Breaking the strings: the signatures of cosmic string loop fragmentation

Author(s)

Auclair, Pierre

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

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The loop fragmentation function for a parent loop $\ell / t = 0.1$ and different assumptions for the small-scale behaviour $\sigma$.

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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 $\lff(y, \ell- y; \ell)$.

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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.

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: Radiation era

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: Matter era : 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.\ $\lff = 0$. Parameters were set to $(C, \alpha, \chi, \Gamma G\mu, \sigma) = (1, 0.1, 0.2, 10^{-7}, 8)$.

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: Radiation era

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: Matter era : 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.\ $\lff = 0$. The parameters were set to $(C, \alpha, \chi, \nu, \xi_c) = (1, 0.1, 1/2, 10^{-2})$.

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: Radiation era

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: Matter era : Loop number density in scaling units for different values of the velocity $\chi$ encoding the strength of the fragmentation model. The black dashed line corresponds to the one-scale model, assuming no fragmentation, i.e.\ $\lff = 0$. The parameters were set to $(C, \alpha, \nu, \Gamma G\mu, \xi_c) = (1, 0.1, 1/2, 10^{-7}, 10^{-3})$.

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: Radiation era

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: Matter era : Loop number density in scaling units $n(\gamma)$ obtained by moving the boundary condition, i.e.\ the position of the loop production scale $\alpha$. We rescaled the value of $C$ to keep $C \alpha$ constant. Parameters were set to $(\chi, \Gamma G\mu, \xi_c, \sigma) = (0.2, 10^{-7}, 10^{-2}, 8)$.