Anisotropic Fractional Cosmology: K-Essence Theory
Abstract
:1. Introduction
2. Brief Review on Fractional Calculus and K-Essence Theory
2.1. Brief Review on Fractional Calculus
2.2. K-Essence Fractional in the Bianchi Type I Scenario
3. Lagrange and Hamilton Formalism
3.1. Exact Solution in the Gauge
3.2. Exact Solution without Gauge N in the Time
3.3. Case for
- Dust Scenario,
4. Quantum Regime
Solution to FDE Associated with the Different State Evolutions
5. Conclusions
- Using the k-essence formalism in a general way, applied to the anisotropic Bianchi type I cosmological model, we found the Hamiltonian density in the scalar field momenta raised to powers of non-integers, which produces in the quantum scheme a fractional differential equation in a natural way. We include the factor-ordering problem in both variables and its momenta , with the order , where , and it was solved in a general way, we include two particular scenarios of our universe.
- We found the solution in the classical scheme employing two gauges, , for two forms of the function in the time t; however, when we let the Lagrange multiplier N, we need to employ a transformed time for solving the classical equation and, only in the dust era, we recover the gauge time .
- In the quantum regime, when we include the factor-ordering problem, the fractional differential equation in the scalar field appears with variable coefficients, and it was necessary to use the fractional series expansion to solve it in a general way.
- In one of our analyses presented on the probability density, we consider the values of the scalar field as significant in the quantum regime, appearing in various scenarios in the behavior of the universe, mainly in those where the universe has a huge behavior; for example, in the actual epoch, where the scalar field appears as a background, the quantum regime appears with big values, but it presents a moderate development in other scenarios with different ordering parameters Q and s.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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State of Evolution | |||
---|---|---|---|
1 | 1 | X | Stiff matter |
2 | Radiation | ||
, | Dust-like | ||
0 | 1, | Inflation | |
−1 | Inflation-like | ||
Inflation-like |
Fractionary Equation | |||
---|---|---|---|
1 | 1 | 1 | |
2 | |||
0 | 0 | ||
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Socorro, J.; Rosales, J.J.; Toledo-Sesma, L. Anisotropic Fractional Cosmology: K-Essence Theory. Fractal Fract. 2023, 7, 814. https://doi.org/10.3390/fractalfract7110814
Socorro J, Rosales JJ, Toledo-Sesma L. Anisotropic Fractional Cosmology: K-Essence Theory. Fractal and Fractional. 2023; 7(11):814. https://doi.org/10.3390/fractalfract7110814
Chicago/Turabian StyleSocorro, José, J. Juan Rosales, and Leonel Toledo-Sesma. 2023. "Anisotropic Fractional Cosmology: K-Essence Theory" Fractal and Fractional 7, no. 11: 814. https://doi.org/10.3390/fractalfract7110814