{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T05:23:31Z","timestamp":1775366611496,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,2]],"date-time":"2020-06-02T00:00:00Z","timestamp":1591056000000},"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>In this paper, we investigate a nonlocal modification of general relativity (GR) with action     S =  1  16 \u03c0 G   \u222b  [ R \u2212 2 \u039b +  ( R \u2212 4 \u039b )   F  ( \u25a1 )    ( R \u2212 4 \u039b )  ]     \u2212 g     d 4  x ,     where     F  ( \u25a1 )  =  \u2211  n = 1   + \u221e    f n   \u25a1 n      is an analytic function of the d\u2019Alembertian \u25a1. We found a few exact cosmological solutions of the corresponding equations of motion. There are two solutions which are valid only if     \u039b \u2260 0 ,  k = 0 ,     and they have no analogs in Einstein\u2019s gravity with cosmological constant    \u039b   . One of these two solutions is     a  ( t )  = A   t    e   \u039b 4   t 2    ,     that mimics properties similar to an interference between the radiation and the dark energy. Another solution is a nonsingular bounce one     a  ( t )  = A   e  \u039b  t 2       . For these two solutions, some cosmological aspects are discussed. We also found explicit form of the nonlocal operator     F ( \u25a1 )    , which satisfies obtained necessary conditions.<\/jats:p>","DOI":"10.3390\/sym12060917","type":"journal-article","created":{"date-parts":[[2020,6,3]],"date-time":"2020-06-03T04:12:09Z","timestamp":1591157529000},"page":"917","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Some Cosmological Solutions of a New Nonlocal Gravity Model"],"prefix":"10.3390","volume":"12","author":[{"given":"Ivan","family":"Dimitrijevic","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11158 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Branko","family":"Dragovich","sequence":"additional","affiliation":[{"name":"Institute of Physics, University of Belgrade, 11080 Belgrade, Serbia"},{"name":"Mathematical Institute, Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexey S.","family":"Koshelev","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica and Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es, Universidade da Beira Interior, 6200 Covilh\u00e3, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zoran","family":"Rakic","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11158 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jelena","family":"Stankovic","sequence":"additional","affiliation":[{"name":"Teacher Education Faculty, University of Belgrade, Kraljice Natalije 43, 11000 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wald, R.M. (1984). General Relativity, University of Chicago Press.","DOI":"10.7208\/chicago\/9780226870373.001.0001"},{"key":"ref_2","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_3","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_4","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.physrep.2011.04.001","article-title":"Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models","volume":"505","author":"Nojiri","year":"2011","journal-title":"Phys. Rep."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.physrep.2008.04.006","article-title":"Bouncing cosmologies","volume":"463","author":"Novello","year":"2008","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1103\/PhysRevLett.99.111301","article-title":"Nonlocal cosmology","volume":"99","author":"Deser","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s10701-014-9780-6","article-title":"Nonlocal models of cosmic acceleration","volume":"44","author":"Woodard","year":"2014","journal-title":"Found. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1088\/1475-7516\/2018\/03\/002","article-title":"Nonlocal gravity. Conceptual aspects and cosmological predictions","volume":"2018","author":"Belgacem","year":"2018","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1088\/1475-7516\/2006\/03\/009","article-title":"Bouncing universes in string-inspired gravity","volume":"603","author":"Biswas","year":"2006","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1088\/1475-7516\/2010\/11\/008","article-title":"Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity","volume":"1011","author":"Biswas","year":"2010","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"31","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_13","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1088\/1475-7516\/2012\/08\/024","article-title":"Stable bounce and inflation in non-local higher derivative cosmology","volume":"8","author":"Biswas","year":"2012","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"159501","DOI":"10.1088\/0264-9381\/31\/15\/159501","article-title":"Generalized gost-free quadratic curvature gravity","volume":"31","author":"Biswas","year":"2014","journal-title":"Class. Quantum Grav."},{"key":"ref_15","first-page":"043533","article-title":"Consistent higher derivative gravitational theories with stable de Sitter and Anti-de Sitter backgrounds","volume":"95","author":"Biswas","year":"2017","journal-title":"Phys. Rev."},{"key":"ref_16","first-page":"251","article-title":"On Nonlocal modified gravity and cosmology","volume":"111","author":"Dragovich","year":"2014","journal-title":"Springer Proc. Math. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1134\/S106377961205019X","article-title":"On bouncing solutions in non-local gravity","volume":"43","author":"Koshelev","year":"2012","journal-title":"Phys. Part. Nuclei"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"155001","DOI":"10.1088\/0264-9381\/30\/15\/155001","article-title":"Stable analytic bounce in non-local Einstein-Gauss-Bonnet cosmology","volume":"30","author":"Koshelev","year":"2013","journal-title":"Class. Quantum Grav."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/JHEP03(2018)071","article-title":"R2 inflation to probe non-perturbative quantum gravity","volume":"1803","author":"Koshelev","year":"2018","journal-title":"JHEP"},{"key":"ref_20","first-page":"1","article-title":"Occurrence of exact R2 inflation in non-local UV-complete gravity","volume":"11","author":"Koshelev","year":"2016","journal-title":"JHEP"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1088\/1475-7516\/2018\/09\/034","article-title":"Classical properties of non-local, ghost- and singularity-free gravity","volume":"2018","author":"Buoninfante","year":"2018","journal-title":"JCAP"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1088\/1475-7516\/2019\/02\/020","article-title":"Towards conformally flat, non-Kasner vacuum solution in infinite derivative gravity","volume":"2019","author":"Koshelev","year":"2019","journal-title":"JCAP"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Elizalde, E., Pozdeeva, E.O., and Vernov, S.Y. (2013). Stability of de Sitter solutions in non-local cosmological models. arXiv.","DOI":"10.1088\/1475-7516\/2013\/07\/034"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"015024","DOI":"10.1088\/0264-9381\/32\/1\/015024","article-title":"Generalised quadratic curvature, non-local infrared modifications of gravity and Newtonian potentials","volume":"32","author":"Conroy","year":"2015","journal-title":"Clas. Quantum Grav."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1134\/S2070046617020017","article-title":"p-Adic mathematical physics: The first 30 years","volume":"9","author":"Dragovich","year":"2017","journal-title":"p-Adic Numbers Ultrametric Anal. Appl."},{"key":"ref_26","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_27","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.nuclphysb.2014.10.015","article-title":"Super-renormalizable and finite gravitational theories","volume":"889","author":"Modesto","year":"2014","journal-title":"Nucl. Phys. B"},{"key":"ref_28","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_29","first-page":"251","article-title":"On modified gravity","volume":"36","author":"Dimitrijevic","year":"2013","journal-title":"Springer Proc. Math. Stat."},{"key":"ref_30","first-page":"550","article-title":"New cosmological solutions in nonlocal modified gravity","volume":"58","author":"Dimitrijevic","year":"2013","journal-title":"Rom. Journ. Phys."},{"key":"ref_31","first-page":"241","article-title":"Some power-law cosmological solutions in nonlocal modified gravity","volume":"111","author":"Dimitrijevic","year":"2014","journal-title":"Springer Proc. Math. Stat."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.2298\/FIL1904163D","article-title":"Cosmology of non-local f(R) gravity","volume":"33","author":"Dimitrijevic","year":"2019","journal-title":"Filomat"},{"key":"ref_33","first-page":"35","article-title":"On nonlocal modified gravity and its cosmological solutions","volume":"191","author":"Dimitrijevic","year":"2016","journal-title":"Springer Proc. Math. Stat."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"619","DOI":"10.2298\/FIL1503619D","article-title":"Some cosmological solutions of a nonlocal modified gravity","volume":"29","author":"Dimitrijevic","year":"2015","journal-title":"Filomat"},{"key":"ref_35","first-page":"195","article-title":"Cosmological solutions in modified gravity with monomial nonlocality","volume":"285","author":"Dimitrijevic","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"53","DOI":"10.2298\/PIM1817053D","article-title":"On nonlocal gravity with constant scalar curvature","volume":"103","author":"Dimitrijevic","year":"2018","journal-title":"Publ. De L\u2019Institut Math. Nouv. S\u00e9rie"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"134848","DOI":"10.1016\/j.physletb.2019.134848","article-title":"Cosmological solutions of a nonlocal square root gravity","volume":"797","author":"Dimitrijevic","year":"2019","journal-title":"Phys. Lett. B"},{"key":"ref_38","first-page":"91","article-title":"Variations of infinite derivative modified gravity","volume":"263","author":"Dimitrijevic","year":"2018","journal-title":"Springer Proc. Math. Stat."},{"key":"ref_39","unstructured":"Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., and Ballardini, M. (2018). Planck 2018 results. VI. Cosmological parameters, Planck collaboration. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/917\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:34:49Z","timestamp":1760175289000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/917"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,2]]},"references-count":39,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12060917"],"URL":"https:\/\/doi.org\/10.3390\/sym12060917","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,2]]}}}