The velocity-dependent one-scale (VOS) model with kinematic constraints of Avgoustidis and Copeland (Phys. Rev. D 81, 063517 (2010)) predicts that multi-tension (p,q) cosmic-superstring networks reach scaling, with species abundances and scaling densities that can exceed the single-tension case by roughly an order of magnitude. However, its macroscopic coefficients are not derived: the junction energy-transfer prescription is only bracketed between two limiting cases, the compactification-dependent binding and intercommutation probabilities and the string-coupling dependence are free inputs, the collision-velocity distribution is an unvalidated Gaussian with universal variance, and the Nambu-Goto-derived kinematic suppression factors SijkS_{ij}^k have an unquantified domain of validity relative to field-theory strings. Later work has resolved the observational consequences of the model for the cosmic microwave background (CMB) and for gravitational waves but still fixes these microscopic inputs by hand. The open problem is to determine these inputs from string-theory microphysics and field-theory or Nambu-Goto simulations — with quantified uncertainties over the compactification and string-coupling parameter space — so that the model's scaling predictions are determined rather than bracketed, and to quantify the validity domain of the Nambu-Goto kinematics underlying SijkS_{ij}^k.

Background

Cosmic strings are linear topological defects that may have formed during symmetry-breaking phase transitions in the early universe; in string theory cosmology, the fundamental string (F-string) and one-dimensional D-brane (D-string) left over from brane inflation (a class of inflationary models in which inflation ends as two branes collide and annihilate) form a stretched cosmic-superstring network. Because the effective string tension depends on the compactification of the extra dimensions, superstrings of different types can bind into (p,q)(p,q) bound states (p F-strings and q D-strings), and collisions between them can create junctions where three string segments of different tension meet. Ordinary field-theory cosmic strings intercommute (reconnect by exchanging the partners of the colliding pair) with probability essentially 1, while superstrings can reconnect with probabilities as low as 10−310^{-3}, an effect that slows energy loss and increases the density of strings surviving at the present epoch — which is why superstrings remain compatible with observational bounds that exclude field-theory strings of comparable tension.

The standard analytical description of string-network evolution is the velocity-dependent one-scale (VOS) model, in which each string species is characterized by a correlation length LiL_i and root-mean-square velocity viv_i obeying a closed system of ordinary differential equations. In the scaling regime, LiL_i grows in proportion to the horizon time so that the network's density per species is constant in comoving units. For multi-tension networks, the VOS equations must be supplemented by terms describing the formation and dynamics of junctions. Avgoustidis and Copeland (2010) derived kinematic constraints on junction formation from Nambu-Goto collisions: the Nambu-Goto action is the leading thin-string approximation in which a string's dynamics is pure free motion constrained only by its fixed tension per unit length. Analyzing the energetically allowed post-collision configurations of colliding string segments, they obtained suppression factors SijkS_{ij}^k that multiply the rate at which collisions between species ii and jj produce a segment of species kk bound at a junction. These kinematic constraints compare favorably, in aggregate, with Abelian-Higgs field-theory lattice simulations (the alternative treatment in which the string is a vortex of a complex scalar field evolved directly), and they enhance predicted scaling densities relative to unconstrained models by roughly an order of magnitude.

The model's predictive power is, however, limited by inputs that are phenomenological rather than derived. First, the kinematics is Nambu-Goto; field-theory strings have a finite core width and radiate differently, and the domain over which the Nambu-Goto-derived SijkS_{ij}^k apply to field-theory (or true super-) strings is unquantified. Second, the coefficients mapping microscopic physics to macroscopic evolution — the compactification-dependent probabilities d~(p,q),(p′,q′)\tilde d_{(p,q),(p',q')} that a collision produces a given bound state, the intercommutation probabilities (e.g. PFFP_{FF} of order 10−210^{-2} to 10−110^{-1}, PDDP_{DD} of order 10−110^{-1} to 1), and their dependence on the string coupling gsg_s — are known only as loose ranges. Third, the averaging over collision velocities uses a Gaussian distribution with universal variance σv2=0.25\sigma_v^2=0.25 independent of tension, and the species hierarchy is truncated at four-string composites (seven distinct tensions) without a convergence test. Fourth, the length of string bound into a new junction is fixed by the prescription ℓijk=LiLj/(Li+Lj)\ell_{ij}^k=L_iL_j/(L_i+L_j), and the binding energy released at junction formation is treated in two limiting cases — fully radiated away (b~ijk=0\tilde b_{ij}^k=0) versus fully redistributed as kinetic energy (b~ijk=d~ijk\tilde b_{ij}^k=\tilde d_{ij}^k) — which bracket rather than determine the answer. The observational side of this program is mature: the CMB anisotropy and gravitational-wave consequences of the kinematically constrained scaling solutions have been computed and used to constrain the tension and coupling from Planck satellite maps, from the Atacama Cosmology Telescope (ACT), and from pulsar-timing data. The unresolved side is the microphysical anchoring of the model itself, which is the subject of this problem.

Problem Statement

Determine the microscopic inputs of the kinematically constrained multi-tension VOS model from first-principles physics, replacing its free phenomenological coefficients with derived or simulation-validated values, and quantify the domain of validity of its Nambu-Goto kinematics. Concretely: (1) derive or measure the junction energy-transfer fractions b~ijk\tilde b_{ij}^k — identifying the actual dissipation channels (radiation of loop-scale and microscopic F-strings, gravitational radiation, kinetic redistribution) and their efficiencies — so that the choice between the bracketing prescriptions b~ijk=0\tilde b_{ij}^k=0 and b~ijk=d~ijk\tilde b_{ij}^k=\tilde d_{ij}^k is settled rather than imposed; (2) determine the compactification-dependent binding probabilities d~(p,q),(p′,q′)\tilde d_{(p,q),(p',q')} and the intercommutation probabilities as functions of the string coupling gsg_s and compactification geometry, with quantified uncertainties, and propagate them through the model to establish whether the order-of-magnitude enhancement of the scaling densities survives; (3) validate or replace the assumed Gaussian collision-velocity distribution with universal variance σv2=0.25\sigma_v^2=0.25, including its possible tension dependence; (4) establish the convergence of the species truncation and the sensitivity of the scaling solution to the assumed produced-link length ℓijk\ell_{ij}^k and to small-scale structure on the strings; and (5) quantify the domain of validity of the Nambu-Goto-derived suppression factors SijkS_{ij}^k with respect to field-theory strings. These five components are jointly required: each addresses a distinct input or validity condition, and the network's scaling predictions are determined rather than bracketed only when all five are fixed. The common objective uniting these components is that the scaling densities and species abundances of the multi-tension network be predicted with quantified uncertainties across the compactification and gsg_s parameter space, rather than bracketed by limiting prescriptions.

References

  1. A. Avgoustidis and E. J. Copeland, "Effect of kinematic constraints on multitension string network evolution", Physical Review D 81, 063517 (2010). DOI: 10.1103/PhysRevD.81.063517; arXiv:0912.4004, https://arxiv.org/abs/0912.4004
  2. A. Pourtsidou, A. Avgoustidis, E. J. Copeland, L. Pogosian, and D. A. Steer, "Scaling configurations of cosmic superstring networks and their cosmological implications", Physical Review D 83, 063525 (2011). DOI: 10.1103/PhysRevD.83.063525; arXiv:1012.5014, https://arxiv.org/abs/1012.5014
  3. A. Avgoustidis, E. J. Copeland, A. Moss, L. Pogosian, A. Pourtsidou, and D. A. Steer, "Constraints on the fundamental string coupling from B-mode experiments", Physical Review Letters 107, 121301 (2011). DOI: 10.1103/PhysRevLett.107.121301; arXiv:1105.6198, https://arxiv.org/abs/1105.6198
  4. T. Charnock, A. Avgoustidis, E. J. Copeland, and A. Moss, "CMB constraints on cosmic strings and superstrings", Physical Review D 93, 123503 (2016). DOI: 10.1103/PhysRevD.93.123503; arXiv:1603.01275, https://arxiv.org/abs/1603.01275
  5. L. Sousa and P. P. Avelino, "Probing Cosmic Superstrings with Gravitational Waves", Physical Review D 94, 063529 (2016). DOI: 10.1103/PhysRevD.94.063529; arXiv:1606.05585, https://arxiv.org/abs/1606.05585
  6. I. Yu. Rybak, A. Avgoustidis, and C. J. A. P. Martins, "Dynamics of junctions and the multitension velocity-dependent one-scale model", Physical Review D 99, 063516 (2019). DOI: 10.1103/PhysRevD.99.063516; arXiv:1812.04584, https://arxiv.org/abs/1812.04584
  7. A. Avgoustidis, E. J. Copeland, A. Moss, and J. Raidal, "The stochastic gravitational wave background from cosmic superstrings", Journal of Cosmology and Astroparticle Physics 07 (2025) 091. DOI: 10.1088/1475-7516/2025/07/091; arXiv:2503.10361, https://arxiv.org/abs/2503.10361
  8. K. Nakamura and M. Yamada, "Delayed Scaling of Multi-Type Cosmic F- and D-strings in VOS Models", arXiv:2512.16821 (2025), https://arxiv.org/abs/2512.16821
  9. J. Raidal, A. Avgoustidis, E. Copeland, and A. Moss, "CMB anisotropies from cosmic (super)strings in light of ACT DR6", arXiv:2602.18272 (2026), https://arxiv.org/abs/2602.18272