Anisotropic fractional cosmology: K-essence theory

TL;DR

This paper explores fractional differential equations arising in anisotropic K-essence cosmology within quantum and classical frameworks, providing exact solutions and analyzing the impact of fractional calculus on cosmological models.

Contribution

It introduces fractional differential equations in the Wheeler-DeWitt equation for anisotropic K-essence cosmology and derives exact classical and quantum solutions, highlighting the role of fractional calculus.

Findings

Fractional differential equations depend on the barotropic parameter.

Exact classical solutions are obtained in different gauges.

Quantum solutions are derived using fractional power series.

Abstract

In the particular configuration of the scalar field K-essence in the Wheeler-DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is $\beta=\frac{2\alpha}{2\alpha - 1}$. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; when $\omega_{X} \in [0,1]$, the order belongs to the interval $1\leq \beta \leq 2$, and when $\omega_{X}\in[-1,0)$, the order belongs to the interval $0< \beta \leq 1$. In the quantum scheme, we introduce the factor ordering problem in the variables $(\Omega,\phi)$ and its corresponding momenta $(\Pi_\Omega, \Pi_\phi)$, obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time $T(\tau)$, however in the dust era we found a closed solution in the gauge time $\tau$. Keywords: Fractional derivative, Fractional Quantum Cosmology; K-essence formalism; Classical and Quantum exact solutions.