Fractional Cosmic String Loops In Expanding Universe

Author(s)

Chaturvedi, Pankaj, Nath, Bikram

Abstract

We study the dynamics of circular cosmic string loops in a spatially flat Friedmann Lemaître Robertson Walker universe within a fractional Polyakov framework that incorporates nonlocal memory effects. Allowing both the loop radius and polar angle to evolve, we obtain a coupled non-autonomous system governed by string tension, cosmological expansion, and an emergent centrifugal contribution. We show that angular dynamics plays a crucial role in determining the loop evolution. In contrast to standard scenarios where loops collapse, we identify a class of solutions exhibiting sustained expansion driven by dynamically generated angular motion. The system also displays nonlinear behavior with signatures of chaos, with the onset of chaotic dynamics closely correlated with expanding solutions. Our results demonstrate that fractional memory effects and angular degrees of freedom qualitatively modify cosmic string loop dynamics, providing new mechanisms for stability in cosmological backgrounds.

Figures

Evolution of the normalized loop radius $R/R_0$ as a function of $y$ for different initial times $\tau_0$. \textbf{Left panel:} $R_0=1$, $\theta_0=\pi/2$, $\alpha=0.75$. Curves correspond to $\tau_0=\{0.1,\,1,\,5,\,10\}$ sec (blue, red, brown, green), collapsing at $y=\{0.691226,\,0.231953,\,0.0623543,\,0.0325816\}$ respectively. \textbf{Right panel:} $R_0=1$, $\theta_0=\pi/4$, $\alpha=0.75$ with the same $\tau_0$ values. Collapse occurs at $y=\{0.774084,\,0.324634,\,0.112622,\,0.0631037\}$.
Caption Evolution of the normalized loop radius $R/R_0$ as a function of $y$ for different initial times $\tau_0$. \textbf{Left panel:} $R_0=1$, $\theta_0=\pi/2$, $\alpha=0.75$. Curves correspond to $\tau_0=\{0.1,\,1,\,5,\,10\}$ sec (blue, red, brown, green), collapsing at $y=\{0.691226,\,0.231953,\,0.0623543,\,0.0325816\}$ respectively. \textbf{Right panel:} $R_0=1$, $\theta_0=\pi/4$, $\alpha=0.75$ with the same $\tau_0$ values. Collapse occurs at $y=\{0.774084,\,0.324634,\,0.112622,\,0.0631037\}$.
Evolution of the normalized loop radius $R/R_0$ as a function of $y$ for different initial times $\tau_0$. \textbf{Left panel:} $R_0=1$, $\theta_0=\pi/2$, $\alpha=0.75$. Curves correspond to $\tau_0=\{0.1,\,1,\,5,\,10\}$ sec (blue, red, brown, green), collapsing at $y=\{0.691226,\,0.231953,\,0.0623543,\,0.0325816\}$ respectively. \textbf{Right panel:} $R_0=1$, $\theta_0=\pi/4$, $\alpha=0.75$ with the same $\tau_0$ values. Collapse occurs at $y=\{0.774084,\,0.324634,\,0.112622,\,0.0631037\}$.
Caption Evolution of the normalized loop radius $R/R_0$ as a function of $y$ for different initial times $\tau_0$. \textbf{Left panel:} $R_0=1$, $\theta_0=\pi/2$, $\alpha=0.75$. Curves correspond to $\tau_0=\{0.1,\,1,\,5,\,10\}$ sec (blue, red, brown, green), collapsing at $y=\{0.691226,\,0.231953,\,0.0623543,\,0.0325816\}$ respectively. \textbf{Right panel:} $R_0=1$, $\theta_0=\pi/4$, $\alpha=0.75$ with the same $\tau_0$ values. Collapse occurs at $y=\{0.774084,\,0.324634,\,0.112622,\,0.0631037\}$.
Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for angularly driven initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Caption Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for angularly driven initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for angularly driven initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Caption Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for angularly driven initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for purely radial evolution initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Caption Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for purely radial evolution initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for purely radial evolution initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Caption Evolution of the normalized physical radius $R/R_0$ as a function of the logarithmic time variable $y$ for purely radial evolution initial conditions. \textbf{Left panel:} Dependence on the fractional time parameter $t$ for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The curves correspond to $t=\{0.01$ (black), $0.1$ (blue), $1$ (red), $5$ (brown), $10$ (green)$\}$.\textbf{Right panel:} Dependence on the initial polar angle $\theta_0$ for fixed $R_0=1$, $\tau_0=1$. The curves correspond to $\theta_0=\{\pi/3$ (blue), $\pi/4$ (red), $\pi/6$ (brown), $\pi/12$ (green)$\}$.
Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
Caption Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
Caption Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
Caption Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
Caption Evolution of the Lyapunov spectrum as a function of integration steps for fixed $R_0=1$, $\alpha=0.75$, and $\theta_0=\pi/3$. The panels correspond to $\tau_0=10^{-16}, 10^{-8}, 10^{-5}, 1$ (top left to bottom right).
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