Breaking the Strings: the signatures of Cosmic String Loop Fragmentation

We study the impact of fragmentation on the cosmic string loop number density, using an approach inspired by the three-scale model and a Boltzmann equation. We build a new formulation designed to be more amenable to numerical resolution and present two complementary numerical methods to obtain the full loop distribution including the effect of fragmentation and gravitational radiation. We show that fragmentation generically predicts a decay of the loop number density on large scales and a deviation from a pure power-law. We expect fragmentation to be crucial for the calibration of loop distribution models.

💡 Research Summary

This paper presents a comprehensive study on the impact of fragmentation on the size distribution of cosmic string loops, offering new theoretical modeling and numerical methods to address long-standing discrepancies in the field.

The authors begin by contextualizing the problem: cosmic strings are hypothetical one-dimensional topological defects that could have formed in the early universe. A network of these strings continuously produces closed loops, which then evolve and potentially emit gravitational waves. Predicting the number density of these loops as a function of their size is crucial for observational signatures but remains theoretically challenging. Existing models (like the LRS and BOS models) disagree, particularly on small scales, and a key hypothesized reason is the treatment of loop fragmentation—the process where a large loop self-intersects and splits into two smaller daughter loops.

To isolate and study fragmentation, the authors construct a simplified model. They neglect loop collisions (arguing they are rare compared to fragmentations) to obtain a linear master equation. The model incorporates three main ingredients: 1) A loop production function P(ℓ,t), which injects loops of a fixed fraction α (~0.1) of the horizon size from the infinite string network. 2) Energy loss via gravitational wave emission, at a rate ˙ℓ = -ΓGμ. 3) A fragmentation function B(y, ℓ-y; ℓ), which gives the rate for a loop of length ℓ to fragment into two loops of lengths y and ℓ-y. A physically motivated form for B is adopted for large daughter loops (y > ξ, where ξ is a network correlation length), proportional to (ξ/y)^(3/2). For very small daughter loops (y < ξ), the unknown behavior is parameterized by an exponent σ, encapsulating theoretical uncertainty.

The core master equation is a linear partial integro-differential equation (IDE) that tracks the loop number density N(t,ℓ) in an expanding universe, balancing loop production, fragmentation loss, and the gain from larger loops fragmenting into the size of interest.

Solving this equation is non-trivial. The paper’s significant technical contribution is the development of two complementary, independent numerical methods:

1. The Unconnected Loop Model (ULM): A stochastic, “microscopic” Monte Carlo approach. It simulates individual fragmentation cascades by randomly drawing loop formation times from the production function, then recursively drawing their fragmentation lifetimes and the sizes of their offspring from probabilistic distributions derived from B(y, ℓ-y; ℓ). The final distribution is built from a large ensemble of such cascades.
2. A custom Integro-Differential Equation (IDE) solver: A deterministic, “macroscopic” approach. By scaling variables (γ = ℓ/t), the IDE is rewritten. The solver exploits the “triangular” structure of the equation—the density at a given scale γ depends only on the density at larger scales Z > γ. This allows for an efficient, top-down resolution starting from the largest loops at γ=α.

The results from both methods agree convincingly, validating the framework. The key findings are:

• Fragmentation generically suppresses the number density of large loops. The larger the fragmentation rate (controlled by parameter χ) or the smaller the correlation length ξ_c, the stronger this suppression.
• It causes a deviation from a pure power-law scaling in the loop distribution n(γ). The cascade of energy from large to small scales distorts the simple scaling expected in non-fragmentation models.
• The behavior at very small scales (γ → 0) is sensitive to the parameter σ, highlighting the need for better understanding of fragmentation physics at the smallest scales.

In conclusion, the paper argues that fragmentation is not a minor effect but a fundamental and unavoidable process in cosmic string network evolution. It must be explicitly included to accurately calibrate loop distribution models. This work provides the necessary tools and formalism to do so, paving the way for more precise predictions of cosmic string observables, particularly their gravitational wave signals, and potentially reconciling differences between various simulation approaches.