Metastable Cosmic Strings in GUT Models

• Metastable cosmic strings are topological defects emerging from multi-stage gauge symmetry breaking that decay through quantum nucleation of monopole–antimonopole pairs.
• Their phenomenology is governed by the decay parameter κ = Mₘ²/μ, with network evolution and fragmentation determined by comparing the decay rate to the Hubble scale.
• Observational predictions include a flat gravitational-wave plateau with a low-frequency turnover, making them key targets for PTA and interferometer experiments.

Metastable cosmic strings are extended topological field configurations that arise in gauge theories with a specific pattern of multi-stage spontaneous symmetry breaking, characterized by a finite but long lifetime due to decay via monopole–antimonopole pair creation. These objects are especially relevant in grand unified theory (GUT) embeddings and are a focus of current research due to their potential to source stochastic gravitational-wave backgrounds detectable by pulsar timing array (PTA) and terrestrial interferometers. The phenomenology of metastable strings is controlled by network evolution, semiclassical decay mechanisms, and the interplay between symmetry-breaking scales and gauge couplings.

1. Symmetry-Breaking Patterns and Topological Origin

Metastable cosmic strings originate in gauge theories where the symmetry-breaking sequence proceeds through at least two steps:

• $G \xrightarrow{v_1} G' \xrightarrow{v_2} H$, with $v_1 \gg v_2$.
• $\pi_1(G/G')=0$ but $\pi_2(G/G')\cong \pi_1(G')/\pi_1(G)\neq 0$, and $\pi_1(G'/H)\neq 0$.

At the higher scale $v_1$, the breaking $G\to G'$ typically produces finite-energy ’t Hooft–Polyakov monopoles due to $\pi_2(G/G')\neq 0$. At the lower scale $v_2$, the breaking $G'\to H$ yields strings (Nielsen–Olesen–type vortices) because $\pi_1(G'/H)\neq 0$. However, since the full theory has $\pi_1(G/H)=0$, the string winding is only locally topologically stable. The charge can be unwound via the creation of a monopole–antimonopole pair, rendering strings metastable (Chitose et al., 16 Jul 2025, Chitose et al., 2023).

Examples include sequences in non-Abelian GUTs:

• $$SO(10) \xrightarrow{\langle 45_H \rangle} SU(3)_C\times SU(2)_L\times SU(2)_R\times U(1)_{B-L} \xrightarrow{\langle 16_H \rangle} SU(3)_C\times SU(2)_L\times U(1)_Y$$,
• $$SU(2)\times U(1) \to U(1)\times U(1) \to U(1)' \to \mathbf{1}$$ (Chitose et al., 18 Jun 2025).

Strings produced at $v_2$ are attached at their endpoints to monopoles from $v_1$, which act as termination points.

2. Decay Mechanism: Quantum Nucleation of Monopoles

The central decay channel for metastable cosmic strings is the Schwinger-like quantum tunneling process: monopole–antimonopole pairs nucleate along the string, breaking it into finite segments. In the thin-wall approximation, the pair-creation rate per unit length is

$$\Gamma_d \simeq \frac{\mu}{2\pi} \exp(-\pi \kappa), \qquad \kappa \equiv \frac{M_m^2}{\mu}$$

where:

• $\mu$ is the string tension,
• $M_m$ is the mass of the confined monopole at $v_1$.

The bounce action for this process is $S_B = \pi\,M_m^2/\mu$, which can be derived from a 1+1D effective theory on the string worldsheet. For large mass hierarchy $v_1\gg v_2$, the thin-wall approximation is robust, but even for moderate hierarchies $V/v \gtrsim 5$, full finite-size analyses confirm $S_B \gtrsim \pi\,\kappa$ as a lower bound (Chitose et al., 2023).

The time at which network-scale decay becomes significant is set by $H(t_{decay}) \sim \sqrt{\Gamma_d}$, at which Hubble-length strings fragment into segments of length $\sim H^{-1}$.

3. Network Structure, Key Parameters, and Dynamics

Several quantitative parameters are essential for the metastable-string phenomenology:

• String tension: $\mu = 2\pi v_s^2$ (Abelian-Higgs, BPS limit), or corrected for couplings and full profiles.
• Monopole mass: $M_m \sim 4\pi v_1/g$, typically higher than $v_s$.
• The parameter $\kappa = M_m^2/\mu$ controls the decay rate.
• Segment dynamics: After fragmentation, string segments are pulled together by a linear potential $V(\ell) = \mu \ell$.
• Oscillation frequency: For a segment of length $\ell$, $\omega \sim \pi/\ell$.

The evolution before decay follows standard scaling: energy density in strings scales with $\rho_s \sim \mu/L^2$, loops are chopped off by intercommutations at rate $\alpha \sim 0.1$, and loops radiate via GWs. After decay onset, the segment length distribution is truncated at $\ell_{\mathrm{max}} \sim H^{-1}$. Network evolution equations must include both Hubble stretching and the exponential decay term $\sim \Gamma_d$ (Buchmuller et al., 2021, Buchmuller et al., 2020).

4. Gravitational-Wave Emission and Spectral Properties

The stochastic GW spectrum from metastable string networks consists primarily of the superposition of radiation from oscillating loops. For segments, GW emission is highly suppressed:

• For loop emission in the scaling regime, the present-day GW energy density fraction per logarithmic frequency is (in the radiation era)

$\Omega_{\mathrm{GW}}(f) \simeq A \, (G \mu)^2$

with $A$ an order-unity factor set by scaling simulations (Buchmuller et al., 2021, Antusch et al., 7 Mar 2025).

• Metastability induces a suppression of the low-frequency ($f \lesssim f_*$) spectrum, with a sharp turnover at

$f_* \sim \sqrt{\Gamma_d}$

The spectrum is typically a broken power law:

$$\Omega_{\mathrm{GW}}(f) \propto \begin{cases} f^{q} & f \ll f_* \ \mathrm{const} & f_* \lesssim f \lesssim f_{\mathrm{cutoff}} \ f^{-1} & f \gg f_{\mathrm{cutoff}} \end{cases}$$

with $q=1$–$2$ (metastability, segment-dominated), with the plateau height set by $G\mu$, and the low-$f$ suppression by $\kappa$ (Buchmuller, 2024).

• For oscillating segments, GW emission of order $P_{GW}\sim G_N \mu^2$ is suppressed by thermal drag acting on the monopoles at the segment ends. Thermal modes on the string, even below the friction-free temperature $T_{fr}$, provide enough resistance to prevent relativistic oscillations. The critical boost $\gamma_m^{(\mathrm{cr})}$ is many orders of magnitude smaller than the Hubble-allowed maximum, so segments shrink without emitting significant GW power (Chitose et al., 16 Jul 2025).
• The total contribution from segments to the GW background is thus negligible; loops dominate the spectrum. For PTAs, the observed background can be $10^{-9}$–$10^{-8}$ for $G\mu \sim 10^{-7}$, $\sqrt{\kappa} \sim 8$ (Antusch et al., 7 Mar 2025, Buchmuller et al., 2023).

5. Parameter Space, Thermal and Cosmological Constraints, and Observational Implications

PTA and interferometer observations severely constrain $G\mu$ and $\kappa$:

• The PTA-preferred window corresponds to $G\mu \sim 10^{-8} \text{ to } 10^{-5}$ and $\sqrt{\kappa} \sim 7.7$–$8.3$ (Antusch et al., 7 Mar 2025), mapped to VEVs $v_{cs}\sim10^{15}$–$10^{16}$ GeV.
• The decay rate's exponential sensitivity to $\kappa$ restricts viable models to a narrow region, with the monopole and string-forming scales within $\sim 10$–$20\%$ of each other (Ahmed et al., 2023).
• Thermal friction sets extremely low critical velocities for monopoles; for standard cosmological temperatures ($T_s \sim 0.1$ MeV), the drag is sufficient to prevent efficient GW oscillations (Chitose et al., 16 Jul 2025).
• Non-gravitational energy loss by the segments dumps most of the string decay energy into massive string fluctuations and the Standard Model thermal bath, with total $E_{\mathrm{in}}Y_{\mathrm{in}} \lesssim 10^{-8}$ per entropy unit, remaining below BBN bounds.

A table summarizes critical parameters:

Parameter	Definition	Benchmark Value (PTA preferred)
String tension	$\mu$	$2\pi v_s^2$
Monopole mass	$M_m$	$4\pi v_1/g$
Decay parameter	$\kappa=M_m^2/\mu$	$\sqrt{\kappa} \sim 8$
Decay rate	$\Gamma_d = (\mu/2\pi) e^{-\pi\kappa}$	$\Gamma_d$ s.t. $H \sim \sqrt{\Gamma_d}$
GW plateau amplitude	$\Omega_{\text{GW}}\sim (G\mu)^2$	$10^{-9}$–$10^{-8}$
Turnover frequency	$f_*$	$f_*\sim10^{-8}$ – $10^{-7}$ Hz

String tension and monopole masses must be tuned such that networks decay before recombination (to avoid CMB constraints), but late enough to produce detectable GW power in the PTA band (Antusch et al., 2024, Buchmuller et al., 2023). High string tension/long-lived networks predict a GW plateau in the LIGO–Virgo band potentially within reach of ground-based interferometers, while a sharp suppression at low frequencies helps evade the strongest PTA bounds that would exclude stable cosmic strings.

6. Realizations in Concrete Gauge Models and Related Theoretical Structures

The metastable string mechanism is realized in numerous concrete particle-physics constructions:

• GUT symmetry breaking chains: $SO(10)$, $SU(5)$, $SU(4)_c \times SU(2)_L \times U(1)_R$, Pati–Salam $SU(4)\times SU(2)_L\times SU(2)_R$, left–right symmetric models $$SU(3)_C\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}$$.
• Flavour-symmetry breaking scenarios ($SU(2)_F\to U(1)_F\to \mathbf{1}$) (Antusch et al., 7 Mar 2025).
• Dark (hidden sector) single-scale models of the form $SU(2)\times U(1)'\to U(1)$, where Z-strings are classically stable in the semi-local regime but decays are induced via monopole tunneling (Ingoldby et al., 11 Nov 2025).

Inflationary cosmology is often embedded to solve the monopole problem (by inflating away monopoles from the higher-scale breaking), ensuring that strings, but not monopoles, survive post-inflation to dominate the GW signatures (Antusch et al., 2024, Afzal et al., 2023). Models consistently reproduce observed relic abundances and are compatible with seesaw neutrino masses, baryogenesis scenarios (e.g., leptogenesis), and dark matter production (Buchmuller et al., 2023, Buchmuller et al., 2020).

A common misconception—that segment oscillations contribute significantly to the GW background—has been definitively refuted for physically relevant parameter ranges. Instead, only the loop network sources observable GW signals (Chitose et al., 16 Jul 2025). Late-time cosmology is unaffected by segment energy injection, which remains subdominant relative to cosmological constraints.

7. Phenomenological Outlook and Experimental Probes

Metastable cosmic strings predict a distinctive GW spectrum: a flat or mildly blue plateau above a turnover frequency $f_*$ set by the decay rate, and a sharp suppression $\propto f^{2}$ or steeper below $f_*$. This shape is consistent with PTA observations indicating a common-spectrum stochastic background at $f\sim 10^{-8}$ Hz. The allowed parameter space encompasses $G\mu\sim10^{-8}$–$10^{-7}$ and $\sqrt{\kappa}\sim8$, mapping to intermediate symmetry-breaking scales of $v\sim10^{15}$–$10^{16}$ GeV. This is further constrained (or will be tested) by upcoming LIGO–Virgo, KAGRA, Einstein Telescope, Cosmic Explorer, LISA, and advanced PTA experiments (Buchmuller, 2024, Buchmuller et al., 2023).

A confirmed GW signal consistent with the metastable-string template would provide unique information on high-scale Higgs VEVs, gauge couplings, and the structure of GUT symmetry breaking—parameters inaccessible to terrestrial colliders or other astrophysical probes. Discrimination between metastable and stable string scenarios is provided by the low-frequency power-law turnover, a hallmark of monopole-induced decay (Buchmuller et al., 2021, Chitose et al., 2023).

In summary, metastable cosmic strings are a generic, calculable consequence of multi-stage gauge symmetry breaking with nontrivial monopole and string homotopy. Their cosmological dynamics, decay via quantum tunneling, and resulting gravitational-wave background are tightly constrained and now directly probed by multi-band gravitational experiments. Their existence and detailed properties, if established, would extend the empirical reach of particle physics to GUT and intermediate scales.