Quantitative calibration and validation of the charge-velocity-dependent one-scale (CVOS) model for current-carrying cosmic string networks

Cosmic strings formed in early-Universe phase transitions can in many particle-physics extensions carry electric currents and charges on their worldsheets ('superconducting' strings). The velocity-dependent one-scale (VOS) model, which describes string networks by a characteristic length and a root-mean-square velocity, was generalized to such current-carrying networks, yielding the charge-velocity-dependent one-scale (CVOS) model. Its evolution equations contain phenomenological functions left unspecified — the loop-chopping bias $g$ , the charge-versus-current energy-loss bias $\rho$ , the chopping efficiency $\tilde{c}$ — and rest on two averaging assumptions that have never been tested against simulations. Later work has classified the model's asymptotic scaling solutions and produced the first large field-theory simulations of current-carrying networks, but the quantitative calibration of the model — measuring its free functions from high-resolution simulations, validating the averaging assumptions, testing possible charge dependence of the momentum parameter $k(v)$ , and extending beyond the small-current regime — remains open. Completing it would convert the CVOS formalism into a quantitatively predictive tool for gravitational-wave and CMB constraints on superconducting cosmic strings.

Background

Cosmic strings are one-dimensional topological defects expected to form during symmetry-breaking phase transitions in the early Universe; they are generic in many extensions of the Standard Model, and their networks evolve towards a 'scaling solution' in which the string energy density stays a fixed fraction of the total cosmic energy density. The velocity-dependent one-scale (VOS) model is the standard semi-analytic description of this evolution: it characterizes a network of Nambu–Goto strings (structureless strings with no internal degrees of freedom) by two averaged quantities, a characteristic length scale $L$ (or correlation length) and a root-mean-square (RMS) velocity $v$ , whose coupled evolution is driven by the cosmological expansion rate and by energy losses, chiefly the production of small closed loops ('loop chopping'), parametrized by a chopping efficiency $c$ and a loop-chopping bias $g$ . A phenomenological momentum parameter $k(v)$ encodes the curvature drive that accelerates strings.

In many realistic models the strings are 'superconducting' (in the sense of Witten's 1985 construction): a charged condensate forms on the string worldsheet, so strings can carry worldsheet currents and charges. Such currents modify the string energy per unit length $U$ and tension $T$ through an equation of state, and introduce new large-scale degrees of freedom. Martins, Peter, Rybak and Shellard generalized the VOS model to this case. The resulting charge-velocity-dependent one-scale (CVOS) model tracks, besides $L$ and $v$ , averaged measures of the timelike current ( $J$ ) and spacelike charge ( $Q$ ), or equivalently the current amplitude $Y=\frac{1}{2}(Q^{2}+J^{2})$ and the chirality parameter $K=Q^{2}-J^{2}$ measuring the distance to the chiral (lightlike-current) limit. The equation of state enters through a generating function $F$ (with derivatives $F'$ , $F''$ ) of the worldsheet state parameter; several microscopic cases are known — a linear one, the Kaluza–Klein case (current arising from string momentum around a compactified extra dimension), and the magnetic/electric Witten models, the latter with a maximum critical current.

The derivation of the CVOS evolution equations rests on two acknowledged averaging assumptions: (A) uncorrelated variables, i.e. replacing averages of functions by functions of averages, $\langle\mathcal{O}^{2}\rangle\approx\langle\mathcal{O}\rangle^{2}$ ; and (B) the neglect of boundary (total-derivative) terms, justified for network-periodic or loop-dominated boundary conditions. Energy conservation additionally requires phenomenological loss terms for the charge and current: a loop-chopping bias $g$ that may depend on the current and charge, a parameter $\rho$ biasing losses between the current and the charge ( $\rho=1/2$ is the unbiased value, $g=1$ the Nambu–Goto value), and an effective chopping efficiency $\tilde{c}$ ; the momentum parameter $k(v)$ is assumed to keep its standard velocity-dependent form, its possible charge dependence being explicitly deferred. These functions are not derived from the microscopic theory; they must be measured from field-theory simulations, exactly as the ordinary VOS parameters were calibrated against Abelian–Higgs simulations.

Subsequent work by the same collaboration has exploited the formalism: the linear equation-of-state case (2021) and the Witten superconducting case (2023) identified when a scaling solution with non-trivial stable current can exist; a classification of all asymptotic scaling solutions (2024, extended to biased cases in 2025) found three classes — decay of charge/current to the Nambu–Goto limit, charge/current domination, and an intermediate regime in which the network together with its charge and current reaches full scaling. On the numerical side, $2048^{3}$ field-theory simulations of a $U(1)_{\mathrm{local}}\times U(1)_{\mathrm{global}}$ model with a current-carrying condensate (2024) provided the first measurements of charge and current coherence lengths and of the condensate equation of state, finding chiral networks with decaying charge and current in the matter era and non-chiral behavior in the radiation era — but only in the small-current regime, and without determining the CVOS phenomenological parameters. What remains is the step that makes the whole program quantitative: calibrating and validating the CVOS model itself against microscopic simulations.

Problem Statement

Convert the charge-velocity-dependent one-scale (CVOS) model for current-carrying cosmic string networks from a formalism with free functions into a quantitatively calibrated and validated description, by using microscopic field-theory simulations to determine its unspecified ingredients and to test its structural assumptions. The common objective uniting the required results is that the calibrated CVOS evolution equations should quantitatively reproduce the large-scale behavior of simulated current-carrying networks (and be falsifiable against them) across cosmological expansion regimes. Specifically, this objective comprises, as jointly supporting parts: (1) measurements, from high-resolution field-theory simulations of current-carrying string networks, of the phenomenological functions left free in the model — the loop-chopping bias $g$ (including any dependence on the averaged current and charge, i.e. on the current amplitude $Y$ and chirality parameter $K$ ), the charge-versus-current loss bias $\rho$ , and the effective chopping efficiency $\tilde{c}$ — together with the equation-of-state inputs $F$ , $F'$ , $F''$ of the worldsheet condensate; (2) a direct quantitative test, against the same simulations, of the two averaging assumptions underpinning the derivation (uncorrelated variables, $\langle\mathcal{O}^{2}\rangle\approx\langle\mathcal{O}\rangle^{2}$ , and the neglect of boundary/total-derivative terms), establishing the domain of validity of each; (3) a determination of whether the momentum parameter $k$ retains its standard velocity-dependent form or acquires charge/current dependence; and (4) an assessment, beyond the small-current linear regime, of whether the model's scaling-solution classes — in particular the full-scaling solution with finite stable charge and current — are realized by networks with realistic microscopic equations of state and loss mechanisms, including the effect of charge source mechanisms such as reconnection-driven charge generation or coupling to external magnetic fields. The four parts (1)–(4) are jointly posed: a CVOS model whose free functions $g$ , $\rho$ and $\tilde{c}$ , together with the equation-of-state inputs $F$ , $F'$ , $F''$ , are fixed by simulation measurements with quantified uncertainties; whose two averaging assumptions are validated, or their domain of validity bounded, over the stated regime; for which the charge/current dependence of the momentum parameter $k$ is determined; and whose predicted scaling behavior — including which of the scaling-solution classes, in particular the full-scaling solution with finite stable charge and current, are realized by networks with realistic microscopic equations of state and loss mechanisms beyond the small-current regime — is quantitatively confirmed or refuted by independent network simulations in at least the radiation and matter eras.

References

C. J. A. P. Martins, Patrick Peter, I. Yu. Rybak, E. P. S. Shellard, "Generalized velocity-dependent one-scale model for current-carrying strings", Phys. Rev. D 103, 043538 (2021). DOI: 10.1103/PhysRevD.103.043538; arXiv:2011.09700, https://arxiv.org/abs/2011.09700
C. J. A. P. Martins, Patrick Peter, I. Yu. Rybak, E. P. S. Shellard, "Charge-velocity-dependent one-scale linear model", Phys. Rev. D 104, 103506 (2021). DOI: 10.1103/PhysRevD.104.103506; arXiv:2108.03147, https://arxiv.org/abs/2108.03147
I. Yu. Rybak, C. J. A. P. Martins, Patrick Peter, E. P. S. Shellard, "Cosmological evolution of Witten superconducting string networks", Phys. Rev. D 107, 123514 (2023). DOI: 10.1103/PhysRevD.107.123514; arXiv:2304.00053, https://arxiv.org/abs/2304.00053
I. Yu. Rybak, C. J. A. P. Martins, Patrick Peter, E. P. S. Shellard, "Cosmic microwave background signatures from current-carrying cosmic strings", Phys. Rev. D 110, 023534 (2024). DOI: 10.1103/PhysRevD.110.023534; arXiv:2403.16332, https://arxiv.org/abs/2403.16332
J. R. C. C. C. Correia, C. J. A. P. Martins, F. C. N. Q. Pimenta, "Evolution of current-carrying string networks", Phys. Lett. B 855, 138788 (2024). DOI: 10.1016/j.physletb.2024.138788; arXiv:2406.03931, https://arxiv.org/abs/2406.03931
F. C. N. Q. Pimenta, C. J. A. P. Martins, "Scaling solutions for current-carrying cosmic string networks", Phys. Rev. D 110, 023540 (2024). DOI: 10.1103/PhysRevD.110.023540; arXiv:2406.03940, https://arxiv.org/abs/2406.03940
F. C. N. Q. Pimenta, C. J. A. P. Martins, "Scaling solutions for current-carrying cosmic string networks. II. Biased solutions", Phys. Rev. D 112, 023535 (2025). DOI: 10.1103/j5kv-czjd; arXiv:2506.21494, https://arxiv.org/abs/2506.21494
P. B. Barbosa, C. J. A. P. Martins, "Current-carrying string network evolution in an external magnetic field", Phys. Rev. D 113, 063542 (2026). DOI: 10.1103/tfvh-7hld; arXiv:2602.22396, https://arxiv.org/abs/2602.22396
Takashi Hiramatsu, Marc Lilley, Daisuke Yamauchi, "Dynamical simulations of colliding superconducting strings", JCAP 06 (2024) 030. DOI: 10.1088/1475-7516/2024/06/030; arXiv:2312.16091, https://arxiv.org/abs/2312.16091
I. Yu. Rybak, L. Sousa, "Emission of gravitational waves by superconducting cosmic strings", JCAP 11 (2022) 024. DOI: 10.1088/1475-7516/2022/11/024; arXiv:2209.01068, https://arxiv.org/abs/2209.01068
I. Yu. Rybak, L. Sousa, "Stochastic gravitational wave background from chiral superconducting cosmic strings", Phys. Rev. D 111, 083502 (2025). DOI: 10.1103/PhysRevD.111.083502; arXiv:2412.17154, https://arxiv.org/abs/2412.17154
Pierre Auclair, Simone Blasi, Vedran Brdar, Kai Schmitz, "Gravitational waves from current-carrying cosmic strings", JCAP 04 (2023) 009. DOI: 10.1088/1475-7516/2023/04/009; arXiv:2207.03510, https://arxiv.org/abs/2207.03510
Jinyoung Jhun, Takashi Hiramatsu, "Gravitational Waves from Superconducting Cosmic Strings", arXiv:2607.25317, https://arxiv.org/abs/2607.25317