The early Universe resembled rapidly cooling water: as the temperature dropped, its fundamental interactions underwent phase transitions—much like liquid turning into ice. In such processes, according to Grand Unified theories, topological defects were born: magnetic monopoles, cosmic strings, and domain walls. Since the prediction of monopoles by Gerard 't Hooft and the experimental discovery of gravitational waves by the LIGO collaboration led by Rainer Weiss, the search for these objects has moved into a practical phase. If defects exist, they must radiate ripples in spacetime, as predicted by Einstein's equations. Inflationary expansion, proposed by Alan Guth, can blow up primordial monopoles, rendering them unobservable, but the remaining strings and walls can become sources of a stochastic gravitational-wave background. Detecting them with laser interferometers and pulsar timing arrays opens a window into physics beyond the known laws.
The authors built a theoretical model with sequential symmetry breaking SU(2) → U(1) → Z₂ → 1, as well as an analogous chain SU(3) → SO(3) → Z₂ → 1. For each stage, the properties of the emerging defects were calculated: magnetic flux of monopoles, string tension, wall dynamics. Then using numerical simulations and analytical estimates (based on Friedmann equations, which describe expansion in accordance with Hubble's law), the gravitational-wave spectra were computed. Effects of friction at early stages, wall collapse, and the transition of the string network to a scaling regime were taken into account. The delay parameter $d_s$ and the time of monopole entry under the horizon $t_M$ allowed modeling three scenarios: effectively stable, quasi-stable, and metastable strings.
The calculations showed that the gravitational-wave background is sensitive to the dimensionless string tension Gμ and the domain wall formation scale $v_{DW}$. For effectively stable strings with Gμ = 10⁻¹⁰ and 10⁻¹² at $v_{DW}$ = 10² GeV, the spectrum falls within the range of ground-based and space-borne detectors. In the case of quasi-stable strings (Gμ ∼ 10⁻⁷, $v_{DW}$ = 10⁸ GeV) with a network formation delay $d_s$ ≳ 10⁷, the signal is compatible with LIGO-Virgo-KAGRA Run 4 data and simultaneously explains the redshift of pulsar timing. The metastable scenario with √κ ≈ 8 is also consistent with observations. Notably, composite strings born from wall collapse carry twice the magnetic flux of the original monopoles—a legacy of the original symmetry. The peak frequency of the background ranges from nanohertz (for pulsar arrays) to hundreds of hertz (ground-based interferometers), and the amplitude $Ω_{GW}$ h² reaches 10⁻¹⁰–10⁻⁸.
This work for the first time unifies predictions for three possible fates of a cosmic string network within a single theoretical framework. This gives experimentalists clear targets: from searching for high-frequency backgrounds on LIGO and the future Einstein Telescope to low-frequency signals in pulsar timing array data. If a signal is detected, we will have direct evidence of phase transitions in the early Universe and possibly be able to look into the physics of gauge theories beyond the Standard Model. Moreover, the defect structure itself sheds light on the topology of symmetry breaking.
Future research could refine the spectra taking into account more realistic cosmological conditions, including the influence of dark matter and adiabatic perturbations. It would also be interesting to consider other symmetry groups, such as SO(10), and their connection to the inflationary model. On the experimental side, key roles will be played by missions like LISA and DECIGO, capable of probing the Gμ ∼ 10⁻¹⁰–10⁻¹² region. Joint analysis of data from multiple detectors will allow decomposing the signal in frequency and separating contributions from different defect types.
The results will impact experimental gravitational-wave astronomy, early-Universe cosmology, and Grand Unified theory, linking symmetry parameters to the observed background.
Next steps include a more accurate accounting of quantum tunneling effects and expansion inhomogeneities, as well as searching for possible correlations with LVK and NANOGrav data.
This article directly addresses the problem of the existence of topological defects and their role in the evolution of the Universe, as well as the question of whether gravitational waves can provide access to physics at Grand Unification energies, inaccessible to accelerators.
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What the work does
The authors built a theoretical model with sequential symmetry breaking SU(2) → U(1) → Z₂ → 1 and SU(3) → SO(3) → Z₂ → 1, calculated the properties of defects—monopoles, strings, and domain walls—and computed gravitational wave spectra for three scenarios of string stability. They showed that the signal could be detected by ground-based and space-based detectors, and that composite strings carry twice the magnetic flux.
Where its strength is
The work unifies several stages of symmetry breaking into a single picture, linking the properties of monopoles and walls to predictions for the gravitational wave background across a wide frequency range. The concrete scenarios (effectively stable, quasistable, metastable) provide ready benchmarks for comparison with LIGO data and pulsar timing arrays.
Where it could go next
Worth trying
Why this may matter
If the gravitational wave spectrum from such defects is detected, it would provide direct evidence of phase transitions in the early Universe and allow us to peer beyond the Standard Model, into energy regimes inaccessible to accelerators.
Nearby works in our archive
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In the model considered, the elementary monopole carries twice the magnetic flux of the elementary string. It's like having a magnet with two north poles, whose field then splits into two smaller magnets — only on cosmic scales.
In the science fiction series Star Trek, cosmic strings sometimes appear as sources of dangerous gravitational disturbances capable of warping spacetime.