Journal of High Energy Physics

, 2016:67 | Cite as

Occurrence of exact R 2 inflation in non-local UV-complete gravity

  • Alexey S. Koshelev
  • Leonardo Modesto
  • Leslaw Rachwal
  • Alexei A. Starobinsky
Open Access
Regular Article - Theoretical Physics


The R + R 2, shortly named “R 2” (“Starobinsky”) inflationary model, represents a fully consistent example of a one-parameter inflationary scenario. This model has a “graceful exit” from inflation and provides a mechanism for subsequent creation and final thermalization of the standard matter. Moreover, it produces a very good fit of the observed spectrum of primordial perturbations. In the present paper we show explicitly that the R 2 inflationary spacetime is an exact solution of a range of weakly non-local (quasi-polynomial) gravitational theories, which provide an ultraviolet completion of the R 2 theory. These theories are ghost-free, super-renormalizable or finite at quantum level, and perturbatively unitary. Their spectrum consists of the graviton and the scalaron that is responsible for driving the inflation. Notably, any further extension of the spectrum leads to propagating ghost degrees of freedom. We are aimed at presenting a detailed construction of such theories in the so called Weyl basis. Further, we give a special account to the cosmological implications of this theory by considering perturbations during inflation. The highlight of the non-local model is the prediction of a modified, in comparison to a local R 2 model, value for the ratio of tensor and scalar power spectra r, depending on the parameters of the theory. The relevant parameters are under control to be successfully confronted with existing observational data. Furthermore, the modified r can surely meet future observational constraints.


Cosmology of Theories beyond the SM Models of Quantum Gravity 


Open Access

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  1. [1]
    A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    A.A. Starobinsky, Nonsingular model of the Universe with the quantum gravitational de Sitter stage and its observational consequences, in Proceedings of the second seminar “Quantum Theory of Gravity”, Moscow, 13-15 October 1981, INR Press, Moscow (1982), pg. 58-72, reprinted in Quantum Gravity, M.A. Markov and P.C. West eds., Plenum Publ. Co., New York (1984), pg. 103-128 [INSPIRE].
  3. [3]
    A.A. Starobinsky, The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy, Sov. Astron. Lett. 9 (1983) 302 [INSPIRE].ADSGoogle Scholar
  4. [4]
    Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE].
  5. [5]
    Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys. 594 (2016) A20 [arXiv:1502.02114] [INSPIRE].
  6. [6]
    BICEP2, Keck Array collaborations, P.A.R. Ade et al., Improved Constraints on Cosmology and Foregrounds from BICEP2 and Keck Array Cosmic Microwave Background Data with Inclusion of 95 GHz Band, Phys. Rev. Lett. 116 (2016) 031302 [arXiv:1510.09217] [INSPIRE].
  7. [7]
    K.S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective action in quantum gravity, CRC Press (1992).Google Scholar
  10. [10]
    M. Asorey, J.L. Lopez and I.L. Shapiro, Some remarks on high derivative quantum gravity, Int. J. Mod. Phys. A 12 (1997) 5711 [hep-th/9610006] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A. Accioly, A. Azeredo and H. Mukai, Propagator, tree-level unitarity and effective nonrelativistic potential for higher-derivative gravity theories in D dimensions, J. Math. Phys. 43 (2002) 473 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F.d.O. Salles and I.L. Shapiro, Do we have unitary and (super)renormalizable quantum gravity below the Planck scale?, Phys. Rev. D 89 (2014) 084054 [arXiv:1401.4583] [INSPIRE].
  13. [13]
    E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146 [INSPIRE].
  14. [14]
    E.T. Tomboulis, Renormalization and unitarity in higher derivative and nonlocal gravity theories, Mod. Phys. Lett. A30 (2015) 1540005.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    E.T. Tomboulis, Nonlocal and quasilocal field theories, Phys. Rev. D 92 (2015) 125037 [arXiv:1507.00981] [INSPIRE].ADSMathSciNetGoogle Scholar
  16. [16]
    T. Biswas, A. Mazumdar and W. Siegel, Bouncing universes in string-inspired gravity, JCAP 03 (2006) 009 [hep-th/0508194] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    F. Briscese, L. Modesto and S. Tsujikawa, Super-renormalizable or finite completion of the Starobinsky theory, Phys. Rev. D 89 (2014) 024029 [arXiv:1308.1413] [INSPIRE].ADSGoogle Scholar
  18. [18]
    S. Deser and R.P. Woodard, Nonlocal Cosmology, Phys. Rev. Lett. 99 (2007) 111301 [arXiv:0706.2151] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    L. Modesto, Super-renormalizable Quantum Gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [INSPIRE].ADSGoogle Scholar
  20. [20]
    L. Modesto and L. Rachwal, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [arXiv:1407.8036] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Talaganis, T. Biswas and A. Mazumdar, Towards understanding the ultraviolet behavior of quantum loops in infinite-derivative theories of gravity, Class. Quant. Grav. 32 (2015) 215017 [arXiv:1412.3467] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    P. Donà, S. Giaccari, L. Modesto, L. Rachwal and Y. Zhu, Scattering amplitudes in super-renormalizable gravity, JHEP 08 (2015) 038 [arXiv:1506.04589] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    L. Modesto, J.W. Moffat and P. Nicolini, Black holes in an ultraviolet complete quantum gravity, Phys. Lett. B 695 (2011) 397 [arXiv:1010.0680] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    T. Biswas, A. Conroy, A.S. Koshelev and A. Mazumdar, Generalized ghost-free quadratic curvature gravity, Class. Quant. Grav. 31 (2014) 015022 [Erratum ibid. 31 (2014) 159501] [arXiv:1308.2319] [INSPIRE].
  26. [26]
    C. Bambi, D. Malafarina and L. Modesto, Terminating black holes in asymptotically free quantum gravity, Eur. Phys. J. C 74 (2014) 2767 [arXiv:1306.1668] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    G. Calcagni, L. Modesto and P. Nicolini, Super-accelerating bouncing cosmology in asymptotically-free non-local gravity, Eur. Phys. J. C 74 (2014) 2999 [arXiv:1306.5332] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    L. Modesto, T. de Paula Netto and I.L. Shapiro, On Newtonian singularities in higher derivative gravity models, JHEP 04 (2015) 098 [arXiv:1412.0740] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    V.P. Frolov, Mass-gap for black hole formation in higher derivative and ghost free gravity, Phys. Rev. Lett. 115 (2015) 051102 [arXiv:1505.00492] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    V.P. Frolov, A. Zelnikov and T. de Paula Netto, Spherical collapse of small masses in the ghost-free gravity, JHEP 06 (2015) 107 [arXiv:1504.00412] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    V.P. Frolov and A. Zelnikov, Head-on collision of ultrarelativistic particles in ghost-free theories of gravity, Phys. Rev. D 93 (2016) 064048 [arXiv:1509.03336] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    J. Edholm, A.S. Koshelev and A. Mazumdar, Behavior of the Newtonian potential for ghost-free gravity and singularity-free gravity, arXiv:1604.01989 [INSPIRE].
  33. [33]
    R.L. Arnowitt, S. Deser and C.W. Misner, Canonical variables for general relativity, Phys. Rev. 117 (1960) 1595 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    L.F. Abbott and S. Deser, Stability of Gravity with a Cosmological Constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    A.A. Starobinsky and H.J. Schmidt, On a general vacuum solution of fourth-order gravity, Class. Quant. Grav. 4 (1987) 695 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    V. Muller, H.J. Schmidt and A.A. Starobinsky, The Stability of the de Sitter Space-time in Fourth Order Gravity, Phys. Lett. B 202 (1988) 198 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    S. Deser and B. Tekin, New energy definition for higher curvature gravities, Phys. Rev. D 75 (2007) 084032 [gr-qc/0701140] [INSPIRE].
  38. [38]
    L. Modesto and L. Rachwal, Universally finite gravitational and gauge theories, Nucl. Phys. B 900 (2015) 147 [arXiv:1503.00261] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    L. Modesto, M. Piva and L. Rachwal, Finite quantum gauge theories, Phys. Rev. D 94 (2016) 025021 [arXiv:1506.06227] [INSPIRE].ADSGoogle Scholar
  40. [40]
    A.A. Starobinsky, Evolution of small perturbations of isotropic cosmological models with one-loop quantum gravitational corrections, JETP Lett. 34 (1981) 438 [INSPIRE].ADSGoogle Scholar
  41. [41]
    I.Ya. Aref’eva, A.S. Koshelev and S.Yu. Vernov, Exact solution in a string cosmological model, Theor. Math. Phys. 148 (2006) 895 [astro-ph/0412619] [INSPIRE].
  42. [42]
    E. Witten, Interacting Field Theory of Open Superstrings, Nucl. Phys. B 276 (1986) 291 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-adic analysis and mathematical physics, Ser. Sov. East Eur. Math. 1 (1994) 1.Google Scholar
  44. [44]
    I.Y. Arefeva, D.M. Belov, A.A. Giryavets, A.S. Koshelev and P.B. Medvedev, Noncommutative field theories and (super)string field theories, hep-th/0111208 [INSPIRE].
  45. [45]
    T. Biswas and A. Mazumdar, Super-Inflation, Non-Singular Bounce and Low Multipoles, Class. Quant. Grav. 31 (2014) 025019 [arXiv:1304.3648] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    D. Chialva and A. Mazumdar, Cosmological implications of quantum corrections and higher-derivative extension, Mod. Phys. Lett. A 30 (2015) 1540008 [arXiv:1405.0513] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    A.O. Barvinsky, Nonlocal action for long distance modifications of gravity theory, Phys. Lett. B 572 (2003) 109 [hep-th/0304229] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    A.O. Barvinsky, Dark energy and dark matter from nonlocal ghost-free gravity theory, Phys. Lett. B 710 (2012) 12 [arXiv:1107.1463] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    A.O. Barvinsky, Aspects of Nonlocality in Quantum Field Theory, Quantum Gravity and Cosmology, Mod. Phys. Lett. A 30 (2015) 1540003 [arXiv:1408.6112] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A. Conroy, T. Koivisto, A. Mazumdar and A. Teimouri, Generalized quadratic curvature, non-local infrared modifications of gravity and Newtonian potentials, Class. Quant. Grav. 32 (2015) 015024 [arXiv:1406.4998] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    A.S. Koshelev, Stable analytic bounce in non-local Einstein-Gauss-Bonnet cosmology, Class. Quant. Grav. 30 (2013) 155001 [arXiv:1302.2140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    T. Biswas, A.S. Koshelev and A. Mazumdar, Gravitational theories with stable (anti-)de Sitter backgrounds, Fundam. Theor. Phys. 183 (2016) 97 [arXiv:1602.08475] [INSPIRE].CrossRefGoogle Scholar
  53. [53]
    B. Craps, T. De Jonckheere and A.S. Koshelev, Cosmological perturbations in non-local higher-derivative gravity, JCAP 11 (2014) 022 [arXiv:1407.4982] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    N.V. Krasnikov, Nonlocal gauge theories, Theor. Math. Phys. 73 (1987) 1184 [INSPIRE].CrossRefGoogle Scholar
  55. [55]
    J. Khoury, Fading gravity and self-inflation, Phys. Rev. D 76 (2007) 123513 [hep-th/0612052] [INSPIRE].ADSMathSciNetGoogle Scholar
  56. [56]
    S. Alexander, A. Marcianò and L. Modesto, The Hidden Quantum Groups Symmetry of Super-renormalizable Gravity, Phys. Rev. D 85 (2012) 124030 [arXiv:1202.1824] [INSPIRE].ADSGoogle Scholar
  57. [57]
    F. Briscese, A. Marcianò, L. Modesto and E.N. Saridakis, Inflation in (Super-)renormalizable Gravity, Phys. Rev. D 87 (2013) 083507 [arXiv:1212.3611] [INSPIRE].ADSGoogle Scholar
  58. [58]
    L. Modesto and S. Tsujikawa, Non-local massive gravity, Phys. Lett. B 727 (2013) 48 [arXiv:1307.6968] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    G. Calcagni and L. Modesto, Nonlocal quantum gravity and M-theory, Phys. Rev. D 91 (2015) 124059 [arXiv:1404.2137] [INSPIRE].ADSMathSciNetGoogle Scholar
  60. [60]
    Q.-G. Huang, A polynomial f(R) inflation model, JCAP 02 (2014) 035 [arXiv:1309.3514] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    Yu.V. Kuzmin, The convergent nonlocal gravitation (in Russian), Sov. J. Nucl. Phys. 50 (1989) 1011 [INSPIRE].Google Scholar
  62. [62]
    P. Van Nieuwenhuizen, On ghost-free tensor lagrangians and linearized gravitation, Nucl. Phys. B 60 (1973) 478 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  63. [63]
    V.A. Alebastrov and G.V. Efimov, A proof of the unitarity of S-matrix in a nonlocal quantum field theory, Commun. Math. Phys. 31 (1973) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    V.A. Alebastrov and G.V. Efimov, Causality in quantum field theory with nonlocal interaction, Commun. Math. Phys. 38 (1974) 11 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  65. [65]
    Y.-D. Li, L. Modesto and L. Rachwal, Exact solutions and spacetime singularities in nonlocal gravity, JHEP 12 (2015) 173 [arXiv:1506.08619] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    T. Biswas, T. Koivisto and A. Mazumdar, Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity, JCAP 11 (2010) 008 [arXiv:1005.0590] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    A.S. Koshelev and S.Yu. Vernov, Cosmological Solutions in Nonlocal Models, Phys. Part. Nucl. Lett. 11 (2014) 960 [arXiv:1406.5887] [INSPIRE].CrossRefGoogle Scholar
  68. [68]
    V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203 [INSPIRE].
  69. [69]
    T. Biswas, A.S. Koshelev, A. Mazumdar and S.Yu. Vernov, Stable bounce and inflation in non-local higher derivative cosmology, JCAP 08 (2012) 024 [arXiv:1206.6374] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    T. Biswas, A.S. Koshelev and A. Mazumdar, Consistent Higher Derivative Gravitational theories with stable de Sitter and Anti-de Sitter Backgrounds, arXiv:1606.01250 [INSPIRE].
  71. [71]
    S.M. Christensen and M.J. Duff, Quantizing Gravity with a Cosmological Constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    J.M. Bardeen, Gauge Invariant Cosmological Perturbations, Phys. Rev. D 22 (1980) 1882 [INSPIRE].ADSMathSciNetGoogle Scholar
  73. [73]
    A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].
  74. [74]
    I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic, Some Cosmological Solutions of a Nonlocal Modified Gravity, arXiv:1508.05583 [INSPIRE].
  75. [75]
    R. Kallosh, A. Linde and D. Roest, Superconformal Inflationary α-Attractors, JHEP 11 (2013) 198 [arXiv:1311.0472] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    A.S. Koshelev and S.Yu. Vernov, On bouncing solutions in non-local gravity, Phys. Part. Nucl. 43 (2012) 666 [arXiv:1202.1289] [INSPIRE].CrossRefGoogle Scholar
  77. [77]
    I. Dimitrijevic, B. Dragovich, J. Grujic, A.S. Koshelev and Z. Rakic, Cosmology of modified gravity with a non-local f(R), arXiv:1509.04254 [INSPIRE].
  78. [78]
    R. Durrer, Cosmological perturbation theory, Lect. Notes Phys. 653 (2004) 31 [astro-ph/0402129] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Alexey S. Koshelev
    • 1
    • 2
  • Leonardo Modesto
    • 3
    • 4
  • Leslaw Rachwal
    • 4
  • Alexei A. Starobinsky
    • 5
    • 6
  1. 1.Departamento de Física and Centro de Matemática e Aplicações (CMA-UBI)Universidade da Beira InteriorCovilhãPortugal
  2. 2.Theoretische Natuurkunde, Vrije Universiteit Brussel, and The International Solvay InstitutesBrusselsBelgium
  3. 3.Department of PhysicsSouthern University of Science and TechnologyShenzhenChina
  4. 4.Department of Physics & Center for Field Theory and Particle PhysicsFudan UniversityShanghaiChina
  5. 5.L.D. Landau Institute for Theoretical Physics RASMoscowRussian Federation
  6. 6.Kazan Federal UniversityKazanRussian Federation

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