This paper studies a simple chain of symmetry breakings in particle physics that can create a sequence of cosmic objects: magnetic monopoles, then strings, and finally domain walls bounded by strings. When those walls collapse they can leave behind composite strings that carry the same magnetic flux as the original monopoles. All of these objects shake space and produce a background of gravitational waves. The authors calculate what that gravitational-wave signal would look like under different cosmological histories.

The authors focus on the breaking pattern SU(2) → U(1) → Z2 → 1. The first step makes ’t Hooft–Polyakov monopoles that carry two units of a U(1) magnetic flux. The second step produces strings that trap the flux in tubes. The final step produces domain walls that end on those strings — so-called walls bounded by strings (WBS). When a WBS collapses, regions with parallel string flux combine into a composite string carrying the full monopole flux. The same chain of topological objects can also appear in a related SU(3) → SO(3) → Z2 → 1 breaking.

At a conceptual level the gravitational-wave signal comes in stages. Initially, ordinary strings radiate as they evolve. Later the wall tension takes over and the WBS network collapses; that collapse produces a burst of radiation and leaves behind composite strings and loops. Two physical scales control the timing: the ratio of string tension to wall tension sets a cross-over time, and the wall tension alone sets when the WBS finally collapse. Whether the composite strings persist or decay depends on quantum tunneling and on how inflation treated the original monopoles.

The paper gives example spectra for three broad scenarios for the composite strings. In an “effectively stable” case they show spectra for dimensionless string tensions Gμ = 10^−10 and 10^−12 with domain-wall scales (vacuum values) around 10^2 and 10^5 GeV. In “quasi-stable” and “metastable” cases they show results for Gμ around 10^−7 and wall scales near 10^8 GeV. For those higher-tension examples the predicted spectra can be compatible with existing upper limits from the LIGO–Virgo–KAGRA Run 4 data and can also be consistent with the stochastic signal reported by recent pulsar timing arrays. The authors compare their spectra to sensitivities of current and planned experiments such as NANOGrav, LVK, LISA, DECIGO, BBO, ET, CE and SKA.