Breaking the Strings: the signatures of Cosmic String Loop Fragmentation

TL;DR

The paper investigates how fragmentation cascades modify the cosmic string loop distribution, using a Boltzmann-like framework inspired by the three-scale model. It introduces a simplified yet tractable fragmentation model with a boundary production term and a fragmentation function, neglecting loop collisions, and derives a linear master equation in both physical and scaling variables. Two independent numerical methods are developed—an Unconnected Loop Model (ULM) that samples fragmentation cascades and a tailored Integro-Differential Equation (IDE) solver that exploits a triangular scaling structure—to obtain the full loop distribution including gravitational-wave losses. The results show that fragmentation reduces large-scale loop density and induces a departure from a pure power-law, with a characteristic asymptotic slope of $-5/2$ above the correlation length in both radiation and matter eras; radiation-era loops are more sensitive to fragmentation, while matter-era small scales remain relatively robust. These findings imply that fragmentation must be accounted for when predicting the stochastic gravitational-wave background and could help reconcile differences between simulations, with implications for current and future GW observatories.

Abstract

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.

Figures (7)

• Figure 1: The loop fragmentation function for a parent loop $\ell / t = 0.1$ and different assumptions for the small-scale behaviour $\sigma$.
• Figure 2: Schematic view of a loop fragmentation cascade. From an initial loop, we draw its lifetime $\Delta t$ randomly and fragment it into two children loops. The sizes of the children loops are also drawn randomly from the loop fragmentation function $\mathcal{B}(y, \ell- y; \ell)$.
• Figure 3: Comparison between the two numerical methods, the Unconnected Loop Model (ULM) in blue and the custom IDE solver in yellow, as we increase the number of fragmentation cascades. The parameters for this figure are $(C, \alpha, \chi, \xi_c, \sigma) = (1, 0.1, 0.2, 10^{-2}, 0)$ in radiation era.
• Figure 4: Loop number density in scaling units $n(\gamma)$ for different values of the correlation length $\xi_c$ obtained with our IDE solver. The black dashed line corresponds to the one-scale model, assuming no fragmentation, i.e. $\mathcal{B} = 0$. Parameters were set to $(C, \alpha, \chi, \Gamma G\mu, \sigma) = (1, 0.1, 0.2, 10^{-7}, 8)$.
• Figure 5: Loop number density in scaling units for different values of the free parameter $\sigma$ encoding our uncertainties about the fragmentation model. The zoom-in region shows how the value of $\sigma$ can help smoothing the sudden drop in the region $]\alpha - \xi_c, \alpha[$ and therefore ease the numerical resolution. The black dashed line corresponds to the one-scale model, assuming no fragmentation, i.e. $\mathcal{B} = 0$. The parameters were set to $(C, \alpha, \chi, \nu, \xi_c) = (1, 0.1, 1/2, 10^{-2})$.