Why do metastable cosmic strings break at early times rather than late times?

Early breaking replaces the standard late-decay picture

The standard picture treats a metastable string network as if it were stable at first: it approaches a scaling regime with roughly one long string per Hubble patch, then decays when the breaking rate per unit length Γ becomes comparable to the Hubble rate squared, H² ≃ Γ. In that picture the decay is driven by quantum tunnelling that nucleates a monopole–antimonopole pair on the string, with rate Γ ≃ (μ/2π) exp(−πκ), where μ is the string tension (energy per unit length), m_M is the monopole mass, and κ ≡ m_M²/μ.

The paper's central claim is that this ordering is wrong for a broad class of formation scenarios. The strings are already broken soon after the network forms, so the network's eventual destruction is usually set by early-time breaking rather than by late-time tunnelling. The resulting segments can be initially super-horizon in size, which is why they can still persist for a significant amount of time and behave effectively like infinite strings until their endpoints re-enter each other's horizons.

Heat and leftover monopoles both cut the strings

The first channel is finite-temperature symmetry restoration. If the post-inflationary temperature exceeds the symmetry-breaking scale, the symmetry is restored and a string network forms when the Universe cools below that scale. At nonzero temperature the tunnelling action is modified: the instanton that nucleates a monopole–antimonopole pair is distorted by the compact Euclidean time direction of period β = 1/T. The paper writes the corrected rate as Γ(T) = C(T) exp[−S_B(T)], with the bounce action S_B given in terms of arcsin(T₀/T) and T₀ = μ/(2m_M) = m_M/(2κ). For values of m_M²/μ ≳ 10 of phenomenological interest, the action scales as m_M/√μ rather than m_M²/μ, so the finite-temperature breaking rate is dramatically faster than the zero-temperature estimate.

The second channel is attachment to monopoles that already exist. During percolation, proto-string segments execute a random walk through space; if the walk encounters a monopole, the string terminates there. The paper considers monopoles produced thermally by freeze-in even when the reheating temperature is below m_M, and monopoles produced by inflationary fluctuations. In both cases the resulting network consists of finite segments and loops rather than truly infinite strings, with the breaking rate scaling as exp(−c m_M/√μ) for an order-one number c.

Super-horizon segments mimic an intact network

A broken network is not automatically a dead network. For a large hierarchy between the monopole mass and the string tension (κ ≫ 1), the finite segments produced by early breaking are extremely super-horizon in length. Until their endpoints re-enter each other's horizons, the segments behave effectively like a network of infinite strings, so the observational signatures resemble those of the conventional metastable scenario.

The paper supports this with a simple analytic model of interactions among finite segments, plus a numerical implementation starting from an exponential distribution of string lengths. In that model, interactions among the segments do not regenerate infinite strings, and the strings are all destroyed once the initial mean length is similar to the Hubble length. The authors are explicit that this is a model result, not a proof about the real network.

NANOGrav survival needs a larger mass hierarchy

The observational consequence is a shift in the required parameter space. For strings to survive to the epochs probed by NANOGrav, the paper finds m_M²/μ ≳ 10³, where m_M is the monopole mass and μ the string tension. That is over an order of magnitude larger than previous estimates, which assumed the zero-temperature tunnelling rate controlled the decay.

The comparison matters because the two numbers answer the same question with different physics. The earlier estimate, m_M²/μ ≃ 60 giving a decay temperature T_d ∼ MeV, follows from the zero-temperature rate of Eq. (2) and the decay-temperature relation T_d ≃ √M_Pl μ^{1/2} exp(−π m_M²/(4μ)). The new threshold follows from early-time breaking setting the destruction time instead. A reader comparing model-building papers should check which breaking mechanism is assumed before comparing their κ values.

The authors also note a model-building benefit: the larger required hierarchy relaxes the tuning needed in the reheating temperature or inflationary Hubble scale to obtain a string-forming transition without a monopole-forming transition.

Late tunnelling is faster at rare high-tension points

The paper also revisits the late-time quantum-tunnelling channel, allowing for the fact that string tension is not uniform in a realistic network. Localised regions such as kinks or cusps can have energy densities exceeding the average by factors of a few. Because the tunnelling rate depends exponentially on μ, even rare high-tension points can dominate the total breaking rate.

The evidence comes from numerical simulations of critically coupled Abelian–Higgs strings on an expanding radiation-dominated background, with lattice size N = 3072 and string-core resolution n_r = 2 or 3. The tension distribution is peaked around the straight-string value μ₀ = 2πv² and has a high-tension tail. The authors report that the shape of the distribution is consistent across resolutions, so the tail is unlikely to be a lattice artifact. Their conclusion is that, for values of m_M²/μ of phenomenological interest, breaking at these points makes the true rate many orders of magnitude larger than the standard estimate of Eq. (2).

The re-percolation assumption carries the conclusion

The key assumption, and by the authors' own account the main uncertainty, is that string loops do not re-percolate to form effectively infinite strings. The paper argues this is plausible because most of the string length is initially in finite segments, though a significant fraction may later move into loops as the network approaches scaling. The authors state they have not proved that loops fail to re-percolate and leave loop dynamics to future work.

Two further scope conditions sit next to the relevant results. The thermal-production analysis assumes instantaneous reheating and thermalisation, which the authors say is unlikely to hold in realistic cosmological histories. The finite-temperature bounce action is computed in the thin-string limit, justified because the phenomenologically interesting regime has √κ ≫ 10. The paper also assumes the simplest realisations of metastable strings, without small parameters separating the monopole mass from the symmetry-breaking scale; in more complex theories the dominant breaking mechanism could change.

Hidden-sector flux tubes are the natural analogue

The paper briefly extends the discussion to flux tubes of a new asymptotically free gauge group in a dark QCD-like hidden sector, with dark-quark masses above the confinement scale. Such flux tubes can behave as metastable cosmic strings, and hidden gauge sectors appear to be common in string-theory compactifications, which the authors present as motivation for treating this as a well-motivated scenario.

The analogy is not exact. In the Abelian Higgs model strings carry an integer winding number, whereas in SU(N_DC) gauge theories flux tubes are labelled by k-ality k = 0, 1, …, N_DC − 1 under the Z_NDC centre symmetry. Two k = 1 strings can, when oriented appropriately, overlap and form a k = 2 string. The authors state that a detailed analysis of flux-tube network structure is left for future work.

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