Inflation and Fractional Quantum Cosmology
Abstract
:1. Introduction
2. Wheeler-DeWitt Equation in Slow Roll Regime
3. Fractional Quantum Cosmology for a Slow Roll Regime
4. Conclusions and Discussions
- According to the arguments presented above, we applied the slow roll regime, and substituted and into generalized WDW Equation (11).
- We then showed that the solution of the simplified differential Equation (12) can be expressed by the Bessel function.
- The model was then examined according to the arguments set out previously; please see the expressions below Equation (15). Concretely, the model was studied assuming the WKB condition. For such a semiclassical regime, we have shown that the scale factor of the universe evolves exponentially at early times, which corresponds to the accelerated de Sitter expansion.
- We have additionally shown that ought to take a value to obtain a sufficient e-folding number required to solve the problems of the standard cosmology.
- We gave a brief overview of fractional quantum mechanics and then explained how a suitable model is constructed for the fractional quantum cosmology. Based on the FQC procedure, we derived the fractional WDW equation and fractional ADM Hamiltonian from those retrieved for the corresponding standard model.
- To solve the fractional WDW equation, we applied the WKB approximation again and set .
- Despite the standard case, we have shown that the scale factor of the universe takes the form of a power-law function of time. It is seen that it not only depends on the Lévy’s fractional parameter , but also, as can be seen from the relation (34), on the volume of the compact 3-space , .
- It is worth mentioning that the standard model and its fractional counterpart yield completely different results for the evolution of the scale factor: for the standard case, not only does the scale factor accelerate with the exponential-law of the time, but also such evolution is only affected by matter (i.e., the constant potential; cf. Equation (18)), and therefore the number of e-folding depends only on the constant L. In the fractional case, but the scale factor is a power-law function of time, which is completely independent of the matter; but instead, only the Lévy’s fractional parameter and the volume of the compact 3-space , , determine how the scale factor evolves (cf. relation (34)). This gives us another dependency for the number of e-folding. Such consequences are interpreted as distinguishing features of the fractional quantum gravity.
- As mentioned, unlike the usual de Sitter solution (18), the evolution of the scale factor associated with the fractional case, i.e., (34), depends on the volume of the compact 3-space , . More concretely, the evolution of the scale factor depends on the global geometry that affects the topology of the universe as a whole. The global geometry of the universe, i.e., the spatial curvature and topology and consequently the shape of the entire universe, are not determined by the Einstein gravitational field equations since they are differential equations that only determine the local features of space-time. Therefore, the 3-volume dependence of the scale factor is entirely a fractional quantum gravity effect. As we see in (34), the value of the scale factor at early time is proportional to the inverse root square of , such that the smaller the value of the 3-volume, the bigger the initial scale factor.Interestingly, the authors of Ref. [61] (for a recent analysis, see [62,63,64]) have shown that the Wilkinson Microwave Anisotropy Probe (WMAP) 3 year data are consistent with the possibility that we live in a ‘small universe’ shaped like a flat 3-torus whose fundamental domain is a cube with side length corresponding to a volume of Gpc. It turns out that the torus model describes the data much better than the best-fit cold dark matter (CDM) model, as it shows the suppression of the CMB anisotropy at large scales first observed by Cosmic Background Explorer (COBE).
- As seen in Equation (28), the Riesz fractional derivative is a non-local operator that can explain non-local processes in the minisuperspace. In fact, non-locality is a general behavior of all fractional derivatives and integrals and hence they describe processes with non-locality in time (memory) and space (large jumps) [65]. The fractional derivative has lagged far behind the integer-order calculus due to the unclear physical meaning, which is a major obstacle. In 1974, it was proposed as an open problem to ask “What are the physical interpretations of fractional calculus?” Using analogous reasoning, a physical explanation for a fractional time derivative was put forth in 2002 [66], although no experiments were performed to support the new time scale. In Ref. [67], the authors have shown that the time fractional order is a memory index. Time disappears in quantum gravity and cosmology, and the space-fractional derivative may play a crucial role. In Refs. [33,59], the authors show that Lévy’s fractional parameter represents the fractal dimension (denoted by D in Equations (34), (35) and (36)) of the black hole horizon or the cosmological horizon. In addition, as we saw in (34), as a result of big jumps in minisuperspace, the initial value of emerged classical scale factor depends on the global geometry (or topology) of the universe. Furthermore, Equation (36) shows that the acceleration of the universe and e-folding of the inflation epoch are direct consequences of its topology.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
FLRW | Friedmann–Lemaître–Robertson–Walker |
FQC | Fractional Quantum Cosmology |
GR | General Relativity |
SE | Schrödinger equation |
WDW | Wheeler–DeWitt |
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Rasouli, S.M.M.; de Oliveira Costa, E.W.; Moniz, P.; Jalalzadeh, S. Inflation and Fractional Quantum Cosmology. Fractal Fract. 2022, 6, 655. https://doi.org/10.3390/fractalfract6110655
Rasouli SMM, de Oliveira Costa EW, Moniz P, Jalalzadeh S. Inflation and Fractional Quantum Cosmology. Fractal and Fractional. 2022; 6(11):655. https://doi.org/10.3390/fractalfract6110655
Chicago/Turabian StyleRasouli, Seyed Meraj Mousavi, Emanuel W. de Oliveira Costa, Paulo Moniz, and Shahram Jalalzadeh. 2022. "Inflation and Fractional Quantum Cosmology" Fractal and Fractional 6, no. 11: 655. https://doi.org/10.3390/fractalfract6110655
APA StyleRasouli, S. M. M., de Oliveira Costa, E. W., Moniz, P., & Jalalzadeh, S. (2022). Inflation and Fractional Quantum Cosmology. Fractal and Fractional, 6(11), 655. https://doi.org/10.3390/fractalfract6110655