Anisotropic fractional cosmology: K-essence theory
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 $β=\frac{2α}{2α- 1}$. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; when $ω_{X} \in [0,1]$, the order belongs to the interval $1\leq β\leq 2$, and when $ω_{X}\in[-1,0)$, the order belongs to the interval $0< β\leq 1$. In the quantum scheme, we introduce the factor ordering problem in the variables $(Ω,ϕ)$ and its corresponding momenta $(Π_Ω, Π_ϕ)$, 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 $T(τ)$, however in the dust era we found a closed solution in the gauge time $τ$. Keywords: Fractional derivative, Fractional Quantum Cosmology; K-essence formalism; Classical and Quantum exact solutions.