The community searches for extensions to the ΛCDM model to resolve some of the mentioned problems. The approaches to face them are divided into two main branches: (i) assume a DE fluid with the capability to accelerate the Universe or (ii) modify General Relativity (GR) to obtain the cosmic acceleration without DE (Motta et al. 2021). This paper will focus on the second point under the formalism known as fractional calculus, which consists of a generalization of the classical integer order calculus, whose derivatives and integrals are of (real or complex) arbitrary order. These fractional operators are not local. In many cases can model real-world phenomena in a better fashion than those obtained by classical calculation. For example, it is coined fractional dynamics as a field of study in physics and mechanics investigating the behaviour of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by us and differentiation of non-integer orders, i.e. by methods in the fractional calculus (see the review by Tarasov 2013). Fractional calculus is a field with multiple applications and a great deal of research activity. Fractional quantum mechanics is employed as a tool within quantum field theory and gravity for fractional spacetime (Calcagni 2010a, b) and the fractional quantum field theory at positive temperature (Lim 2006; Lim & Eab 2019) and other applications of quantum cosmology (Moniz & Jalalzadeh 2020a, b; Jalalzadeh, da Silva & Moniz 2021; Rasouli, Jalalzadeh & Moniz 2021). Recently, the community explores the fractional calculus to tackle problems associated in cosmology (Shchigolev 2011, 2013a, b, 2016, 2021; Calcagni 2013, 2017a, 2021a, b; Calcagni & De Felice 2020; Jalalzadeh, Costa & Moniz 2022), stochastic GW background (Calcagni & Kuroyanagi 2021), luminosity distance (Calcagni et al. 2019), inflation and CMB spectrum (Calcagni, Kuroyanagi & Tsujikawa 2016; Calcagni 2017b), Fractional Action Cosmology (El-Nabulsi 2012, 2016a; Jamil, Momeni & Rashid 2012), fractional geodesic equation, complex GR, and discrete gravity El-Nabulsi (2013b), minimal couplings (El-Nabulsi 2013a), phantom (Rami 2015), Ornstein–Uhlenbeck-like fractional differential equation in cosmology (El-Nabulsi 2016b), a variable Order Parameter (El-Nabulsi 2017a), wormholes in fractional action cosmology (El-Nabulsi 2017c). New metrics were considered (El-Nabulsi 2017b), as well as some DE models in emergent, logamediate, and intermediate scenarios of the universe (Debnath, Jamil & Chattopadhyay 2012; Debnath, Chattopadhyay & Jamil 2013). For instance, Shchigolev (2016, 2021) found α = 0.926 (where α is the order of the Riemann–Liouville fractional integral). In Shchigolev (2011,2013a, b) were obtained several exact solutions for cosmological models, which differ significantly from the standard model due to the fractal nature of spacetime (Calcagni 2010a, b). Jalalzadeh et al. (2022) explore the interval 1  α  2 using Riesz’s fractional derivative (that is not related to the index of Riemann–Liouville fractional integral) to obtain the non-boundary and tunnelling wave functions for a closed de Sitter geometry. Another example, Barrientos, Mendoza & Padilla (2021), studies the Universe dynamics without DM and DE components by modifying the mathematical structure of Friedmann equations with fractional calculus. Another approach is calculating the value of the Λ (due to the well-known ultraviolet divergence in the standard quantum field theory), which needs to restructure the theory using the fractional calculus (Calcagni 2021c). Giusti (2020) and Torres et al. (2020) explore Modified Newtonian Dynamics Theories and quantum cosmology in this fractional approach (Barrientos et al. 2021). Finally, notice that there are several definitions of fractional derivatives and fractional integrals, such as those of Riemann–Liouville, Caputo, Riesz, Hadamard, Marchand, and Griinwald–Letnikov, among other more recent ones (see Kilbas, Srivastava & Trujillo 2006; Podlubny 1998, and references therein). Even though these operators are already well studied, some of the usual features related to function differentiation fails, such as Leibniz’s rule, the chain rule, and the semigroup property (Podlubny 1998; Kilbas et al. 2006).
Citation: El-Nabulsi; 2016a
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