Predictions of the stochastic gravitational-wave background (SGWB) from cosmic strings — including the metastable strings formed in D-term hybrid inflation studied by El Bourakadi et al. (2024) — are computed from analytic loop number densities, gravitational-wave emission efficiencies, and monopole-nucleation decay rates. The resulting mapping from the dimensionless string tension Gμ (and, for metastable strings, the decay parameter κ = m²/μ) to the observable spectrum Ω_GW(f) is not robust: observational collaborations quote Gμ constraints that differ by one to two orders of magnitude across benchmark loop-distribution models, and 2024–2026 work is still revising the decay inputs themselves (lattice bounce actions, early-time network breaking). The problem is to determine the loop production and size distribution, the loop gravitational-wave emission normalization, and the metastable-decay dynamics from first principles with quantified uncertainty, so that predicted spectra and the observational constraints derived from them (pulsar timing arrays, ground-based interferometers, and LISA forecasts) become insensitive to network-modeling choices.

Background

Cosmic strings are one-dimensional topological defects that can form at a symmetry-breaking phase transition in the very early universe, most plausibly at a grand-unified-theory (GUT) scale. Hybrid inflation models in supergravity (supersymmetric extensions of gravity coupled to matter fields), such as the D-term model studied in the source paper — where the inflationary energy is tied to a Fayet–Iliopoulos term (a constant, model-independent contribution to the D-term, the auxiliary-field part of the supergravity scalar potential) and inflation ends when this term triggers a waterfall transition (the rapid roll of a second field that terminates the slow-roll phase) — generically produce cosmic strings at the end of inflation. A string network evolves toward a scaling regime in which a horizon-sized fraction of the universe is always filled by long strings, while closed loops are continuously chopped off by string self-intersections. Each oscillating loop of length ℓ radiates gravitational waves (GWs) in harmonics n with a power spectrum P_n, losing length at a rate dℓ/dt = −ΓGμ with Γ ≈ 50 a dimensionless efficiency. Summing over the loop population, weighted by the loop number density n(ℓ,t), and integrating over redshift along the line of sight gives today's stochastic gravitational-wave background (SGWB) energy-density fraction Ω_GW(f) — a stochastic, unresolved background potentially observable by pulsar timing arrays (PTAs, sensitive near nanohertz frequencies), the LIGO–Virgo–KAGRA (LVK) ground-based interferometers (tens of hertz), and the future space detector LISA (millihertz). The key microscopic parameter is the dimensionless tension Gμ (G is Newton's constant, μ the mass per unit length), constrained by cosmic-microwave-background data to Gμ ≲ 10⁻⁷.

In theories where strings and monopoles coexist (as in most GUTs), strings are metastable: they decay by the spontaneous nucleation of monopole–antimonopole pairs, with a decay rate per unit length Γ_d = (μ/2π)exp(−πκ) governed by the dimensionless ratio κ = m²/μ of the squared monopole mass to the string tension (the subscript distinguishes Γ_d from the loop radiation efficiency Γ above). Buchmuller, Domcke and Schmitz (2020) turned this into a PTA-era template: metastable strings contribute an SGWB whose high-frequency break encodes the decay redshift set by κ. El Bourakadi et al. (2024) used exactly this formalism (their Eqs. 41–43), together with the 'smoothed' loop power spectrum of Blanco-Pillado and Olum (2017), to connect D-term inflation parameters, reheating, gravitino production, and the cosmic-string SGWB. Their predictions rest on assumed network inputs: the analytic radiation-era loop density (a number of the form 0.18/[t^{3/2}(ℓ+ΓGμt)^{5/2}]), the decay parameter κ_cs, and the numerical factors Γ and the loop-density normalization.

The catch is that these inputs are not known from first principles, and different choices change the answer substantially. Two families of loop model dominate the literature. The BOS distribution (from simulations of Nambu–Goto strings — idealized, infinitesimally thin strings whose dynamics depend only on their tension, with no field-theory internal structure — by Blanco-Pillado, Olum and Shlaer) has loops born at a sizable fraction of the horizon size; the LRS distribution (Lorenz, Ringeval and Sakellariadou, built on the one-scale/velocity-dependent-one-scale approach, in which network evolution is described by correlation-length and velocity variables) is dominated by small loops near the gravitational-backreaction scale (backreaction being the smoothing of small-scale string structure by the strings' own gravitational field). LVK analyses label these benchmark models A and B, with an interpolating model C. Because small loops radiate at higher frequencies for the same tension, the inferred bound on Gμ differs by one to two orders of magnitude between models. Field-theory simulations — which evolve the full string-forming fields of the simplest string-forming theory, the Abelian–Higgs model, rather than idealized thin strings — add a further wrinkle: many small loops there shed energy into particles rather than GWs, motivating a phenomenological fraction f_NG of Nambu-Goto-like loops. For metastable strings, the decay inputs are also under revision: the bounce action (the Euclidean action of the tunneling solution whose exponential sets the nucleation rate) controlling exp(−πκ) was until recently computed only in the thin-string approximation (treating the string as an infinitesimally thin classical defect rather than solving the full field theory), and early-universe thermal and percolation effects can break the network long before quantum tunneling does. The unresolved question is whether the mapping from (Gμ, κ) to Ω_GW(f) can be made robust against these network-modeling choices.

Problem Statement

Establish the mapping from cosmic-string network parameters — the dimensionless tension Gμ, and for metastable strings the decay parameter κ = m²/μ — to the observable stochastic gravitational-wave spectrum Ω_GW(f) from first principles, to the point that the prediction and the constraints derived from it are robust against network-modeling choices. The network-modeling inputs whose choice currently controls this mapping are: (i) the loop production function and loop-size distribution n(ℓ,t) across the radiation- and matter-dominated eras, including the large-loop BOS-type versus small-loop LRS-type discrepancy and the Nambu-Goto versus Abelian–Higgs loop-dynamics fraction f_NG; (ii) the gravitational-wave emission efficiency per loop (Γ and the harmonic power spectrum P_n) and its normalization; and (iii) for metastable strings, the decay rate — the bounce action entering exp(−πκ), including thin-string corrections and early-time thermal/percolation network breaking — and the resulting network lifetime. The criterion for success is that the spread in the predicted Ω_GW(f) — equivalently, in the bound on Gμ (and, for metastable strings, on κ) inferred from a given data set — across the benchmark loop distributions and decay prescriptions be reduced to within that data set's quoted observational uncertainty, with any irreducible residual dependence quantified; all three inputs (i)–(iii) are required and none is context only.

References

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