Delayed Scaling of Multi-Type Cosmic F- and D-strings in VOS Models

We investigate the velocity-dependent one-scale (VOS) model to the case of one cosmic F-string and two D-strings as color flux tubes in pure Spin($4N$) gauge theory. We analytically calculate the scaling string density as a function of the reconnection probabilities, and confirm our results with numerical calculations. We also determine the timescale at which the string density reaches the scaling regime, and find that for certain values of the reconnection probability, the scaling time can become extremely large, by many orders of magnitude. This leads to a characteristic suppression signature of the gravitational-wave signal at high frequencies, which may become observable in the frequency range of future interferometric gravitational-wave observations.

💡 Research Summary

The paper investigates the dynamics of a three‑species cosmic string network that arises in a pure Spin(4N) Yang‑Mills theory without any Higgs fields or fermions. In this hidden‑sector gauge theory the confinement scale Λ (taken to be ≲10¹² GeV) produces macroscopic color‑flux tubes which behave as cosmic strings. Because the one‑form center symmetry of Spin(4N) is Z₂ × Z₂, there are three non‑trivial center charges: one vector‑type charge (the “F‑string”) and two spinor‑type charges (the “D₁‑ and D₂‑strings”). Their tensions satisfy μ_F ∼ Λ² and μ_D₁ = μ_D₂ ∼ N Λ², so the D‑strings are heavier by a factor of N. Large‑N arguments suggest that the reconnection probabilities are strongly suppressed: p_F ∼ N⁻² for F‑strings and p_D ∼ e^{‑cN} for D‑strings, where c is an O(1) constant. The authors treat these probabilities as free parameters because a precise first‑principles calculation is not yet available.

To describe the evolution of such a network the authors extend the velocity‑dependent one‑scale (VOS) model, which traditionally tracks a single correlation length ξ and RMS velocity v for a homogeneous string network. In the multi‑species case the energy densities ρ_i (i = F, D₁, D₂) and velocities v_i obey coupled differential equations (Eqs. 3.1–3.2). The equations contain: (i) Hubble dilution, (ii) energy loss to small loops, modeled by an effective loop‑chopping efficiency \tilde{c} multiplied by the appropriate reconnection probability, (iii) “zipping” interactions where two different string species collide and produce a third species while consuming portions of the original strings, and (iv) curvature‑driven acceleration terms with the momentum‑parameter k(v) taken from Nambu‑Goto simulations. The correlation length ξ_i is identified with the average inter‑string distance L_i, while the curvature radius R_i is set equal to ξ_i.

Analytically, the authors look for scaling solutions where ξ_i ∝ t and v_i approach constant values. In the limit of unit reconnection probability the standard VOS scaling (ξ ≈ 0.3 t, v ≈ 0.6) is recovered. However, when p_i ≪ 1 the loop‑loss term becomes negligible and the network can no longer shed length efficiently. The analytic treatment shows that the scaling solution still exists but the approach to it is delayed: the time required for ξ_i/t to reach its asymptotic value scales roughly as t_s ∝ p_i^{‑α} with α≈1–2 depending on the hierarchy of tensions. In particular, if the product p_F p_D is below ≈10⁻⁶, the scaling time can exceed the age of the universe by several orders of magnitude. This “prolonged scaling” is confirmed by numerical integration of the full VOS system, where the authors vary p_F and p_D over many decades and map out the region of parameter space that yields delayed scaling.

The phenomenological consequence of delayed scaling is a suppression of the loop production rate n_loop ∝ ξ⁻³. Since loops are the primary source of stochastic gravitational‑wave (GW) background in Nambu‑Goto string models, a reduced loop density translates into a lowered GW spectrum at frequencies corresponding to loops formed before scaling is achieved. The authors compute Ω_GW(f) using the standard loop‑distribution formalism, considering both the case of equal tensions (μ_F = μ_D) and a hierarchical case (μ_D ≫ μ_F). In both scenarios the high‑frequency tail (f ≳ 10 Hz, relevant for ground‑based interferometers) is strongly attenuated, while the low‑frequency part (f ≲ nHz, probed by pulsar‑timing arrays) remains essentially unchanged because those frequencies are sourced by loops formed after scaling has been reached. The transition frequency f_trans, where the spectrum bends downward, is directly linked to the scaling time t_s; measuring such a break would therefore provide an indirect probe of the reconnection probabilities and the existence of multiple string species.

The paper also includes several appendices. Appendix A gives a detailed derivation of the scaling solution for the extended VOS model, distinguishing the behavior of the correlation length for the light (F) and heavy (D) strings. Appendix B reviews the conventional single‑species VOS model with a phenomenological p‑dependence, showing that it cannot capture the dramatic delay found in the multi‑species case. Appendix C repeats the analytic scaling analysis within the conventional framework for completeness.

In the discussion, the authors stress that while the model is motivated by pure Yang‑Mills confinement, the same formalism applies to cosmic superstrings in string theory, where small reconnection probabilities arise from quantum effects. They point out that future GW observatories such as the Einstein Telescope, Cosmic Explorer, and space‑based LISA will be sensitive to the high‑frequency suppression predicted here, offering a novel way to test hidden‑sector gauge dynamics. They also note that a more precise determination of the reconnection and zipping probabilities would require large‑N lattice simulations or holographic calculations, which they leave for future work.

Overall, the study provides a comprehensive theoretical framework for multi‑type string networks with suppressed reconnection, demonstrates that delayed scaling can be many orders of magnitude longer than the Hubble time, and translates this dynamical effect into a distinctive gravitational‑wave signature that could be observable with next‑generation detectors.