Fractional Time-Delayed differential equations: Applications in Cosmological Studies

TL;DR

The paper develops a fractional time-delayed framework to model cosmological evolution with memory effects, deriving Caputo-based equations for the Hubble parameter that include delayed bulk-viscosity terms and solving them analytically and numerically. By analyzing first-, fractional-, and higher-order delay differential equations, it establishes characteristic structures via Laplace transforms and Mittag-Leffler functions, and demonstrates a de Sitter attractor at late times. The work extends to fractional cosmology with multiple delays, introduces mollifier-based smoothing for stable numerics, and provides generalization to arbitrary delay configurations. Overall, it offers a mathematically rigorous, memory-informed approach to viscous cosmology that can replicate accelerated expansion without invoking scalar fields, with practical numerical schemes for exploration and validation.

Abstract

Fractional differential equations model processes with memory effects, providing a realistic perspective on complex systems. We examine time-delayed differential equations, discussing first-order and fractional Caputo time-delayed differential equations. We derive their characteristic equations and solve them using the Laplace transform. We derive a modified evolution equation for the Hubble parameter incorporating a viscosity term modeled as a function of the delayed Hubble parameter within Eckart's theory. We extend this equation using the last-step method of fractional calculus, resulting in Caputo's time-delayed fractional differential equation. This equation accounts for the finite response times of cosmic fluids, resulting in a comprehensive model of the Universe's behavior. We then solve this equation analytically. Due to the complexity of the analytical solution, we also provide a numerical representation. Our solution reaches the de Sitter equilibrium point. Additionally, we present some generalizations.

Figures (9)

• Figure 1: Analytical solution $H(t)$, for the cases $\gamma=4/3,1$. The other parameters are $\eta_0=0.2$, $T=20$, $H_0=1$, $y_0= H_0-H_B$. The dashed line represents the de Sitter solution.
• Figure 2: Analytical $q(t)$ and $\omega_\text{eff}(t)$ given by \ref{['q_2']} and \ref{['weff_2']} respectively, for the cases $\gamma=4/3,1$. The other parameters are $\eta_0=0.2$, $T=20$ and $H_0=1$. The minimun values of $q(t)$ and $\omega_\text{eff}(t)$ are $(q=-18.9,\omega_\text{eff}=-12.9)$ for the case $\gamma=4/3$ (radiation), and $(q=-10.7,\omega_\text{eff}=-7.47)$ for the case $\gamma=1$ (matter).
• Figure 3: Analytical $H_m(t)$ for $m=500$ from \ref{['solB']}, with $\alpha=0.9$, $\gamma=4/3,1$ and $\eta_0=0.2$, $T=20$, and $H_0=1$. Using $H_n^{\text{corrected}}(t)$ defined by \ref{['H_summation']} for $n=114$ achieves high accuracy with fewer terms.
• Figure 4: $H_n^{\text{corrected}}(t)$ defined by \ref{['H_summation']} for $n=114$, with $\alpha=0.9$, $\gamma=4/3,1$ and $\eta_0=0.2$, $T=20$, and $H_0=1$ compared with Mittag-Leffler $H_B +( H_0 - H_B ) E (\alpha , - 2 \eta_0 t \alpha )$.
• Figure 5: $H(t)$ for the numerical solution using the formula for the fractional Euler method \ref{['fract-Euler-Method']}, for $\alpha=0.9$ and $\gamma=4/3,1$, and Mittag-Leffler function $H_B+E\left(\alpha,-2\eta_0 t^\alpha\right)$. The other parameters are $\eta_0=0.2$, $T=20$ and $H_0=1$. The dashed line represents the de Sitter solution.

Theorems & Definitions (32)

• Remark 1
• Remark 2
• Proposition 1
• proof
• Remark 3
• Proposition 2
• proof
• Remark 4
• Proposition 3
• proof