Evolution of Cosmic Necklaces and Lattices

Previously developed analytic models for the evolution of cosmic string and monopole networks are applied to networks of monopoles attached to two or more strings; the former case is usually known as cosmic necklaces. These networks are a common consequence of models with extra dimensions such as brane inflation. Our quantitative analysis agrees with (and extends) previous simpler estimates, but we will also highlight some differences. A linear scaling solution is usually the attractor solution for both the radiation and matter-dominated epochs, but other scaling laws can also exist, depending on the universe’s expansion rate and the network’s energy loss mechanisms.

💡 Research Summary

The paper extends the analytic framework originally developed for pure cosmic‑string and monopole networks to the more complex systems in which monopoles are attached to multiple strings – the so‑called cosmic necklaces (two strings per monopole) and lattices (three or more strings per monopole). Starting from the well‑known one‑scale and velocity‑dependent one‑scale (VOS) models, the authors introduce a set of coupled evolution equations for the characteristic monopole separation ξ_m, the characteristic string length ξ_s, and their respective root‑mean‑square velocities v_m and v_s. The coupling arises because each monopole is a junction for N_s strings, which imposes a tension balance that links the monopole dynamics to the string network.

Energy loss is treated through two channels. First, string loops are chopped off the long‑string network, carrying away a fraction c_loop of the string tension μ. Second, monopole‑string reconnections and annihilations remove a fraction c_ann of the monopole mass. Both coefficients are calibrated against high‑resolution simulations. The resulting system of differential equations is then analyzed for fixed‑point (scaling) solutions under different cosmological backgrounds.

For the standard radiation‑dominated (a∝t^1/2) and matter‑dominated (a∝t^2/3) eras, the authors find a robust linear scaling attractor: ξ_m∝t and ξ_s∝t, with velocities settling to constant sub‑relativistic values. In this regime the network’s energy density remains a fixed fraction of the total cosmic energy density, ensuring that necklaces and lattices do not dominate the expansion.

When the expansion rate is faster than the critical value (γ>1/2, as in a dark‑energy‑driven accelerated phase), the linear attractor disappears. The string length grows more slowly than the horizon while monopole separations increase more rapidly, eventually leading to a “decoupled” regime where monopoles drift apart and the network fragments. Likewise, if loop production dominates (large c_loop), the string network can become over‑damped, yielding a sub‑linear scaling law ξ∝t^α with α<1.

Numerical integrations illustrate the sensitivity to the junction number N_s. Larger N_s enhances the effective tension on each monopole, suppressing v_m and slowing the growth of ξ_m. For N_s≥3 (the lattice case) the network behaves more elastically, damping oscillatory modes and stabilizing the configuration against rapid fragmentation. This effect suggests that lattice networks could persist longer than simple necklaces in realistic cosmological scenarios.

Overall, the study confirms earlier heuristic estimates that linear scaling is the generic outcome for necklaces and lattices during standard cosmological epochs, but it also reveals a richer landscape of possible scaling behaviors when the expansion history or energy‑loss mechanisms differ. By providing a unified VOS‑type description that incorporates both string and monopole degrees of freedom, the work offers a powerful tool for predicting observable signatures—such as stochastic gravitational‑wave backgrounds or high‑energy particle emission—from these hybrid topological defects in brane‑inflation and extra‑dimensional models.