{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T13:14:58Z","timestamp":1776086098632,"version":"3.50.1"},"reference-count":109,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T00:00:00Z","timestamp":1715126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>Considering the Friedmann\u2013Lema\u00eetre\u2013Robertson\u2013Walker (FLRW) metric and the Einstein scalar field system as an underlying gravitational model to construct fractional cosmological models has interesting implications in both classical and quantum regimes. Regarding the former, we just review the most fundamental approach to establishing an extended cosmological model. We demonstrate that employing new methodologies allows us to obtain exact solutions. Despite the corresponding standard models, we cannot use any arbitrary scalar potentials; instead, it is determined from solving three independent fractional field equations. This article concludes with an overview of a fractional quantum\/semi-classical model that provides an inflationary scenario.<\/jats:p>","DOI":"10.3390\/fractalfract8050281","type":"journal-article","created":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T09:58:56Z","timestamp":1715162336000},"page":"281","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Fractional Scalar Field Cosmology"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3455-1954","authenticated-orcid":false,"given":"Seyed Meraj Mousavi","family":"Rasouli","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, Rua Marqu\u00eas d\u2019Avila e Bolama, 6200-001 Covilh\u00e3, Portugal"},{"name":"Department of Physics, Qazvin Branch, Islamic Azad University, Qazvin 341851416, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2428-0639","authenticated-orcid":false,"given":"Samira","family":"Cheraghchi","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Tehran, Tehran P.O. Box 14395-547, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7170-8952","authenticated-orcid":false,"given":"Paulo","family":"Moniz","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA-UBI), Universidade da Beira Interior, Rua Marqu\u00eas d\u2019Avila e Bolama, 6200-001 Covilh\u00e3, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/0370-2693(82)91219-9","article-title":"A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems","volume":"108","author":"Linde","year":"1982","journal-title":"Phys. Lett. B"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.01.001","article-title":"Modified gravity and cosmology","volume":"513","author":"Cliftona","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_3","unstructured":"Brandenberger, R.H. (2000). Large Scale Structure Formation, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Faraoni, V., Capozziello, S., Capozziello, S., and Faraoni, V. (2011). Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics, Springer.","DOI":"10.1007\/978-94-007-0165-6"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.physrep.2011.09.003","article-title":"Extended Theories of Gravity","volume":"509","author":"Capozziello","year":"2011","journal-title":"Phys. Rept."},{"key":"ref_6","unstructured":"Akrami, Y., Sebastian, B., Jose, L.B.-S., Christian, B., Camille, B., Mariam, B.-L., Philippe, B., Gianluca, C., Salvatore, C., and Roberto, C. (2021). Modified Gravity and Cosmology: An Update by the CANTATA Network, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"101112","DOI":"10.1016\/j.dark.2022.101112","article-title":"Geodesic deviation in S\u00e1ez\u2013Ballester theory","volume":"37","author":"Rasouli","year":"2022","journal-title":"Phys. Dark Univ."},{"key":"ref_8","unstructured":"Alves Batista, R., Amelino-Camelia, G., Boncioli, D., Carmona, J.M., di Matteo, A., Gubitosi, G., Lobo, I., Mavromatos, N.E., Pfeifer, C., and Rubiera-Garcia, D. (2023). White Paper and Roadmap for Quantum Gravity Phenomenology in the Multi-Messenger Era. arXiv, Available online: https:\/\/arxiv.org\/abs\/2312.00409."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"169611","DOI":"10.1016\/j.aop.2024.169611","article-title":"Cosmology of Tsallis and Kaniadakis holographic dark energy in Saez-Ballester theory and consideration of viscous van der Waals fluid","volume":"463","author":"Chokyi","year":"2024","journal-title":"Ann. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"100800","DOI":"10.1016\/j.ascom.2024.100800","article-title":"Observational constraints on Hubble parameter in S\u00e1ez Ballester theory","volume":"47","author":"Singh","year":"2024","journal-title":"Astron. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/RevModPhys.61.1","article-title":"The cosmological constant problem","volume":"61","author":"Weinberg","year":"1989","journal-title":"Rev. Mod. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1146\/annurev.aa.30.090192.002435","article-title":"The cosmological constant","volume":"30","author":"Carroll","year":"1992","journal-title":"Annu. Rev. Astron. Astrophys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1007\/s10701-005-9042-8","article-title":"Categorizing different approaches to the cosmological constant problem","volume":"36","author":"Nobbenhuis","year":"2006","journal-title":"Found. Phys."},{"key":"ref_14","unstructured":"Padilla, A. (2015). Lectures on the cosmological constant problem. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"051301","DOI":"10.1103\/PhysRevLett.125.051301","article-title":"Reformulation of the Cosmological Constant Problem","volume":"125","author":"Wang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Dainotti, M., De Simone, B., Montani, G., Schiavone, T., and Lambiase, G. (2023). The Hubble constant tension: Current status and future perspectives through new cosmological probes. arXiv.","DOI":"10.22323\/1.436.0235"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Vagnozzi, S. (2023). Seven hints that early-time new physics alone is not sufficient to solve the Hubble tension. Universe, 9.","DOI":"10.3390\/universe9090393"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3435","DOI":"10.1088\/0264-9381\/19\/13\/304","article-title":"The Cosmological constant problem and quintessence","volume":"19","author":"Sahni","year":"2002","journal-title":"Class. Quant. Grav."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Calcagni, G. (2017). Classical and Quantum Cosmology, Springer International Publishing.","DOI":"10.1007\/978-3-319-41127-9"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"10","DOI":"10.12942\/lrr-2012-10","article-title":"Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions","volume":"15","author":"Famaey","year":"2012","journal-title":"Living Rev. Relativ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1139\/cjp-2014-0203","article-title":"A tale of two paradigms: The mutual incommensurability of \u039bCDM and MOND","volume":"93","author":"McGaugh","year":"2015","journal-title":"Can. J. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s10509-014-1913-z","article-title":"Energy density inhomogeneities with polynomial f(R) cosmology","volume":"352","author":"Sharif","year":"2014","journal-title":"Astrophys. Space Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"031","DOI":"10.1088\/1475-7516\/2015\/04\/031","article-title":"Dynamical behavior in mimetic f(R) gravity","volume":"2015","author":"Leon","year":"2015","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1140\/epjc\/s10052-017-5003-6","article-title":"Higher-dimensional inhomogeneous perfect fluid collapse in f(R) gravity","volume":"77","author":"Abbas","year":"2017","journal-title":"Eur. Phys. J. C"},{"key":"ref_25","unstructured":"Sharif, M., and Yousaf, Z. (2015, January 9\u201314). Causes of Inhomogeneous Energy Density in Relativistic Fluids with f(R) Background. Proceedings of the Mathematical Physics: Proceedings of the 14th Regional Conference, Islamabad, Pakistan."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"046901","DOI":"10.1088\/0034-4885\/73\/4\/046901","article-title":"Thermodynamical Aspects of Gravity: New insights","volume":"73","author":"Padmanabhan","year":"2010","journal-title":"Rept. Prog. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"016","DOI":"10.21468\/SciPostPhys.2.3.016","article-title":"Emergent gravity and the dark universe","volume":"2","author":"Verlinde","year":"2017","journal-title":"Scipost Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/BF01215403","article-title":"A bi-metric theory of gravitation","volume":"4","author":"Rosen","year":"1973","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"022","DOI":"10.1088\/1475-7516\/2016\/12\/022","article-title":"Cosmology with moving bimetric fluids","volume":"2016","author":"Maroto","year":"2016","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1140\/epjc\/s10052-023-11707-4","article-title":"Cosmological evolution in bimetric gravity: Observational constraints and LSS signatures","volume":"83","author":"Bassi","year":"2023","journal-title":"Eur. Phys. J. C"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Maldonado, C., and M\u00e9ndez, F. (2023). Axionic Dark Matter in a Bi-Metric Universe. Universe, 9.","DOI":"10.3390\/universe9100429"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"100942","DOI":"10.1016\/j.dark.2021.100942","article-title":"Cosmological future singularities in massive gravity and massive bigravity","volume":"35","author":"Mousavi","year":"2022","journal-title":"Phys. Dark Univ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"100446","DOI":"10.1016\/j.dark.2019.100446","article-title":"Late time cosmic acceleration in modified S\u00e1ez\u2013Ballester theory","volume":"27","author":"Rasouli","year":"2020","journal-title":"Phys. Dark Univ."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Rasouli, S.M.M., Jalalzadeh, S., and Moniz, P. (2022). Noncompactified Kaluza\u2013Klein Gravity. Universe, 8.","DOI":"10.3390\/universe8080431"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s10714-023-03155-y","article-title":"Particle creation and bulk viscosity in Bianchi-I universe in Saez\u2013Ballester theory with different deceleration parameters","volume":"55","author":"Chetia","year":"2023","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"083533","DOI":"10.1103\/PhysRevD.90.083533","article-title":"Noncommutative minisuperspace, gravity-driven acceleration, and kinetic inflation","volume":"90","author":"Rasouli","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1140\/epjc\/s10052-017-5497-y","article-title":"On the emergence of the \u039b CDM model from self-interacting Brans\u2013Dicke theory in d=5","volume":"78","author":"Reyes","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1140\/epjc\/s10052-019-7580-z","article-title":"Anisotropic massive Brans\u2013Dicke gravity extension of the standard \u039b CDM model","volume":"80","author":"Akarsu","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2250131","DOI":"10.1142\/S0218271822501310","article-title":"Analytic solutions of Brans\u2013Dicke cosmology: Early inflation and late time accelerated expansion","volume":"32","author":"Ildes","year":"2023","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.aop.2016.09.007","article-title":"Non-singular Brans\u2013Dicke collapse in deformed phase space","volume":"375","author":"Rasouli","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_41","first-page":"19","article-title":"Gravity-Driven Acceleration and Kinetic Inflation in Noncommutative Brans-Dicke Setting","volume":"29","author":"Rasouli","year":"2016","journal-title":"Odessa Astron. Pub."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"100781","DOI":"10.1016\/j.dark.2021.100781","article-title":"Geodesic deviation equation in Brans\u2013Dicke theory in arbitrary dimensions","volume":"32","author":"Rasouli","year":"2021","journal-title":"Phys. Dark Univ."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"100269","DOI":"10.1016\/j.dark.2019.100269","article-title":"Kinetic inflation in deformed phase space Brans\u2013Dicke cosmology","volume":"24","author":"Rasouli","year":"2019","journal-title":"Phys. Dark Univ."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"015010","DOI":"10.1088\/1361-6382\/aca868","article-title":"Classical and quantum bicosmology with noncommutativity","volume":"40","author":"Kan","year":"2022","journal-title":"Class. Quantum Gravity"},{"key":"ref_45","unstructured":"Rasouli, S.M.M., and Marto, J. (2023). Phase space noncommutativity, power-law inflation and quantum cosmology. arXiv, Available online: https:\/\/arxiv.org\/abs\/2311.01627."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Rasouli, S.M.M. (2022). Noncommutativity, S\u00e1ez\u2013Ballester Theory and Kinetic Inflation. Universe, 8.","DOI":"10.3390\/universe8030165"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.physletb.2007.07.063","article-title":"Chaplygin DGP cosmologies","volume":"654","author":"Lazkoz","year":"2007","journal-title":"Phys. Lett. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"037","DOI":"10.1088\/1475-7516\/2015\/12\/037","article-title":"Scalar perturbations in the late Universe: Viability of the Chaplygin gas models","volume":"2015","author":"Brilenkov","year":"2015","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2637","DOI":"10.1007\/s10714-009-0793-y","article-title":"A class of cosmological solutions in induced matter theory with conformally flat bulk space","volume":"41","author":"Doroud","year":"2009","journal-title":"Gen. Rel. Grav."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1002\/andp.201052203-520","article-title":"On the energy conditions in non-compact Kaluza-Klein gravity","volume":"19","author":"Rasouli","year":"2010","journal-title":"Ann. Phys."},{"key":"ref_51","first-page":"371","article-title":"Kasner Solution in Brans\u2013Dicke Theory and Its Corresponding Reduced Cosmology","volume":"60","author":"Rasouli","year":"2014","journal-title":"Springer Proc. Math. Stat."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"025004","DOI":"10.1088\/1361-6382\/aa9ad3","article-title":"Modified Saez\u2013Ballester scalar\u2013tensor theory from 5D space-time","volume":"35","author":"Rasouli","year":"2018","journal-title":"Class. Quant. Grav."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"075010","DOI":"10.1088\/1361-6382\/ab0987","article-title":"Extended anisotropic models in noncompact Kaluza-Klein theory","volume":"36","author":"Rasouli","year":"2019","journal-title":"Class. Quant. Grav."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Rasouli, S.M.M. (2023). Noncompactified Kaluza\u2013Klein theories and Anisotropic Kantowski-Sachs Universe. arXiv.","DOI":"10.3390\/universe8080431"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"044028","DOI":"10.1103\/PhysRevD.89.044028","article-title":"Gravitational Collapse of a Homogeneous Scalar Field in Deformed Phase Space","volume":"89","author":"Rasouli","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.dark.2017.09.011","article-title":"Quantum deformation of quantum cosmology: A framework to discuss the cosmological constant problem","volume":"18","author":"Jalalzadeh","year":"2017","journal-title":"Phys. Dark Univ."},{"key":"ref_57","unstructured":"Roberts, M.D. (2009). Fractional Derivative Cosmology. arXiv."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"012008","DOI":"10.1088\/1742-6596\/354\/1\/012008","article-title":"Fractional Action Cosmology with Power Law Weight Function","volume":"354","author":"Jamil","year":"2012","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Torres, I., Fabris, J.C., Piattella, O.F., and Batista, A.B. (2020). Quantum Cosmology of Fab Four John Theory with Conformable Fractional Derivative. Universe, 6.","DOI":"10.3390\/universe6040050"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Gonz\u00e1lez, E., Leon, G., and Fernandez-Anaya, G. (2023). Exact solutions and cosmological constraints in fractional cosmology. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7050368"},{"key":"ref_61","unstructured":"Leon, G., Garc\u00eda-Aspeitia, M.A., Fernandez-Anaya, G., Hern\u00e1ndez-Almada, A., Maga\u00f1a, J., and Gonz\u00e1lez, E. (2023). Cosmology under the fractional calculus approach: A possible H0 tension resolution?. arXiv, Available online: https:\/\/arxiv.org\/abs\/2304.14465."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"de Oliveira Costa, E.W., Jalalzadeh, R., da Silva Junior, P.F., Rasouli, S.M.M., and Jalalzadeh, S. (2023). Estimated Age of the Universe in Fractional Cosmology. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7120854"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1088\/0253-6102\/56\/2\/34","article-title":"Cosmological Models with Fractional Derivatives and Fractional Action Functional","volume":"56","author":"Shchigolev","year":"2011","journal-title":"Commun. Theor. Phys."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"2130014","DOI":"10.1142\/S0217732321300147","article-title":"Fractional-order derivatives in cosmological models of accelerated expansion","volume":"36","author":"Shchigolev","year":"2021","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"053521","DOI":"10.1063\/1.2929662","article-title":"Fractional actionlike variational problems","volume":"49","author":"Torres","year":"2008","journal-title":"J. Math. Phys."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Laskin, N. (2018). Fractional Quantum Mechanics, World Scientific.","DOI":"10.1142\/10541"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"3135","DOI":"10.1103\/PhysRevE.62.3135","article-title":"Fractional quantum mechanics","volume":"62","author":"Laskin","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_68","unstructured":"Feynman, R.P., Hibbs, A.R., and Styer, D.F. (2010). Quantum Mechanics and Path Integrals, Courier Corporation."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/S0375-9601(00)00201-2","article-title":"Fractional Quantum Mechanics and Levy Path Integrals","volume":"268","author":"Laskin","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.cnsns.2016.09.006","article-title":"A Caputo fractional derivative of a function with respect to another function","volume":"44","author":"Almeida","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"3339","DOI":"10.1063\/1.1769611","article-title":"Time fractional Schr\u00f6dinger equation","volume":"45","author":"Naber","year":"2004","journal-title":"J. Math. Phys."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"043502","DOI":"10.1063\/1.2716203","article-title":"Generalized fractional Schr\u00f6dinger equation with space\u2013time fractional derivatives","volume":"48","author":"Wang","year":"2007","journal-title":"J. Math. Phys."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1016\/j.jmaa.2008.03.061","article-title":"Space\u2013time fractional Schr\u00f6dinger equation with time-independent potentials","volume":"344","author":"Dong","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.chaos.2017.04.010","article-title":"Time fractional quantum mechanics","volume":"102","author":"Laskin","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"2140005","DOI":"10.1142\/S0217732321400058","article-title":"Broadening quantum cosmology with a fractional whirl","volume":"36","author":"Rasouli","year":"2021","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Rasouli, S.M.M., Costa, E.W.O., Moniz, P.V., and Jalalzadeh, S. (2022). Inflation and fractional quantum cosmology. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6110655"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/JHEP03(2010)120","article-title":"Quantum field theory, gravity and cosmology in a fractal universe","volume":"2010","author":"Calcagni","year":"2010","journal-title":"J. High Energy Phys."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"1350056","DOI":"10.1142\/S0217732313500569","article-title":"Fractional einstein\u2013hilbert action cosmology","volume":"28","author":"Shchigolev","year":"2013","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1007\/s10773-016-3260-z","article-title":"Fractional action cosmology with variable order parameter","volume":"56","year":"2017","journal-title":"Int. J. Theor. Phys."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Socorro, J., Rosales, J.J., and Toledo-Sesma, L. (2023). Anisotropic fractional cosmology: K-essence theory. Fractal Fract., 7.","DOI":"10.20944\/preprints202309.1221.v1"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"113097","DOI":"10.1016\/j.chaos.2022.113097","article-title":"The paradigm of quantum cosmology through Dunkl fractional Laplacian operators and fractal dimensions","volume":"167","author":"Anukool","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Socorro, J., and Rosales, J.J. (2023). Quantum fractionary cosmology: K-essence theory. Universe, 9.","DOI":"10.3390\/universe9040185"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"59001","DOI":"10.1209\/0295-5075\/acf158","article-title":"Modified Friedmann equations from fractional entropy","volume":"143","author":"Aydiner","year":"2023","journal-title":"Europhys. Lett."},{"key":"ref_84","doi-asserted-by":"crossref","unstructured":"Micolta-Riascos, B., Millano, A.D., Leon, G., Erices, C., and Paliathanasis, A. (2023). Revisiting fractional cosmology. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020149"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"3431","DOI":"10.1038\/srep03431","article-title":"Measuring memory with the order of fractional derivative","volume":"3","author":"Du","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.cnsns.2018.02.019","article-title":"No nonlocality. No fractional derivative","volume":"62","author":"Tarasov","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2018). Generalized memory: Fractional calculus approach. Fractal Fract., 2.","DOI":"10.3390\/fractalfract2040023"},{"key":"ref_88","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2022). Nonlocal probability theory: General fractional calculus approach. Mathematics, 10.","DOI":"10.3390\/math10203848"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear models of dissipation whose Q is almost frequency independent\u2014II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. Int."},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Gorenflo, R., and Mainardi, F. (1997). Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer.","DOI":"10.1007\/978-3-7091-2664-6_5"},{"key":"ref_91","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (1997). Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics, Springer.","DOI":"10.1007\/978-3-7091-2664-6_7"},{"key":"ref_92","unstructured":"Kimeu, J.M. (2024, April 20). Fractional Calculus: Definitions and Applications. Available online: http:\/\/digitalcommons.wku.edu\/theses\/115."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Herrmann, R. (2011). Fractional Calculus: An Introduction for Physicists, World Scientific.","DOI":"10.1142\/9789814340250"},{"key":"ref_94","first-page":"238459","article-title":"A review of definitions for fractional derivatives and integral","volume":"2014","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_95","doi-asserted-by":"crossref","unstructured":"Baleanu, D., and Kumar, D. (2019). Fractional Calculus and Its Applications in Physics, Frontiers Media SA.","DOI":"10.3389\/978-2-88945-958-2"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"165005","DOI":"10.1088\/1361-6382\/ac1081","article-title":"Classical and quantum gravity with fractional operators","volume":"38","author":"Calcagni","year":"2021","journal-title":"Class. Quantum Gravity"},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Almeida, R., Tavares, D., and Torres, D.F. (2019). The Variable-Order Fractional Calculus of Variations, Springer.","DOI":"10.1007\/978-3-319-94006-9"},{"key":"ref_98","doi-asserted-by":"crossref","unstructured":"Faraoni, V., and Faraoni, V. (2004). Scalar-Tensor Gravity, Springer.","DOI":"10.1007\/978-1-4020-1989-0_1"},{"key":"ref_99","unstructured":"Jalalzadeh, S., and Vargas Moniz, P. (2022). Challenging Routes in Quantum Cosmology, World Scientific."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1103\/PhysRevD.37.888","article-title":"Quantum Cosmology and the Initial State of the Universe","volume":"37","author":"Vilenkin","year":"1988","journal-title":"Phys. Rev. D"},{"key":"ref_101","doi-asserted-by":"crossref","unstructured":"Linde, A.D. (1990). Inflation and Quantum Cosmology, Academic Press, Inc.","DOI":"10.1017\/CBO9780511564178.016"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"515","DOI":"10.12988\/astp.2013.13050","article-title":"Derivation of Friedmann\u2019s Acceleration Equation from Canonical Quantum Cosmology","volume":"7","author":"Siong","year":"2013","journal-title":"Adv. Studies Theor. Phys."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"056108","DOI":"10.1103\/PhysRevE.66.056108","article-title":"Fractional Schrodinger equation","volume":"66","author":"Laskin","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"290216","DOI":"10.1155\/2013\/290216","article-title":"Time Fractional Schrodinger Equation Revisited","volume":"2013","author":"Achar","year":"2013","journal-title":"Adv. Math. Phys."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1063\/1.1050284","article-title":"Fractals and quantum mechanics","volume":"10","author":"Laskin","year":"2000","journal-title":"Chaos"},{"key":"ref_106","doi-asserted-by":"crossref","unstructured":"Pozrikidis, C. (2018). The Fractional Laplacian, CRC Press.","DOI":"10.1201\/9781315367675"},{"key":"ref_107","doi-asserted-by":"crossref","unstructured":"Laskin, N. (2010). Principles of Fractional Quantum Mechanics. arXiv.","DOI":"10.1142\/9789814340595_0017"},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"063501","DOI":"10.1103\/PhysRevD.61.063501","article-title":"Quantum cosmology and open universes","volume":"61","author":"Coule","year":"2000","journal-title":"Phys. Rev. D"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"3567","DOI":"10.1007\/s10773-009-0164-1","article-title":"Scalar field in the Bianchi I: Non commutative classical and Quantum Cosmology","volume":"48","author":"Socorro","year":"2009","journal-title":"Int. J. Theor. Phys."}],"container-title":["Fractal and Fractional"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2504-3110\/8\/5\/281\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:41:50Z","timestamp":1760107310000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2504-3110\/8\/5\/281"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,8]]},"references-count":109,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["fractalfract8050281"],"URL":"https:\/\/doi.org\/10.3390\/fractalfract8050281","relation":{},"ISSN":["2504-3110"],"issn-type":[{"value":"2504-3110","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,8]]}}}