{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T08:57:34Z","timestamp":1767085054746,"version":"build-2065373602"},"reference-count":55,"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":["Symmetry"],"abstract":"<jats:p>We examine the key aspects of gravitational theories that incorporate non-local terms, particularly in the context of cosmology and spherical symmetry. We thus explore various extensions of General Relativity, including non-local effects in the action through the function F(R,\u25a1\u22121R), where R denotes the Ricci curvature scalar and the operator \u25a1\u22121 introduces non-locality. By selecting the functional forms using Noether Symmetries, we identify exact solutions within a cosmological framework. We can thus reduce the dynamics of these chosen models and obtain analytical solutions for the equations of motion. Therefore, we study the capability of the selected models in matching cosmological observations by evaluating the phase space and the fixed points; this allows one to further constrain the non-local model selected by symmetry considerations. Furthermore, we also investigate gravitational non-local effects on astrophysical scales. In this context, we seek symmetries within the framework of f(R,\u25a1\u22121R) gravity and place constraints on the free parameters. Specifically, we analyze the impact of non-locality on the orbits of the S2 star orbiting SgrA*.<\/jats:p>","DOI":"10.3390\/sym16050579","type":"journal-article","created":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T12:27:11Z","timestamp":1715171231000},"page":"579","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Non-Local Cosmology: From Theory to Observations"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0333-6234","authenticated-orcid":false,"given":"Francesco","family":"Bajardi","sequence":"first","affiliation":[{"name":"Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Via Cinthia 21, 80126 Napoli, Italy"}]},{"given":"Salvatore","family":"Capozziello","sequence":"additional","affiliation":[{"name":"Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Via Cinthia 21, 80126 Napoli, Italy"},{"name":"Department of Physics \u201cE. Pancini\u201d, University of Naples \u201cFederico II\u201d, Via Cinthia 21, 80126 Napoli, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4","DOI":"10.12942\/lrr-2014-4","article-title":"The Confrontation between General Relativity and Experiment","volume":"17","author":"Will","year":"2014","journal-title":"Living Rev. Relativ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/0550-3213(86)90193-8","article-title":"The Ultraviolet Behavior of Einstein Gravity","volume":"266","author":"Goroff","year":"1986","journal-title":"Nucl. Phys. B"},{"key":"ref_3","unstructured":"Birrell, N.D., and Davies, P.C.W. (1984). Quantum Fields in Curved Space, Cambridge Univercity Press."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"061102","DOI":"10.1103\/PhysRevLett.116.061102","article-title":"Observation of Gravitational Waves from a Binary Black Hole Merger","volume":"116","author":"Abbott","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","unstructured":"The Event Horizon Telescope Collaboration, Akiyama, K., Alberdi, A., Alef, W., Asada, K., Azuly, R., Baczko, A.-K., Ball, D., Balokovi\u0107, M., and Barrett, J. (2019). First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J. Lett., 875, L1."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"243001","DOI":"10.1088\/0264-9381\/32\/24\/243001","article-title":"Testing General Relativity with Present and Future Astrophysical Observations","volume":"32","author":"Berti","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"010001","DOI":"10.1103\/PhysRevD.86.010001","article-title":"Review of Particle Physics (RPP)","volume":"86","author":"Beringer","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.1086\/113063","article-title":"21-cm line studies of spiral galaxies. 2. The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types","volume":"86","author":"Bosma","year":"1981","journal-title":"Astron. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1146\/annurev.astro.46.060407.145243","article-title":"Dark Energy and the Accelerating Universe","volume":"46","author":"Frieman","year":"2008","journal-title":"Ann. Rev. Astron. Astrophys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1086\/300499","article-title":"Observational evidence from supernovae for an accelerating universe and a cosmological constant","volume":"116","author":"Riess","year":"1998","journal-title":"Astron. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1103\/PhysRevD.16.953","article-title":"Renormalization of Higher Derivative Quantum Gravity","volume":"16","author":"Stelle","year":"1977","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/0370-2693(87)91011-2","article-title":"Scalar Fields in Cosmology with an Exponential Potential","volume":"185","author":"Halliwell","year":"1987","journal-title":"Phys. Lett. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"123510","DOI":"10.1103\/PhysRevD.59.123510","article-title":"Cosmological scaling solutions of nonminimally coupled scalar fields","volume":"59","author":"Uzan","year":"1999","journal-title":"Phys. Rev. D"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.01.001","article-title":"Modified Gravity and Cosmology","volume":"513","author":"Clifton","year":"2012","journal-title":"Phys. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1103\/RevModPhys.82.451","article-title":"f(R) Theories Of Gravity","volume":"82","author":"Sotiriou","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3","DOI":"10.12942\/lrr-2010-3","article-title":"f(R) theories","volume":"13","author":"Felice","year":"2010","journal-title":"Living Rev. Relativ."},{"key":"ref_18","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. Rep."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1134\/S0021364007150027","article-title":"Disappearing cosmological constant in f(R) gravity","volume":"86","author":"Starobinsky","year":"2007","journal-title":"JETP Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"086005","DOI":"10.1103\/PhysRevD.74.086005","article-title":"Modified f(R) gravity consistent with realistic cosmology: From matter dominated epoch to dark energy universe","volume":"74","author":"Nojiri","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.06.001","article-title":"Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution","volume":"692","author":"Nojiri","year":"2017","journal-title":"Phys. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1142\/S0218271802002025","article-title":"Curvature quintessence","volume":"11","author":"Capozziello","year":"2002","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0370-2693(80)90670-X","article-title":"A New Type of Isotropic Cosmological Models Without Singularity","volume":"91","author":"Starobinsky","year":"1980","journal-title":"Phys. Lett. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physletb.2005.10.010","article-title":"Modified Gauss\u2013Bonnet theory as gravitational alternative for dark energy","volume":"631","author":"Nojiri","year":"2005","journal-title":"Phys. Lett. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"084007","DOI":"10.1103\/PhysRevD.73.084007","article-title":"Dark energy in modified Gauss\u2013Bonnet gravity: Late-time acceleration and the hierarchy problem","volume":"73","author":"Cognola","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10714-023-03092-w","article-title":"Late-time constraints on modified Gauss\u2013Bonnet cosmology","volume":"55","author":"Bajardi","year":"2023","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bajardi, F., and Capozziello, S. (2022). Noether Symmetries in Theories of Gravity, Cambridge University Press.","DOI":"10.1017\/9781009208727"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Bajardi, F., Capozziello, S., Salvo, T.D., and Spinnato, F. (2023). The Noether Symmetry Approach: Foundation and Applications: The Case of Scalar-Tensor Gauss\u2013Bonnet Gravity. Symmetry, 15.","DOI":"10.3390\/sym15091625"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2050215","DOI":"10.1142\/S0219887820502151","article-title":"The Noether\u2013Bessel-Hagen symmetry approach for dynamical systems","volume":"17","author":"Urban","year":"2020","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_30","unstructured":"Modesto, L. (2012, January 1\u20137). Super-renormalizable Gravity. Proceedings of the 13th Marcel Grossmann Meeting, Stockholm, Sweden."},{"key":"ref_31","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_32","doi-asserted-by":"crossref","first-page":"083507","DOI":"10.1103\/PhysRevD.87.083507","article-title":"Inflation in (Super-)renormalizable Gravity","volume":"87","author":"Briscese","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"111301","DOI":"10.1103\/PhysRevLett.99.111301","article-title":"Nonlocal Cosmology","volume":"99","author":"Deser","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","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_35","doi-asserted-by":"crossref","first-page":"1540003","DOI":"10.1142\/S0217732315400039","article-title":"Aspects of Nonlocality in Quantum Field Theory, Quantum Gravity and Cosmology","volume":"30","author":"Barvinsky","year":"2015","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1016\/j.physletb.2007.12.001","article-title":"Modified non-local-F(R) gravity as the key for the inflation and dark energy","volume":"659","author":"Nojiri","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1140\/epjc\/s10052-017-5283-x","article-title":"Constraining Generalized Non-local Cosmology from Noether Symmetries","volume":"77","author":"Bahamonde","year":"2017","journal-title":"Eur. Phys. J. C"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/j.physletb.2008.04.054","article-title":"Phantom and non-phantom dark energy: The Cosmological relevance of non-locally corrected gravity","volume":"663","author":"Jhingan","year":"2008","journal-title":"Phys. Lett. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1850188","DOI":"10.1142\/S0219887818501888","article-title":"De Sitter and power-law solutions in non-local Gauss\u2013Bonnet gravity","volume":"15","author":"Elizalde","year":"2018","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"136907","DOI":"10.1016\/j.physletb.2022.136907","article-title":"Non-local curvature gravity cosmology via Noether symmetries","volume":"826","author":"Acunzo","year":"2022","journal-title":"Phys. Lett. B"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"137475","DOI":"10.1016\/j.physletb.2022.137475","article-title":"The phase-space view of non-local gravity cosmology","volume":"834","author":"Capozziello","year":"2022","journal-title":"Phys. Lett. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"101346","DOI":"10.1016\/j.dark.2023.101346","article-title":"Reconstructing the distortion function of non-local cosmology: A model-independent approach","volume":"42","author":"Capozziello","year":"2023","journal-title":"Phys. Dark Univ."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"043531","DOI":"10.1103\/PhysRevD.90.043531","article-title":"Precision cosmology with Pad\u00e9 rational approximations: Theoretical predictions versus observational limits","volume":"90","author":"Aviles","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2576","DOI":"10.1093\/mnras\/staa871","article-title":"High-redshift cosmography: Auxiliary variables versus Pad\u00e9 polynomials","volume":"494","author":"Capozziello","year":"2020","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"A173","DOI":"10.1051\/0004-6361\/202037986","article-title":"Populations of filaments from the distribution of galaxies in numerical simulations","volume":"641","author":"Aghanim","year":"2020","journal-title":"Astron. Astrophys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1088\/0004-637X\/692\/2\/1075","article-title":"Monitoring stellar orbits around the Massive Black Hole in the Galactic Center","volume":"692","author":"Gillessen","year":"2009","journal-title":"Astrophys. J."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"044053","DOI":"10.1103\/PhysRevD.99.044053","article-title":"Constraining nonlocal gravity by S2 star orbits","volume":"99","author":"Dialektopoulos","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"11","DOI":"10.2298\/FUPCT1901011B","article-title":"Constraining Scalar-Tensor gravity models by S2 star orbits around the Galactic Center","volume":"17","author":"Borka","year":"2019","journal-title":"Facta Univ. Ser. Phys. Chem. Tech."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"104067","DOI":"10.1103\/PhysRevD.97.104067","article-title":"Analysis of the Yukawa gravitational potential in f(R) gravity I: Semiclassical periastron advance","volume":"97","author":"Martino","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1950047","DOI":"10.1142\/S0219887819500476","article-title":"Weak field limit and gravitational waves in higher-order gravity","volume":"16","author":"Capozziello","year":"2019","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"135821","DOI":"10.1016\/j.physletb.2020.135821","article-title":"Considerations on gravitational waves in higher-order local and non-local gravity","volume":"810","author":"Capozziello","year":"2020","journal-title":"Phys. Lett. B"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"175008","DOI":"10.1088\/1361-6382\/ac1720","article-title":"Gravitational waves in non-local gravity","volume":"38","author":"Capozziello","year":"2021","journal-title":"Class. Quantum Gravity"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1140\/epjc\/s10052-018-6035-2","article-title":"Renormalization group in super-renormalizable quantum gravity","volume":"78","author":"Modesto","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1540005","DOI":"10.1142\/S0217732315400052","article-title":"Renormalization and unitarity in higher derivative and nonlocal gravity theories","volume":"30","author":"Tomboulis","year":"2015","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"628","DOI":"10.1140\/epjc\/s10052-017-5210-1","article-title":"Nonlocal Teleparallel Cosmology","volume":"77","author":"Bahamonde","year":"2017","journal-title":"Eur. Phys. J. 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