On Nonlocal Modified Gravity and Its Cosmological Solutions

  • Ivan Dimitrijevic
  • Branko Dragovich
  • Jelena Stankovic
  • Alexey S. Koshelev
  • Zoran Rakic
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 191)


During hundred years of General Relativity (GR), many significant gravitational phenomena have been predicted and discovered. General Relativity is still the best theory of gravity. Nevertheless, some (quantum) theoretical and (astrophysical and cosmological) phenomenological difficulties of modern gravity have been motivation to search more general theory of gravity than GR. As a result, many modifications of GR have been considered. One of promising recent investigations is Nonlocal Modified Gravity. In this article we present a brief review of some nonlocal gravity models with their cosmological solutions, in which nonlocality is expressed by an analytic function of the d’Alembert-Beltrami operator \(\Box \). Some new results are also presented.


Dark Matter Dark Energy Cosmological Solution Constant Scalar Curvature Nonlocal Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Work on this paper was partially supported by Ministry of Education, Science and Technological Development of the Republic of Serbia, grant No 174012. B.D. thanks Prof. Vladimir Dobrev for invitation to participate and give a talk on nonlocal gravity, as well as for hospitality, at the XI International Workshop “Lie Theory and its Applications in Physics”, 15–21 June 2015, Varna, Bulgaria. B.D. also thanks a support of the ICTP - SEENET-MTP project PRJ-09 “Cosmology and Strings” during preparation of this article. AK is supported by the FCT Portugal fellowship SFRH/BPD/105212/2014 and in part by FCT Portugal grant UID/MAT/00212/2013 and by RFBR grant 14-01-00707.


  1. 1.
    Abbott, B. P., et al., (LIGO Scientific Collaboration and Virgo Collaboration): Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016)Google Scholar
  2. 2.
    Ade, P. A. R., Aghanim, N., Armitage-Caplan, C., et al. (Planck Collaboration): Planck 2013 results. XVI. Cosmological parameters. [arXiv:1303.5076v3]
  3. 3.
    Aref’eva, I.Ya., Nonlocal string tachyon as a model for cosmological dark energy. AIP Conf. Proc. 826, 301 (2006). [astro-ph/0410443].Google Scholar
  4. 4.
    Aref’eva, I.Ya., Joukovskaya, L.V., Vernov, S.Yu.: Bouncing and accelerating solutions in nonlocal stringy models. JHEP 0707, 087 (2007) [hep-th/0701184]Google Scholar
  5. 5.
    Aref’eva, I.Ya., Volovich, I.V., On the null energy condition and cosmology. Theor. Math. Phys. 155, 503 (2008). [hep-th/0612098].Google Scholar
  6. 6.
    Aref’eva, I.Ya., Volovich, I.V., Cosmological Daemon. JHEP 1108, 102 (2011).Google Scholar
  7. 7.
    Barnaby, N., Biswas, T., Cline, J.M.: \(p\)-Adic inflation. JHEP 0704, 056 (2007) [hep-th/0612230]Google Scholar
  8. 8.
    Barvinsky, A.O.: Dark energy and dark matter from nonlocal ghost-free gravity theory. Phys. Lett. B 710, 12–16 (2012). [arXiv:1107.1463 [hep-th]]
  9. 9.
    Biswas, T., Gerwick, E., Koivisto, T., Mazumdar, A.: Towards singularity and ghost free theories of gravity. Phys. Rev. Lett. 108, 031101 (2012). [arXiv:1110.5249v2 [gr-qc]]
  10. 10.
    Biswas, T., Conroy, A., Koshelev, A.S., Mazumdar, A.: Generalized gost-free quadratic curvature gravity. [arXiv:1308.2319 [hep-th]]
  11. 11.
    Biswas, T., Mazumdar, A., Siegel, W: Bouncing universes in string-inspired gravity. J. Cosmology Astropart. Phys. 0603, 009 (2006). [arXiv:hep-th/0508194]
  12. 12.
    Biswas, T., Koivisto, T., Mazumdar, A.: Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity. J. Cosmology Astropart. Phys. 1011, 008 (2010). [arXiv:1005.0590v2 [hep-th]].
  13. 13.
    Biswas, T., Koshelev, A.S., Mazumdar, A., Vernov, S.Yu.: Stable bounce and inflation in non-local higher derivative cosmology. J. Cosmology Astropart. Phys. 08, 024 (2012). [arXiv:1206.6374 [astro-ph.CO]]
  14. 14.
    Brandenberger, R.H.: The matter bounce alternative to inflationary cosmology. [arXiv:1206.4196 [astro-ph.CO]]
  15. 15.
    Brekke, L., Freund, P.G.O.: \(p\)-Adic numbers in physics. Phys. Rep. 233, 1–66 (1993).Google Scholar
  16. 16.
    Briscese, F., Marciano, A., Modesto, L., Saridakis, E.N.: Inflation in (super-)renormalizable gravity. Phys. Rev. D 87, 083507 (2013). [arXiv:1212.3611v2 [hep-th]]
  17. 17.
    Calcagni, G., Modesto, L., Nicolini, P.: Super-accelerting bouncing cosmology in assymptotically-free non-local gravity. [arXiv:1306.5332 [gr-qc]]
  18. 18.
    Calcagni, G., Nardelli, G.: Nonlocal gravity and the diffusion equation. Phys. Rev. D 82, 123518 (2010). [arXiv:1004.5144 [hep-th]]
  19. 19.
    Calcagni, G., Montobbio, M., Nardelli, G.: A route to nonlocal cosmology. Phys. Rev. D 76, 126001 (2007). [arXiv:0705.3043v3 [hep-th]]
  20. 20.
    Capozziello S., Elizalde E., Nojiri S., Odintsov S.D.: Accelerating cosmologies from non-local higher-derivative gravity. Phys. Lett. B 671, 193 (2009). [arXiv:0809.1535]
  21. 21.
    Chicone, C., Mashhoon, B.: Nonlocal gravity in the solar system. Class. Quantum Grav. 33, 075005 (2016).[arXiv:1508.01508 [gr-qc]]
  22. 22.
    Clifton,T., Ferreira, P.G., Padilla, A., Skordis, C.: Modified gravity and cosmology. Phys. Rep. 513, 1–189 (2012). [arXiv:1106.2476v2 [astro-ph.CO]]
  23. 23.
    Conroy, A., Koshelev, A. S., Mazumdar, A., Geodesic completeness and homogeneity condition for cosmic inflation. Phys. Rev. D 90, no. 12, 123525 (2014). [arXiv:1408.6205 [gr-qc]].
  24. 24.
    Craps, B., de Jonckheere, T., Koshelev, A.S.: Cosmological perturbations in non-local higher-derivative gravity. [arXiv:1407.4982 [hep-th]]
  25. 25.
    Cusin, G., Foffa, S., Maggiore, M., Michele Mancarella, M.: Conformal symmetry and nonlinear extensions of nonlocal gravity. Phys. Rev. D 93, 083008 (2016). [arXiv:1602.01078 [hep-th]]
  26. 26.
    Deffayet, C., Woodard, R.P.: Reconstructing the distortion function for nonlocal cosmology. JCAP 0908, 023 (2009). [arXiv:0904.0961 [gr-qc]]
  27. 27.
    Deser, S., Woodard, R.P.: Nonlocal cosmology. Phys. Rev. Lett. 99, 111301 (2007). [arXiv:0706.2151 [astro-ph]]
  28. 28.
    Dimitrijevic, I.: Cosmological solutions in modified gravity with monomial nonlocality. Appl. Math. Comput. 195–203 (2016). [arXiv:1604.06824 [gr-qc]]
  29. 29.
    Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: On modified gravity. Springer Proceedings in Mathematics & Statistics 36, 251–259 (2013). [arXiv:1202.2352 [hep-th]]
  30. 30.
    Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: New cosmological solutions in nonlocal modified gravity. Rom. Journ. Phys. 58 (5-6), 550–559 (2013). [arXiv:1302.2794 [gr-qc]]
  31. 31.
    Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: A new model of nonlocal modified gravity. Publications de l’Institut Mathematique 94 (108), 187–196 (2013)Google Scholar
  32. 32.
    Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: Some pawer-law cosmological solutions in nonlocal modified gravity. Springer Proceedings in Mathematics & Statistics 111, 241–250 (2014)Google Scholar
  33. 33.
    Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: Some cosmological solutions of a nonlocal modified gravity. Filomat 29 (3), 619–628. arXiv:1508.05583 [hep-th]
  34. 34.
    Dimitrijevic, I., Dragovich, B., Grujic J., Koshelev A. S., Rakic, Z.: Cosmology of modified gravity with a non-local \(f(R)\). arXiv:1509.04254 [hep-th]
  35. 35.
    Dimitrijevic, I., Dragovich, B., Grujic J., Koshelev A. S., Rakic, Z.: Paper in preparation.Google Scholar
  36. 36.
    Dirian, Y., Foffa, S., Khosravi, N., Kunz, M., Maggiore, M.: Cosmological perturbations and structure formation in nonlocal infrared modifications of general relativity. [arXiv:1403.6068 [astro-ph.CO]]
  37. 37.
    Dragovich, B.: On nonlocal modified gravity and cosmology. Springer Proceedings in Mathematics & Statistics 111, 251–262 (2014). arXiv:1508.06584 [gr-qc]
  38. 38.
    Dragovich, B.: Nonlocal dynamics of \(p\)-adic strings. Theor. Math. Phys. 164 (3), 1151–1155 (2010). [arXiv:1011.0912v1 [hep-th]]
  39. 39.
    Dragovich, B.: Towards \(p\)-adic matter in the universe. Springer Proceedings in Mathematics and Statistics 36, 13–24 (2013). [arXiv:1205.4409 [hep-th]]
  40. 40.
    Dragovich, B., Khrennikov, A. Yu., Kozyrev, S. V., Volovich, I. V.: On p-adic mathematical physics. p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 1–17 (2009). [arXiv:0904.4205 [math-ph]]
  41. 41.
    Edholm, J., Koshelev, A. S., Mazumdar, A.: Universality of testing ghost-free gravity. [arXiv:1604.01989 [gr-qc]]
  42. 42.
    Elizalde, E., Pozdeeva, E.O., Vernov, S.Yu.: Stability of de Sitter solutions in non-local cosmological models. PoS(QFTHEP2011) 038, (2012). [arXiv:1202.0178 [gr-qc]]
  43. 43.
    Elizalde, E., Pozdeeva, E.O., Vernov, S.Yu., Zhang, Y.: Cosmological solutions of a nonlocal model with a perfect fluid. J. Cosmology Astropart. Phys. 1307, 034 (2013). [arXiv:1302.4330v2 [hep-th]]
  44. 44.
    Golovnev, A., Koivisto, T., Sandstad, M.: Effectively nonlocal metric-affine gravity. Phys. Rev. D 93, 064081 (2016). [arXiv:1509.06552v2 [gr-qc]]
  45. 45.
    Jhingan, S., Nojiri, S., Odintsov, S.D., Sami, Thongkool M.I., Zerbini, S.: Phantom and non-phantom dark energy: The Cosmological relevance of non-locally corrected gravity. Phys. Lett. B 663, 424-428 (2008). [arXiv:0803.2613 [hep-th]]
  46. 46.
    Koivisto, T.S.: Dynamics of nonlocal cosmology. Phys. Rev. D 77, 123513 (2008). [arXiv:0803.3399 [gr-qc]]
  47. 47.
    Koivisto, T.S.: Newtonian limit of nonlocal cosmology. Phys. Rev. D 78, 123505 (2008). [arXiv:0807.3778 [gr-qc]]
  48. 48.
    Koshelev, A.S., Vernov, S.Yu.: On bouncing solutions in non-local gravity. [arXiv:1202.1289v1 [hep-th]]
  49. 49.
    Koshelev, A.S., Vernov, S.Yu.: Cosmological solutions in nonlocal models. [arXiv:1406.5887v1 [gr-qc]]
  50. 50.
    Koshelev, A.S.: Modified non-local gravity. [arXiv:1112.6410v1 [hep-th]]
  51. 51.
    Koshelev, A.S.: Stable analytic bounce in non-local Einstein-Gauss-Bonnet cosmology. [arXiv:1302.2140 [astro-ph.CO]]
  52. 52.
    Koshelev, A.S., Vernov, S.Yu.: Analysis of scalar perturbations in cosmological models with a non-local scalar field. Class. Quant. Grav. 28, 085019 (2011). [arXiv:1009.0746v2 [hep-th]]
  53. 53.
    Koshelev, A.S., Modesto, L., Rachwal, L., Starobinsky, A.A.: Occurrence of exact \(R^2\) inflation in non-local UV-complete gravity. [arXiv:1604.03127v1 [hep-th]]
  54. 54.
    Lehners, J.-L., Steinhardt, P.J.: Planck 2013 results support the cyclic universe. arXiv:1304.3122 [astro-ph.CO]
  55. 55.
    Li, Y-D., Modesto, L., Rachwal, L.: Exact solutions and spacetime singularities in nonlocal gravity. JHEP 12, 173 (2015). [arXiv:1506.08619 [hep-th]]
  56. 56.
    Modesto, L.: Super-renormalizable quantum gravity. Phys. Rev. D 86, 044005 (2012). [arXiv:1107.2403 [hep-th]]
  57. 57.
    Modesto, L., Rachwal, L.: Super-renormalizable and finite gravitational theories. Nucl. Phys. B 889, 228 (2014). [arXiv:1407.8036 [hep-th]]
  58. 58.
    Modesto, L., Tsujikawa, S.: Non-local massive gravity. Phys. Lett. B 727, 48–56 (2013). [arXiv:1307.6968 [hep-th]]
  59. 59.
    Moffat, J.M.: Ultraviolet complete quantum gravity. Eur. Phys. J. Plus 126, 43 (2011). [arXiv:1008.2482 [gr-qc]]
  60. 60.
    Nojiri, S., Odintsov, S.D.: Unified cosmic history in modified gravity: from \(F(R)\) theory to Lorentz non-invariant models. Phys. Rep. 505, 59–144 (2011). [arXiv:1011.0544v4 [gr-qc]]
  61. 61.
    Nojiri, S., Odintsov, S.D.: Modified non-local-F(R) gravity as the key for inflation and dark energy. Phys. Lett. B 659, 821–826 (2008). [arXiv:0708.0924v3 [hep-th]
  62. 62.
    Novello, M., Bergliaffa, S.E.P.: Bouncing cosmologies. Phys. Rep. 463, 127–213 (2008). [arXiv:0802.1634 [astro-ph]]
  63. 63.
    T. P. Sotiriou, V. Faraoni, \(f(R)\) theories of gravity. Rev. Mod. Phys. 82 (2010) 451–497. [arXiv:0805.1726v4 [gr-qc]]
  64. 64.
    Stelle, K.S.: Renormalization of higher derivative quantum gravity. Phys. Rev. D 16, 953 (1977)Google Scholar
  65. 65.
    Vladimirov, V.S., Volovich, I.V., Zelenov, E.I., p-adic Analysis and Mathematical Physics, 1994Google Scholar
  66. 66.
    Woodard, R.P.: Nonlocal models of cosmic acceleration. [arXiv:1401.0254 [astro-ph.CO]]
  67. 67.
    Zhang, Y.-li., Sasaki, M.: Screening of cosmological constant in non-local cosmology. Int. J. Mod. Phys. D 21, 1250006 (2012). [arXiv:1108.2112 [gr-qc]]
  68. 68.
    Zhang, Y.-li., Koyama, K., Sasaki, M., Zhao, G-B.: Acausality in nonlocal gravity theory. JHEP 1603, 039(2016). [arXiv:1601.03808v2 [hep-th]]

Copyright information

© Springer Nature Singapore Pte Ltd. 2016

Authors and Affiliations

  • Ivan Dimitrijevic
    • 1
  • Branko Dragovich
    • 2
    • 3
  • Jelena Stankovic
    • 4
  • Alexey S. Koshelev
    • 5
    • 6
  • Zoran Rakic
    • 1
  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia
  2. 2.Institute of PhysicsUniversity of BelgradeBelgradeSerbia
  3. 3.Mathematical Institute SANUBelgradeSerbia
  4. 4.Teacher Education FacultyUniversity of BelgradeBelgradeSerbia
  5. 5.Departamento de Física and Centro de Matemática e AplicaçõesUniversidade da Beira InteriorCovilhãPortugal
  6. 6.Theoretische Natuurkunde, Vrije Universiteit Brussel, and The International Solvay InstitutesBrusselsBelgium

Personalised recommendations