run 4a (LV4a) \cite{LIGOScientific:2025bgj, LIGOScientific:2025kry}, and NANOGrav 15-year (NG15) data.
Caption Spectra for the best-fit cases of standard and current-carrying MSCSs with and without vortons. The standard MSCS spectrum is represented by the solid line for $\log (G\mu) = -5.873$ and $\sqrt{\kappa_d} = 7.943$. The spectrum of current-carrying MSCSs without vortons is shown by the dash-dotted line for $\log (G\mu) = -6.230$, $\sqrt{\kappa_d} = 8.089$, and $Y=0.543$. The spectrum of current-carrying MSCSs with vortons is shown by the dashed line for $\log (G\mu) = -5.860$, $\sqrt{\kappa_d} = 8.037$, $Y=0.472$, and $P_{\rm vort}=1$. We also show the power-law integrated sensitivity curves \cite{Thrane:2013oya, Schmitz:2020syl} for several proposed experiments, namely SKA \cite{Janssen:2014dka}, LISA \cite{Bartolo:2016ami}, DECIGO \cite{Sato_2017}, BBO \cite{Crowder:2005nr, Corbin:2005ny}, HLVK \cite{KAGRA:2013rdx}, ET \cite{Mentasti:2020yyd}, and CE \cite{Regimbau:2016ike}, as well as bounds from big bang nucleosynthesis (BBN) \cite{Maggiore:1999vm}, LVK Run 4a (LV4a) \cite{LIGOScientific:2025bgj, LIGOScientific:2025kry}, and NANOGrav 15-year (NG15) data.
Vorton abundance $\Omega_{\rm vort}$ for current-carrying MSCSs, as given by Eq.~\eqref{VortAbund}. The left upper panel shows the vorton abundance for $Y=0.8$ and $P_{\rm vort}=1$ for different values of $\kappa_d$ and $G\mu$. The right upper panel shows the vorton abundance for $\sqrt{\kappa_d}=7$ and $P_{\rm vort}=1$ for different values of $G\mu$ and $Y$. The lower panel shows the vorton abundance for fixed $Y=0.1$.
Caption Vorton abundance $\Omega_{\rm vort}$ for current-carrying MSCSs, as given by Eq.~\eqref{VortAbund}. The left upper panel shows the vorton abundance for $Y=0.8$ and $P_{\rm vort}=1$ for different values of $\kappa_d$ and $G\mu$. The right upper panel shows the vorton abundance for $\sqrt{\kappa_d}=7$ and $P_{\rm vort}=1$ for different values of $G\mu$ and $Y$. The lower panel shows the vorton abundance for fixed $Y=0.1$.
Vorton abundance $\Omega_{\rm vort}$ for current-carrying MSCSs, as given by Eq.~\eqref{VortAbund}. The left upper panel shows the vorton abundance for $Y=0.8$ and $P_{\rm vort}=1$ for different values of $\kappa_d$ and $G\mu$. The right upper panel shows the vorton abundance for $\sqrt{\kappa_d}=7$ and $P_{\rm vort}=1$ for different values of $G\mu$ and $Y$. The lower panel shows the vorton abundance for fixed $Y=0.1$.
Caption Vorton abundance $\Omega_{\rm vort}$ for current-carrying MSCSs, as given by Eq.~\eqref{VortAbund}. The left upper panel shows the vorton abundance for $Y=0.8$ and $P_{\rm vort}=1$ for different values of $\kappa_d$ and $G\mu$. The right upper panel shows the vorton abundance for $\sqrt{\kappa_d}=7$ and $P_{\rm vort}=1$ for different values of $G\mu$ and $Y$. The lower panel shows the vorton abundance for fixed $Y=0.1$.
Results of the analysis with \texttt{PTArcade} for MSCSs with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $\log(\alpha) \in \left[ -3, 0 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Caption Results of the analysis with \texttt{PTArcade} for MSCSs with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $\log(\alpha) \in \left[ -3, 0 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Results of the analysis with \texttt{PTArcade} for MSCSs with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $\log(\mathcal{F}) \in \left[ -5, 0 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Caption Results of the analysis with \texttt{PTArcade} for MSCSs with uniform prior distributions $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$ and $\log(\mathcal{F}) \in \left[ -5, 0 \right]$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Results of the analysis with \texttt{PTArcade} for MSCSs with a current and nonzero coupling $e^2$ to the associated field. We adopt uniform priors on $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$, and $\log(e^2) \in \left[ -8, -1 \right]$ with $Y$ fixed to $0.1$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
Caption Results of the analysis with \texttt{PTArcade} for MSCSs with a current and nonzero coupling $e^2$ to the associated field. We adopt uniform priors on $\sqrt{\kappa_d} \in \left[7,9\right]$, $\log(G\mu) \in \left[-10,-4\right]$, and $\log(e^2) \in \left[ -8, -1 \right]$ with $Y$ fixed to $0.1$. The LVK constraint at 95\% credibility, $\Omega_{\rm gw}(25\,{\rm Hz}) < 2.8 \times 10^{-9}$ \cite{LIGOScientific:2025bgj}, is shown by the dashed line and is given by \eqref{LVK_Y_Constr}.
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