New gravitational-wave templates for metastable cosmic strings: Loop breaking versus network collapse

Metastable cosmic strings are a common prediction of grand unified theories and act as a source of a gravitational-wave background (GWB) that can explain the 2023 pulsar timing array (PTA) signal. In this paper, we revisit the GWB signal from metastable strings, emphasizing the need to carefully distinguish between two different time scales: (i) t_LB, the time scale of loop breaking because of spontaneous monopole nucleation on closed string loops, and (ii) t_NC, the time scale of network collapse when string segments attached to monopoles begin to enter the Hubble horizon. We discuss under which conditions these two time scales are similar or far apart from each other and illustrate the resulting consequences for the GWB signal. In doing so, we generalize the description of the GWB signal from metastable strings to a three-parameter model in terms of the string tension G\mu and the time scales t_LB and t_NC, which allows us to unify the modeling of standard metastable strings with what is known as quasi-stable strings. In the limit of a large t_LB/t_NC ratio, we, moreover, derive a compact analytical expression for the predicted GWB spectrum in excellent agreement with numerical results in the literature. We thus conclude that our new templates for the GWB spectrum from metastable strings can be readily used in the analysis of future PTA data sets.