This paper revisits how a network of metastable cosmic strings would produce a background of gravitational waves. The authors point out that two different time scales control the signal: the time when closed string loops break because magnetic monopoles appear on them (loop breaking, t_LB), and the time when finite string segments with monopoles at their ends start to fit inside the observable Hubble patch and the whole network collapses (network collapse, t_NC). Treating these times as separate leads to a wider variety of predicted gravitational-wave spectra than earlier models allowed. The authors package these possibilities into a three-parameter model based on the string tension Gμ and the two times t_LB and t_NC, and they derive an analytic formula for the spectrum in the limit where loop breaking happens much later than network collapse.

Cosmic strings are tube-like defects that can form in the early Universe in some extensions of particle physics, including certain grand unified theories (GUTs). Metastable strings are unstable because they can nucleate monopoles, a kind of point defect, and the loops of string that form during the network’s evolution radiate gravitational waves. Pulsar timing array (PTA) collaborations reported evidence in 2023 for a stochastic gravitational-wave background at nanohertz frequencies. That signal is not yet a >=5σ discovery (for example, recent NANOGrav data give evidence at about 3–4σ), and competing explanations exist, including supermassive black-hole binaries. Still, metastable strings remain a viable possibility that motivates better templates for data analysis.

In practical terms, the authors explain what t_LB and t_NC mean and why they need not be the same. t_NC marks when long string segments first become visible to local observers and the network stops producing new large loops. t_LB marks when closed loops break because monopoles nucleate on them. Previous work commonly set t_LB equal to t_NC. The paper shows that physical effects—such as enhanced breaking at finite temperature shortly after network formation or a small thermal abundance of monopoles—can make loop breaking start earlier or later than network collapse. To capture these possibilities the authors introduce a hierarchy parameter pκ that controls loop-breaking times and allow t_NC to vary independently.