Completing the varying-tension growth mechanism for cosmic strings: loop production, percolation into networks, multi-modulus dynamics, and field-theoretic strings

The physical size of cosmic string loops grows when the string tension decreases in time, as happens in string compactifications where the tension is set by rolling moduli. Brunelli, Cicoli and Pedro (JCAP 07 (2025) 010) mapped the phase-space conditions for this growth beyond the originally proposed kination regime, finding growth for fundamental strings and for EFT strings from branes wrapped on fibration cycles, but under the test-string (no-backreaction) approximation and without a model for how the initial loop population is produced. Subsequent work developed loop-tracker solutions, homogeneous-fluid backreaction including gravitational-wave friction, and conditional gravitational-wave spectra. However, the initial production mechanism for loops (number density, length distribution, correlations), the resulting percolation threshold for network formation, multi-rolling-modulus dynamics, field-theoretic strings with moduli-dependent tension, and self-consistent loop-distribution backreaction remain open, as explicitly acknowledged in the latest follow-up work. The problem of completing the varying-tension growth mechanism into a quantitative network-formation scenario in realistic compactifications is therefore open.

Background

Cosmic strings are one-dimensional topological defects that can form in the early universe; a network of them sources a stochastic gravitational-wave background (SGWB) searched for by pulsar-timing arrays and by future detectors such as LISA. In string theory the corresponding objects are cosmic superstrings — fundamental (F-)strings, D-strings and their $(p,q)$ bound states (bound states of $p$ F-strings and $q$ D-strings) — together with effective-field-theory (EFT) strings obtained when a $p$-brane wraps a $(p-1)$-cycle of the compact extra dimensions. Unlike field-theoretic defects, their tension $\mu$ is not constant: it depends on the expectation values of moduli, the scalar fields parameterising the size and shape of the compactification, such as the volume modulus of type IIB compactifications and the dilaton controlling the string coupling. Since moduli can roll after inflation, $\mu$ can be time-dependent. A Nambu–Goto string (a string whose dynamics is fixed by its tension per unit length) with decreasing tension has its physical loop size grow as $\ell(t)=\ell_i\sqrt{\mu_i/\mu(t)}$, and when $2H+\dot{\mu}/\mu<0$, with $H$ the Hubble rate, loops grow even in comoving coordinates, so an initially isolated population of small loops can meet and percolate into a network. This growth condition is satisfied for F-strings in a kination background — an epoch in which the kinetic energy of the rolling volume modulus dominates, $\rho\propto a^{-6}$ — which arises naturally after volume-modulus inflation in the Large Volume Scenario (LVS), a class of type IIB compactifications with exponentially large internal volume, whose post-inflationary modulus evolution is plagued by the overshoot problem (a rolling modulus whose post-inflation kinetic energy carries it past its potential minimum, threatening the compactification). The mechanism was proposed by Conlon–Copeland–Hardy–Sánchez González (2024), and Revello–Villa (2024) generalised the velocity-one-scale model (the standard phenomenological framework describing a string network by a correlation length and a root-mean-square string velocity) to time-varying tension. Brunelli–Cicoli–Pedro (2025) then recast the growth analysis as an autonomous dynamical system (a set of first-order evolution equations in the number of e-foldings whose fixed points classify the asymptotic cosmological behaviours) for a spatially flat universe containing a barotropic fluid (a component whose pressure is a fixed function of its energy density, such as radiation or matter) and a single rolling modulus with tension $\mu\sim e^{\xi\phi}$. They derived the growth region of phase space, showed that kination is an unstable fixed point supporting at most an $\mathcal{O}(10)$-e-fold transient of growth while stable scaling fixed points (where the modulus energy density tracks the fluid) can sustain growth for long epochs, established growth for F-strings and for EFT strings from D3-branes wrapped on the base 2-cycle and NS5-branes wrapped on the fibre 4-cycle of fibred Calabi–Yau threefolds (six-dimensional compact complex manifolds with vanishing first Chern class, the standard internal spaces of string compactifications, here admitting a torus-fibration structure), and showed that EFT strings from blow-up cycles (cycles resolving point-like singularities, with sub-Planckian field range) cannot sustain appreciable growth. Throughout, the strings are treated as test objects that do not backreact on the background, and the initial loop population is assumed rather than derived. The unresolved target is to complete this mechanism: quantify how the initial loops are produced and whether they percolate into a network, extend the analysis to several rolling moduli and to field-theoretic strings whose tension depends on moduli, and make the backreaction of the string energy density self-consistent.

Problem Statement

The problem is to complete the time-varying-tension growth mechanism for cosmic strings so that it yields definite statements about network formation in realistic string compactifications, by resolving the extensions that Brunelli–Cicoli–Pedro (2025) identified as open and that subsequent work has not closed. All components serve one objective: deciding, with controlled approximations, whether the varying-tension growth mechanism actually produces cosmic string networks, and what gravitational-wave signal follows.

(1) Loop production and percolation: quantify the cosmological mechanism that produces the initial population of string loops in the post-inflationary, rolling-moduli background — their number density, size distribution, and spatial correlations — and use it to determine the minimum number of e-foldings of loop-size growth required for the loops to meet and percolate into a network. This is the input needed to convert the already-computed phase-space growth durations into statements about whether networks form, and it is the prerequisite for precise predictions of the stochastic gravitational-wave background of the formed network (and for assessing how growth alters a gas of highly excited strings, including its Hagedorn phase, the high-temperature regime of string theory in which the exponentially growing number of internal string states prevents a conventional thermal description).

(2) Multi-modulus backgrounds: extend the dynamical-systems growth analysis to scenarios with more than one rolling modulus, taking the simultaneous evolution of the volume modulus and the dilaton as the required scope and treating generalisation to further rolling moduli as optional, where coexisting string species (F-strings, D-strings, (p,q)-strings) have tensions with different dependences on the rolling fields, and the canonical normalisations, effective tension-modulus couplings, and autonomous-system structure all change.

(3) Field-theoretic strings: analyse the dynamics and possible growth of field-theoretic cosmic strings formed by spontaneous breaking of a gauge symmetry with a U(1) factor, whose Higgs vacuum expectation value — and hence tension — depends on bulk moduli and on brane-localised open-string moduli with different couplings to the bulk.

(4) Self-consistent backreaction: complete the existing homogeneous-fluid treatments of the string energy density by evolving the full loop length distribution together with its backreaction on the Friedmann background and the modulus dynamics, connecting the growth phase to network-level evolution.

Components (1)-(4) are posed jointly: no single one of them alone resolves the question, and the gravitational-wave signal named in the objective is a consequence of these components rather than a separate problem.

Progress already made — backreaction at the homogeneous single-length-fluid level, and gravitational-wave spectra for phenomenologically assumed loop populations — does not by itself answer the question, since those results are conditional on the uncomputed production model and on monochromatic or parametrised loop distributions.

The sixth issue raised by the source — analytic control of EFT strings from blow-up cycles — is treated as settled at the level relevant here, since the source itself bounds their growth to fewer than about 0.1 e-folding; a complete analytic treatment in realistic compactifications is a secondary aspect that a solution may but need not address.

References

1. Luca Brunelli, Michele Cicoli, Francisco G. Pedro, Growth of cosmic strings beyond kination, JCAP 07 (2025) 010, DOI:10.1088/1475-7516/2025/07/010, arXiv:2503.11293, https://arxiv.org/abs/2503.11293
2. Joseph P. Conlon, Edmund J. Copeland, Edward Hardy, Noelia Sánchez González, Percolating cosmic string networks from kination, Phys. Rev. D 110 (2024) 083537, DOI:10.1103/PhysRevD.110.083537, arXiv:2406.12637, https://arxiv.org/abs/2406.12637
3. Filippo Revello, Gonzalo Villa, Cosmic (super)strings with a time-varying tension, JCAP 04 (2025) 049, DOI:10.1088/1475-7516/2025/04/049, arXiv:2411.04186, https://arxiv.org/abs/2411.04186
4. Noelia Sánchez González, Joseph P. Conlon, Edmund J. Copeland, Edward Hardy, Dynamical systems and superstring phases in the early universe, JHEP 10 (2025) 121, DOI:10.1007/JHEP10(2025)121, arXiv:2505.14187, https://arxiv.org/abs/2505.14187
5. Luca Brunelli, Michele Cicoli, Francisco Gil Pedro, Dynamics of cosmic superstrings and the overshoot problem, JCAP 05 (2026) 042, DOI:10.1088/1475-7516/2026/05/042, arXiv:2510.06359, https://arxiv.org/abs/2510.06359
6. Luca Brunelli, Michele Cicoli, Muhammad Hassan, Seyed Ehsan Qoreishi, Francisco Gil Pedro, Gravitational Waves from Multiple Cosmic Superstrings and the Overshoot Problem, arXiv:2607.26124, https://arxiv.org/abs/2607.26124
7. Anish Ghoshal, Filippo Revello, Gonzalo Villa, Cosmic superstrings in large volume compactifications: PTAs, LISA and time-varying tension, JCAP 08 (2025) 092, DOI:10.1088/1475-7516/2025/08/092, arXiv:2504.20994, https://arxiv.org/abs/2504.20994
8. Joseph P. Conlon, Edmund J. Copeland, Edward Hardy, Noelia Sánchez González, High-frequency gravitational waves from superstring phases in the early universe, JHEP 08 (2026) 092, DOI:10.1007/JHEP08(2026)092, arXiv:2511.16404, https://arxiv.org/abs/2511.16404
9. Luca Brunelli, Filippo Revello, Gonzalo Villa, The gravitational wave landscape of cosmic string networks with varying tension, arXiv:2601.10790, https://arxiv.org/abs/2601.10790
10. Hubert Lau Sze Chun, Joseph P. Conlon, Evolution of Generic Varying-Tension Cosmic Strings in Expanding Spacetimes, Int. J. Mod. Phys. A, DOI:10.1142/S0217751X26501587, arXiv:2512.05170, https://arxiv.org/abs/2512.05170
11. Dibya Chakraborty, Ahmed Rakin Kamal, Fire at the Tip of the Throat: Hagedorn Phase after brane-antibrane inflation?, arXiv:2606.08767, https://arxiv.org/abs/2606.08767