{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T11:44:49Z","timestamp":1774352689734,"version":"3.50.1"},"reference-count":67,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,5]],"date-time":"2022-11-05T00:00:00Z","timestamp":1667606400000},"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>The Wheeler\u2013DeWitt equation for a flat and compact Friedmann\u2013Lema\u00eetre\u2013Robertson\u2013Walker cosmology at the pre-inflation epoch is studied in the contexts of the standard and fractional quantum cosmology. Working within the semiclassical regime and applying the Wentzel-Kramers-Brillouin (WKB) approximation, we show that some fascinating consequences are obtained for our simple fractional scenario that are completely different from their corresponding standard counterparts: (i) The conventional de Sitter behavior of the inflationary universe for constant potential is replaced by a power-law inflation. (ii) The non-locality of the Riesz\u2019s fractional derivative produces a power-law inflation that depends on the fractal dimension of the compact spatial section of space-time, independent of the energy scale of the inflaton.<\/jats:p>","DOI":"10.3390\/fractalfract6110655","type":"journal-article","created":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T11:56:36Z","timestamp":1667908596000},"page":"655","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Inflation and Fractional Quantum Cosmology"],"prefix":"10.3390","volume":"6","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\/0000-0002-4388-0445","authenticated-orcid":false,"given":"Emanuel W.","family":"de Oliveira Costa","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal de Pernambuco, Recife 50670-901, PE, Brazil"}]},{"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4854-2960","authenticated-orcid":false,"given":"Shahram","family":"Jalalzadeh","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal de Pernambuco, Recife 50670-901, PE, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1103\/PhysRevD.23.347","article-title":"The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems","volume":"23","author":"Guth","year":"1981","journal-title":"Phys. Rev. D"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1103\/PhysRevLett.48.1220","article-title":"Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking","volume":"48","author":"Albrecht","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"1888","DOI":"10.1103\/PhysRevD.7.1888","article-title":"Radiative Corrections as the Origin of Spontaneous Symmetry Breaking","volume":"7","author":"Coleman","year":"1973","journal-title":"Phys. Rev. D"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2162","DOI":"10.1103\/PhysRevD.29.2162","article-title":"A Prescription for Successful New Inflation","volume":"29","author":"Steinhardt","year":"1984","journal-title":"Phys. Rev. D"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1016\/0550-3213(83)90592-8","article-title":"Primordial Supersymmetric Inflation","volume":"221","author":"Ellis","year":"1983","journal-title":"Nucl. Phys. B"},{"key":"ref_7","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_8","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_9","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1140\/epjp\/s13360-021-02007-5","article-title":"Hubble tension bounds the GUP and EUP parameters","volume":"136","author":"Aghababaei","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"584","DOI":"10.1140\/epjp\/s13360-021-01587-6","article-title":"Complete noncommutativity in a cosmological model with radiation","volume":"136","author":"Marcon","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"123505","DOI":"10.1103\/PhysRevD.103.123505","article-title":"Bimetric universe with matter","volume":"103","author":"Maldonado","year":"2021","journal-title":"Phys. Rev. D"},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1038\/246396a0","article-title":"Is the universe a vacuum fluctuation","volume":"246","author":"Tryon","year":"1973","journal-title":"Nature"},{"key":"ref_14","unstructured":"Jalalzadeh, S., and Vargas Moniz, P. (2022). Challenging Routes in Quantum Cosmology, World Scientific."},{"key":"ref_15","unstructured":"Kilbas, A., Srivastava, H., and Trujillo, J. (2006). Theory and Applications Of Fractional Differential Equations, Elsevier Science."},{"key":"ref_16","unstructured":"Podlubny, I. (1999). Fractional Differential Equations; Mathematics in Science and Engineering, Elsevier."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1088\/1126-6708\/2005\/10\/050","article-title":"Fractal spacetime structure in asymptotically safe gravity","volume":"10","author":"Lauscher","year":"2005","journal-title":"JHEP"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5","DOI":"10.12942\/lrr-2006-5","article-title":"The Asymptotic Safety Scenario in Quantum Gravity","volume":"9","author":"Niedermaier","year":"2006","journal-title":"Living Rev. Relativ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1007\/JHEP12(2011)012","article-title":"Fractal space-times under the microscope: A Renormalization Group view on Monte Carlo data","volume":"12","author":"Reuter","year":"2011","journal-title":"JHEP"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/978-3-642-33036-0_8","article-title":"Asymptotic Safety, Fractals, and Cosmology","volume":"863","author":"Reuter","year":"2013","journal-title":"Lect. Notes Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"171301","DOI":"10.1103\/PhysRevLett.95.171301","article-title":"Spectral dimension of the universe","volume":"95","author":"Ambjorn","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"104040","DOI":"10.1103\/PhysRevD.81.104040","article-title":"Spectral dimension of a quantum universe","volume":"81","author":"Modesto","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.physrep.2012.03.007","article-title":"Nonperturbative Quantum Gravity","volume":"519","author":"Ambjorn","year":"2012","journal-title":"Phys. Rept."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"131303","DOI":"10.1103\/PhysRevLett.107.131303","article-title":"Spectral dimension as a probe of the ultraviolet continuum regime of causal dynamical triangulations","volume":"107","author":"Sotiriou","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"242002","DOI":"10.1088\/0264-9381\/26\/24\/242002","article-title":"Fractal Structure of Loop Quantum Gravity","volume":"26","author":"Modesto","year":"2009","journal-title":"Class. Quant. Grav."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3","DOI":"10.12942\/lrr-2013-3","article-title":"The Spin Foam Approach to Quantum Gravity","volume":"16","author":"Perez","year":"2013","journal-title":"Living Rev. Rel."},{"key":"ref_27","unstructured":"Rovelli, C. (2004). Quantum Gravity, Cambridge University Press. Cambridge Monographs on Mathematical Physics."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"161301","DOI":"10.1103\/PhysRevLett.102.161301","article-title":"Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point","volume":"102","author":"Horava","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"012034","DOI":"10.1088\/1742-6596\/283\/1\/012034","article-title":"Horava-Lifshitz gravity: A status report","volume":"283","author":"Sotiriou","year":"2011","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"031101","DOI":"10.1103\/PhysRevLett.108.031101","article-title":"Towards singularity and ghost free theories of gravity","volume":"108","author":"Biswas","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"124059","DOI":"10.1103\/PhysRevD.91.124059","article-title":"Nonlocal quantum gravity and M-theory","volume":"91","author":"Calcagni","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"044005","DOI":"10.1103\/PhysRevD.86.044005","article-title":"Super-renormalizable Quantum Gravity","volume":"86","author":"Modesto","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"L121901","DOI":"10.1103\/PhysRevD.105.L121901","article-title":"de Sitter fractional quantum cosmology","volume":"105","author":"Jalalzadeh","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_34","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_35","doi-asserted-by":"crossref","unstructured":"Moniz, P.V., and Jalalzadeh, S. (2020). From Fractional Quantum Mechanics to Quantum Cosmology: An Overture. Mathematics, 8.","DOI":"10.3390\/math8030313"},{"key":"ref_36","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. Quant. Grav."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1350092","DOI":"10.1142\/S0217751X13500929","article-title":"Multi-fractional spacetimes, asymptotic safety and Ho\u0159ava\u2013Lifshitz gravity","volume":"28","author":"Calcagni","year":"2013","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"064057","DOI":"10.1103\/PhysRevD.95.064057","article-title":"Multiscale spacetimes from first principles","volume":"95","author":"Calcagni","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1007\/JHEP08(2022)024","article-title":"Gravitational potential and galaxy rotation curves in multi-fractional spacetimes","volume":"8","author":"Calcagni","year":"2022","journal-title":"JHEP"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Garc\u00eda-Aspeitia, M.A., Fernandez-Anaya, G., Hern\u00e1ndez-Almada, A., Leon, G., and Maga\u00f1a, J. (2022). Cosmology under the fractional calculus approach. arXiv.","DOI":"10.1093\/mnras\/stac3006"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"957863","DOI":"10.1155\/2014\/957863","article-title":"Fractional quantum field theory: From lattice to continuum","volume":"2014","author":"Tarasov","year":"2014","journal-title":"Adv. High Energy Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"165006","DOI":"10.1088\/1361-6382\/ac103c","article-title":"Quantum scalar field theories with fractional operators","volume":"38","author":"Calcagni","year":"2021","journal-title":"Class. Quantum Grav."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/j.physa.2005.08.005","article-title":"Fractional derivative quantum fields at positive temperature","volume":"363","author":"Lim","year":"2006","journal-title":"Phys. A: Stat. Mech. Appl"},{"key":"ref_44","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_45","doi-asserted-by":"crossref","unstructured":"Linde, A.D. (1990). Inflation and Quantum Cosmology, Academic Press, Inc.","DOI":"10.1017\/CBO9780511564178.016"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1088\/1475-7516\/2004\/10\/004","article-title":"Creation of a compact topologically nontrivial inflationary universe","volume":"10","author":"Linde","year":"2004","journal-title":"JCAP"},{"key":"ref_47","unstructured":"Cooke, M.A.L. (2022, September 28). An Introduction to Quantum Cosmology. Available online: https:\/\/www.semanticscholar.org\/paper\/An-Introduction-to-Quantum-Cosmology-Cooke\/453c364f02087ee31283b7d4362ab550a6f3c094."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"El-Nabulsi, R.A., Moaaz, O., and Bazighifan, O. (2020). New results for oscillatory behavior of fourth-order differential equations. Symmetry, 12.","DOI":"10.3390\/sym12010136"},{"key":"ref_49","unstructured":"Feynman, R.P., Hibbs, A.R., and Styer, D. (2010). Quantum Mechanics and Path Integrals, Dover Publications."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/S0375-9601(00)00201-2","article-title":"Fractional quantum mechanics and Levy paths integrals","volume":"268","author":"Laskin","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_51","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_52","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_53","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_54","doi-asserted-by":"crossref","unstructured":"Laskin, N. (2010). Principles of Fractional Quantum Mechanics. arXiv.","DOI":"10.1142\/9789814340595_0017"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Laskin, N. (2018). Fractional Quantum Mechanics, World Scientific.","DOI":"10.1142\/10541"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02395016","article-title":"L\u2019int\u00e9grale de Riemann-Liouville et le probl\u00e8me de Cauchy","volume":"81","author":"Riesz","year":"1949","journal-title":"Acta Math."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"7612490","DOI":"10.1155\/2018\/7612490","article-title":"Fractional Derivative Regularization in QFT","volume":"2018","author":"Tarasov","year":"2018","journal-title":"Adv. High Energy Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"549","DOI":"10.4310\/ATMP.2012.v16.n2.a5","article-title":"Geometry of fractional spaces","volume":"16","author":"Calcagni","year":"2012","journal-title":"Adv. Theor. Math. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1140\/epjc\/s10052-021-09438-5","article-title":"Prospecting black hole thermodynamics with fractional quantum mechanics","volume":"81","author":"Jalalzadeh","year":"2021","journal-title":"Eur. Phys. J. C"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Pozrikidis, C. (2018). The Fractional Laplacian, Chapman and Hall\/CRC.","DOI":"10.1201\/9781315367675"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"125006","DOI":"10.1088\/0264-9381\/25\/12\/125006","article-title":"Do we Live in a Small Universe?","volume":"25","author":"Aurich","year":"2008","journal-title":"Class. Quant. Grav."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1088\/1475-7516\/2013\/08\/009","article-title":"The Topology and Size of the Universe from CMB Temperature and Polarization Data","volume":"8","author":"Aslanyan","year":"2013","journal-title":"JCAP"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"225017","DOI":"10.1088\/0264-9381\/25\/22\/225017","article-title":"A spatial correlation analysis for a toroidal universe","volume":"25","author":"Aurich","year":"2008","journal-title":"Class. Quant. Grav."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"A26","DOI":"10.1051\/0004-6361\/201321546","article-title":"Planck 2013 results. XXVI. Background geometry and topology of the Universe","volume":"571","author":"Ade","year":"2014","journal-title":"Astron. Astrophys."},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Herrmann, R. (2018). Fractional Calculus, World Scientific. [3rd ed.].","DOI":"10.1142\/11107"},{"key":"ref_66","unstructured":"Podlubny, I. (2001). Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. arXiv."},{"key":"ref_67","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."}],"container-title":["Fractal and Fractional"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2504-3110\/6\/11\/655\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:17Z","timestamp":1760145077000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2504-3110\/6\/11\/655"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,5]]},"references-count":67,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["fractalfract6110655"],"URL":"https:\/\/doi.org\/10.3390\/fractalfract6110655","relation":{},"ISSN":["2504-3110"],"issn-type":[{"value":"2504-3110","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,5]]}}}