Fractional Gravity/Cosmology in Classical and Quantum Regimes

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 6268

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Departamento de Física, Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, Rua Marquês d’Avila e Bolama, 6200-001 Covilhã, Portugal
Interests: general relativity; quantum field theory; gravitational physics; quantum mechanics; theoretical particle; physics; high energy physics
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Departamento de Física, Universidade Federal de Pernambuco, Recife 50670-901, Brazil
Interests: theoretical physics; quantum gravity; quantum cosmology; foundations of quantum mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In fractional calculus, the power of the differentiation operator (which is not local) is any rational (or even real or complex) number. Recently, this has been widely used in various branches of physics. Such frameworks have played an important role in understanding complex systems in both the classical and quantum regimes. In what follows, we will focus on fractional gravity and cosmology.

Regarding the classical regime, the fractional derivative cosmology has been established by two different methods: (i) The last-step modification method is the simplest one, in which the given cosmological field equations for a specific model are replaced by the corresponding fractional field equations. (ii) The first-step modification method can be considered a more fundamental methodology. In this method one, starts by establishing fractional derivative geometry. More concretely, the variational principle for the fractional action is applied to establish a modified cosmological model. 

The main objective of the mentioned methods is the investigation of open problems in gravity/cosmology.  

In the context of fractional quantum mechanics, the fractional Schrödinger equation (SE) has been obtained with space, time and space–time fractional derivatives. These frameworks have been applied to solve various problems with different potentials. Moreover, fractional quantum mechanics has been applied as a tool within quantum field theory and gravity for fractional spacetime. Inspired by the modified SE mentioned above, the fractional Wheeler–DeWitt equation associated with the fractional quantum cosmology was also set up. 

New ideas, current developments, future perspectives and review articles on the above fractional proposals relevant to gravitation and cosmology are the focus of this Issue and are welcome.

Dr. Seyed Meraj Mousavi Rasouli
Dr. Shahram Jalalzadeh
Guest Editors

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Keywords

  • fractional calculus
  • fractional derivatives
  • Riesz fractional operator
  • Caputo fractional operator
  • fractional Brownian motion
  • Lévy path integrals
  • fractional action-like variational approach
  • fractional quantum mechanics
  • fractional classical cosmology
  • fractional quantum cosmology
  • non-local operators

Published Papers (4 papers)

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Research

21 pages, 1332 KiB  
Article
Estimated Age of the Universe in Fractional Cosmology
by Emanuel Wallison de Oliveira Costa, Raheleh Jalalzadeh, Pedro Felix da Silva Júnior, Seyed Meraj Mousavi Rasouli and Shahram Jalalzadeh
Fractal Fract. 2023, 7(12), 854; https://doi.org/10.3390/fractalfract7120854 - 30 Nov 2023
Viewed by 988
Abstract
Our proposed cosmological framework, which is based on fractional quantum cosmology, aims to address the issue of synchronicity in the age of the universe. To achieve this, we have developed a new fractional ΛCDM cosmological model. We obtained the necessary formalism by [...] Read more.
Our proposed cosmological framework, which is based on fractional quantum cosmology, aims to address the issue of synchronicity in the age of the universe. To achieve this, we have developed a new fractional ΛCDM cosmological model. We obtained the necessary formalism by obtaining the fractional Hamiltonian constraint in a general minisuperspace. This formalism has allowed us to derive the fractional Friedmann and Raychaudhuri equations for a homogeneous and isotropic cosmology. Unlike the traditional de Sitter phase, our model exhibits a power-law accelerated expansion in the late-time universe, when vacuum energy becomes dominant. By fitting the model’s parameters to cosmological observations, we determined that the fractional parameter of Lévy equals α=1.986. Additionally, we have calculated the age of the universe to be 13.8196 Gyr. Furthermore, we have found that the ratio of the age to Hubble time from the present epoch to the distant future is finite and confined within the interval 0.9858Ht<95.238. Full article
(This article belongs to the Special Issue Fractional Gravity/Cosmology in Classical and Quantum Regimes)
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19 pages, 1729 KiB  
Article
Anisotropic Fractional Cosmology: K-Essence Theory
by José Socorro, J. Juan Rosales and Leonel Toledo-Sesma
Fractal Fract. 2023, 7(11), 814; https://doi.org/10.3390/fractalfract7110814 - 09 Nov 2023
Viewed by 871
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 [...] Read more.
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=2α2α1. This fractional equation belongs to different intervals depending on the value of the barotropic parameter; when ωX[0,1], the order belongs to the interval 1β2, and when ωX[1,0), the order belongs to the interval 0<β1. In the quantum scheme, we introduce the factor ordering problem in the variables (Ω,ϕ) and its corresponding momenta (ΠΩ,Πϕ), 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. Full article
(This article belongs to the Special Issue Fractional Gravity/Cosmology in Classical and Quantum Regimes)
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36 pages, 1095 KiB  
Article
Exact Solutions and Cosmological Constraints in Fractional Cosmology
by Esteban González, Genly Leon and Guillermo Fernandez-Anaya
Fractal Fract. 2023, 7(5), 368; https://doi.org/10.3390/fractalfract7050368 - 29 Apr 2023
Cited by 5 | Viewed by 1679
Abstract
This paper investigates exact solutions of cosmological interest in fractional cosmology. Given μ, the order of Caputo’s fractional derivative, and w, the matter equation of state, we present specific exact power-law solutions. We discuss the exact general solution of the Riccati [...] Read more.
This paper investigates exact solutions of cosmological interest in fractional cosmology. Given μ, the order of Caputo’s fractional derivative, and w, the matter equation of state, we present specific exact power-law solutions. We discuss the exact general solution of the Riccati Equation, where the solution for the scale factor is a combination of power laws. Using cosmological data, we estimate the free parameters. An analysis of type Ia supernovae (SNe Ia) data and the observational Hubble parameter data (OHD), also known as cosmic chronometers, and a joint analysis with data from SNe Ia + OHD leads to best-fit values for the free parameters calculated at 1σ, 2σ and 3σ confidence levels (CLs). On the other hand, these best-fit values are used to calculate the age of the Universe, the current deceleration parameter (both at 3σ CL) and the current matter density parameter at 1σ CL. Finding a Universe roughly twice as old as the one of ΛCDM is a distinction of fractional cosmology. Focusing our analysis on these results, we can conclude that the region in which μ>2 is not ruled out by observations. This parameter region is relevant because fractional cosmology gives a power-law solution without matter, which is accelerated for μ>2. We present a fractional origin model that leads to an accelerated state without appealing to Λ or dark energy. Full article
(This article belongs to the Special Issue Fractional Gravity/Cosmology in Classical and Quantum Regimes)
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32 pages, 2395 KiB  
Article
Revisiting Fractional Cosmology
by Bayron Micolta-Riascos, Alfredo D. Millano, Genly Leon, Cristián Erices and Andronikos Paliathanasis
Fractal Fract. 2023, 7(2), 149; https://doi.org/10.3390/fractalfract7020149 - 03 Feb 2023
Cited by 8 | Viewed by 1872
Abstract
Recently, the research community has been exploring fractional calculus to address problems related to cosmology; in this approach, the gravitational action integral is altered, leading to a modified Friedmann equation, then the resulting theory is compared against observational data. In this context, dynamical [...] Read more.
Recently, the research community has been exploring fractional calculus to address problems related to cosmology; in this approach, the gravitational action integral is altered, leading to a modified Friedmann equation, then the resulting theory is compared against observational data. In this context, dynamical systems can be used along with an analysis the phase spaces for different values of the fractional order of the derivative and their different matter contents. The equilibrium points are classified, providing a range for the order of the fractional derivative in order to investigate whether the cosmological history can be reconstructed and a late-time accelerating power-law solution obtained for the scale factor. In this paper, we discuss the physical interpretation of the corresponding cosmological solutions with particular emphasis on the influence of the fractional order of the derivative in a theory of gravity that includes a scalar field minimally coupled to gravity. The presented results improve and extend those obtained previously, further demonstrating that fractional calculus can play a relevant role in cosmology. Full article
(This article belongs to the Special Issue Fractional Gravity/Cosmology in Classical and Quantum Regimes)
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