Validated dynamics of cosmic string networks with time-varying tension: calibrating the generalized VOS model, the momentum parameter k(v), and the post-percolation fate

In string compactifications the tension of cosmic (super)strings is set by moduli, which can roll in the early universe, making the tension time-dependent, $\mu(t) \propto t^{-m}$. Revello and Villa (JCAP 04 (2025) 049) generalized the Velocity-dependent One-Scale (VOS) model to such networks, predicting modified scaling attractors, an ultra-relativistic radiation-like attractor when $2H + \dot{\mu}/\mu < 0$, growing loops, and percolation. Follow-up work through 2026 has computed the gravitational-wave spectra of these networks and exhibited concrete string-theory realizations, but the underlying network dynamics remain unvalidated: no network-scale numerical simulation implementing $\mu(t)$ exists, the velocity dependence and sign flip of the momentum parameter $k(v)$ at $v^2 = 1/2$ lack a general argument beyond perturbative configurations, and the network's fate after loop-driven percolation is unsettled. The open problem is to place the dynamics of varying-tension string networks on firm quantitative ground: calibrate the chopping efficiency $\tilde{c}(v)$ and momentum parameter $k(v)$ and confirm or refute the predicted attractors by direct network simulation, establish $k(v)$'s sign structure for generic microscopic configurations, and determine the post-percolation regime.

Background

Cosmic strings are linelike topological defects that can form during symmetry-breaking phase transitions in the early universe; their energy per unit length, the tension $\mu$, is usually treated as a constant. In string theory, fundamental strings and strings from branes wrapped on internal cycles ('cosmic superstrings' and 'effective strings') acquire tensions that depend on the expectation values of the compactification moduli, so if moduli roll in the early universe the tension becomes a function of time, $\mu(t)$ — for instance $\mu \propto t^{-m}$ during a kination epoch (a period when the kinetic energy of a rolling scalar dominates the energy density). Earlier one-scale analyses of time-dependent-tension strings (Yamaguchi, Phys. Rev. D 72 (2005) 043533; Ichikawa, Takahashi and Yamaguchi, Phys. Rev. D 74 (2006) 063526) already found scaling behaviour below a critical rate of tension variation, and Conlon, Copeland, Hardy and Sanchez Gonzalez (arXiv:2406.12637) showed that when the tension decreases fast enough loops grow, percolate, and can form a network. The standard quantitative description of string networks is the Velocity-dependent One-Scale (VOS) model of Martins and Shellard, a mean-field model in which the network is described by a correlation length $L$ (with long-string energy density $\rho = \mu/L^2$) and a root-mean-square velocity $v$, obeying two coupled ordinary differential equations with phenomenological parameters: the loop-chopping efficiency $\tilde{c}$, which controls energy loss into small loops, and the momentum parameter $k(v)$, which controls curvature-driven acceleration of the strings. In the standard constant-tension ansatz, $k(v)$ is positive for $v^2 \leq 1/2$ and negative for $v^2 > 1/2$; both parameters have been calibrated against Nambu-Goto and field-theory numerical simulations only for constant $\mu$. Revello and Villa (JCAP 04 (2025) 049) generalized the VOS model to time-dependent tension, $\mu \propto t^{-m}$, in a background with energy density $\rho_b \propto a^{-n}$, where $a$ is the cosmological scale factor (so $n = 4$ is radiation, $n = 3$ matter, $n = 6$ scalar-field kination). They found that the usual scaling attractor ($L = \xi t$, constant $v$) persists with modified $\xi$ and $v$, that when $2H + \dot{\mu}/\mu < 0$ (where $H = \dot{a}/a$ is the Hubble rate) no fixed point exists at $v^2 \leq 1/2$ and, for $mn > 4$, a new ultra-relativistic attractor with $v^2 > 1/2$ (where $k(v)$ must be negative) can appear, in which the network redshifts like radiation; and that loops can grow and percolate. Their microscopic evidence for the $k(v)$ sign flip relies on perturbative calculations for special configurations (a static infinite string and a circular loop) and numerical interpolation valid only for $v^2 \leq 1/2$. The model is now the dynamical input for a rapidly growing literature computing gravitational-wave forecasts for string compactifications (e.g. arXiv:2504.20994 and arXiv:2601.10790), yet its phenomenological ingredients have never been tested against a simulation of a network with time-varying tension.

Problem Statement

For a cosmic string network whose tension decreases as a power law $\mu \propto t^{-m}$ in an expanding background with $\rho_b \propto a^{-n}$, place the generalized VOS description on firm quantitative ground by resolving three linked questions about one common objective — the validated dynamics of varying-tension string networks. (1) Direct network-scale validation: simulate a string network that implements $\mu(t)$ while preserving worldsheet reparametrization invariance (invariance under relabeling of the worldsheet coordinates, which constrains how $\mu(t)$ may enter the equations of motion), and determine whether the predicted attractors are realized — the modified scaling solution $L = \xi t$ with constant $v$, and, when $2H + \dot{\mu}/\mu < 0$ and $mn > 4$, the ultra-relativistic radiation-like attractor with $v^2 > 1/2$ — and calibrate the velocity-dependent loop-chopping efficiency $\tilde{c}(v)$ and momentum parameter $k(v)$ over the full velocity range, including the regime $v^2 > 1/2$ never probed by constant-tension simulations. (2) The momentum parameter: establish by a general argument (or refute by an explicit configuration) that $k(v)$ changes sign at $v^2 = 1/2$ when $\mu$ varies with time, going beyond the perturbative straight-string and circular-loop calculations on which the current ansatz rests, or else derive its actual functional behaviour for generic microscopic string configurations. (3) Post-percolation fate: after tension-driven loop growth causes the loop population to percolate, determine the subsequent evolution of the network — whether it settles into a new scaling or quasi-scaling regime (for example with an effective chopping efficiency tending to zero), what the energy budget between long strings, loops, and radiation becomes, and whether the network can ever dominate the energy density of the universe.

References

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