R2 inflation to probe non-perturbative quantum gravity

  • Alexey S. KoshelevEmail author
  • K. Sravan Kumar
  • Alexei A. Starobinsky
Open Access
Regular Article - Theoretical Physics


It is natural to expect a consistent inflationary model of the very early Universe to be an effective theory of quantum gravity, at least at energies much less than the Planck one. For the moment, R + R2, or shortly R2, inflation is the most successful in accounting for the latest CMB data from the PLANCK satellite and other experiments. Moreover, recently it was shown to be ultra-violet (UV) complete via an embedding into an analytic infinite derivative (AID) non-local gravity. In this paper, we derive a most general theory of gravity that contributes to perturbed linear equations of motion around maximally symmetric space-times. We show that such a theory is quadratic in the Ricci scalar and the Weyl tensor with AID operators along with the Einstein-Hilbert term and possibly a cosmological constant. We explicitly demonstrate that introduction of the Ricci tensor squared term is redundant. Working in this quadratic AID gravity framework without a cosmological term we prove that for a specified class of space homogeneous space-times, a space of solutions to the equations of motion is identical to the space of backgrounds in a local R2 model. We further compute the full second order perturbed action around any background belonging to that class. We proceed by extracting the key inflationary parameters of our model such as a spectral index (ns), a tensor-to-scalar ratio (r) and a tensor tilt (nt). It appears that ns remains the same as in the local R2 inflation in the leading slow-roll approximation, while r and nt get modified due to modification of the tensor power spectrum. This class of models allows for any value of r < 0.07 with a modified consistency relation which can be fixed by future observations of primordial B-modes of the CMB polarization. This makes the UV complete R2 gravity a natural target for future CMB probes.


Cosmology of Theories beyond the SM Models of Quantum Gravity 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    R.M. Wald, General relativity, University of Chicago press, Chicago, U.S.A. (1984).CrossRefzbMATHGoogle Scholar
  2. [2]
    Virgo, LIGO Scientific collaboration, B.P. Abbott et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
  3. [3]
    K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    K.S. Stelle, Classical gravity with higher derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. 91B (1980) 99 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    A. A. Starobinsky, Nonsingular model of the Universe with the quantum gravitational de Sitter stage and its observational consequences, in the proceedings of the Second Seminar “Quantum Theory of Gravity”, October 13–15, Mopscow, Russia (1981).Google Scholar
  7. [7]
    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
  8. [8]
    Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE].
  9. [9]
    Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys. 594 (2016) A20 [arXiv:1502.02114] [INSPIRE].
  10. [10]
    BICEP2, Keck Array collaboration, 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].
  11. [11]
    M. Ostrogradsky, Mémoires sur les équations différentielles relatives au problème des isopérimètres, Mem. Ac. St. Petersbourg 6 (1850) 385.Google Scholar
  12. [12]
    R.P. Woodard, Avoiding dark energy with 1/r modifications of gravity, Lect. Notes Phys. 720 (2007) 403 [astro-ph/0601672].
  13. [13]
    A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].
  14. [14]
    C. Deffayet and D.A. Steer, A formal introduction to Horndeski and Galileon theories and their generalizations, Class. Quant. Grav. 30 (2013) 214006 [arXiv:1307.2450] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A.A. Starobinsky, Evolution of small perturbations of isotropic cosmological models with one-loop quantum gravitational corrections, JETP Lett. 34 (1981) 438.ADSGoogle Scholar
  16. [16]
    E. Witten, Interacting field theory of open superstrings, Nucl. Phys. B 276 (1986) 291 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    I.Y. Arefeva, D.M. Belov, A.A. Giryavets, A.S. Koshelev and P.B. Medvedev, Noncommutative field theories and (super)string field theories, in the proceedingso f the 11th Jorge Andre Swieca Summer School, January 14–27, Campos do Jordao, Sao Paulo, Brazil (2001), hep-th/0111208 [INSPIRE].
  18. [18]
    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
  19. [19]
    E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146 [INSPIRE].
  20. [20]
    T. Biswas, A. Mazumdar and W. Siegel, Bouncing universes in string-inspired gravity, JCAP 03 (2006) 009 [hep-th/0508194] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    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
  22. [22]
    L. Modesto, Super-renormalizable quantum gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [INSPIRE].ADSGoogle Scholar
  23. [23]
    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
  24. [24]
    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
  25. [25]
    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
  26. [26]
    A. Conroy, A.S. Koshelev and A. Mazumdar, Defocusing of null rays in infinite derivative gravity, JCAP 01 (2017) 017 [arXiv:1605.02080] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    J. Edholm, A.S. Koshelev and A. Mazumdar, Behavior of the Newtonian potential for ghost-free gravity and singularity-free gravity, Phys. Rev. D 94 (2016) 104033 [arXiv:1604.01989] [INSPIRE].ADSMathSciNetGoogle Scholar
  28. [28]
    A.S. Koshelev and A. Mazumdar, Do massive compact objects without event horizon exist in infinite derivative gravity?, Phys. Rev. D 96 (2017) 084069 [arXiv:1707.00273] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    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].
  30. [30]
    I. Ya. Aref’eva and A.S. Koshelev, Cosmic acceleration and crossing of w = −1 barrier from cubic superstring field theory, JHEP 02 (2007) 041 [hep-th/0605085] [INSPIRE].
  31. [31]
    I. Ya. Aref’eva and A.S. Koshelev, Cosmological signature of tachyon condensation, JHEP 09 (2008) 068 [arXiv:0804.3570] [INSPIRE].
  32. [32]
    G. Calcagni, Cosmological tachyon from cubic string field theory, JHEP 05 (2006) 012 [hep-th/0512259] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    G. Calcagni, M. Montobbio and G. Nardelli, A route to nonlocal cosmology, Phys. Rev. D 76 (2007) 126001 [arXiv:0705.3043] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    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].MathSciNetCrossRefGoogle Scholar
  35. [35]
    L. Modesto and L. Rachwal, Universally finite gravitational and gauge theories, Nucl. Phys. B 900 (2015) 147 [arXiv:1503.00261] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    L. Modesto and L. Rachwal, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [arXiv:1407.8036] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    L. Modesto, L. Rachwal and I.L. Shapiro, Renormalization group in super-renormalizable quantum gravity, arXiv:1704.03988 [INSPIRE].
  38. [38]
    A.S. Koshelev, K. Sravan Kumar, L. Modesto and L. Rachwal, Finite quantum gravity in (A)dS, arXiv:1710.07759 [INSPIRE].
  39. [39]
    A.H. Guth, The inflationary universe: a possible solution to the horizon and flatness problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].ADSzbMATHGoogle Scholar
  40. [40]
    V. Mukhanov, Physical foundations of cosmology, Cambridge University Press, Cambridge U.K. (2005).CrossRefzbMATHGoogle Scholar
  41. [41]
    L. Kofman, A.D. Linde and A.A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73 (1994) 3195 [hep-th/9405187] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    L. Kofman, A.D. Linde and A.A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56 (1997) 3258 [hep-ph/9704452] [INSPIRE].
  43. [43]
    J. Martin, C. Ringeval, R. Trotta and V. Vennin, The best inflationary models after Planck, JCAP 03 (2014) 039 [arXiv:1312.3529] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    V. Vennin, K. Koyama and D. Wands, Encyclopædia curvatonis, JCAP 11 (2015) 008 [arXiv:1507.07575] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    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
  46. [46]
    A.S. Koshelev, L. Modesto, L. Rachwal and A.A. Starobinsky, Occurrence of exact R 2 inflation in non-local UV-complete gravity, JHEP 11 (2016) 067 [arXiv:1604.03127] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    T. Biswas, T. Koivisto and A. Mazumdar, Nonlocal theories of gravity: the flat space propagator, in the proceedings of the Barcelona Postgrad Encounters on Fundamental Physics, October 17–19, Barcelona, Spain (2012), arXiv:1302.0532 [INSPIRE].
  48. [48]
    T. Biswas, A.S. Koshelev and A. Mazumdar, Consistent higher derivative gravitational theories with stable de Sitter and Anti-de Sitter backgrounds, Phys. Rev. D 95 (2017) 043533 [arXiv:1606.01250] [INSPIRE].ADSMathSciNetGoogle Scholar
  49. [49]
    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
  50. [50]
    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].
  51. [51]
    S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Crm Proceedings & Lecture Notes. American Mathematical Society, U.S.A. (2001).Google Scholar
  52. [52]
    V. Balasubramanian, J. de Boer and D. Minic, Notes on de Sitter space and holography, Class. Quant. Grav. 19 (2002) 5655 [hep-th/0207245] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    J. Edholm, UV completion of the Starobinsky model, tensor-to-scalar ratio and constraints on nonlocality, Phys. Rev. D 95 (2017) 044004 [arXiv:1611.05062] [INSPIRE].ADSMathSciNetGoogle Scholar
  54. [54]
    D.J. Brooker, S.D. Odintsov and R.P. Woodard, Precision predictions for the primordial power spectra from f(R) models of inflation, Nucl. Phys. B 911 (2016) 318 [arXiv:1606.05879] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  55. [55]
    D. Baumann, H. Lee and G.L. Pimentel, High-scale inflation and the tensor tilt, JHEP 01 (2016) 101 [arXiv:1507.07250] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    D. Müller, A. Ricciardone, A.A. Starobinsky and A. Toporensky, Anisotropic cosmological solutions in R + R 2 gravity, arXiv:1710.08753 [INSPIRE].
  57. [57]
    J. Ellis, D.V. Nanopoulos and K.A. Olive, Starobinsky-like inflationary models as avatars of no-scale supergravity, JCAP 10 (2013) 009 [arXiv:1307.3537] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    A. Kehagias, A. Moradinezhad Dizgah and A. Riotto, Remarks on the Starobinsky model of inflation and its descendants, Phys. Rev. D 89 (2014) 043527 [arXiv:1312.1155] [INSPIRE].ADSGoogle Scholar
  59. [59]
    A.S. Koshelev, K. Sravan Kumar and P. Vargas Moniz, Effective models of inflation from a nonlocal framework, Phys. Rev. D 96 (2017) 103503 [arXiv:1604.01440] [INSPIRE].ADSGoogle Scholar
  60. [60]
    C.P. Burgess, M. Cicoli, S. de Alwis and F. Quevedo, Robust inflation from fibrous strings, JCAP 05 (2016) 032 [arXiv:1603.06789] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  61. [61]
    R. Kallosh, A. Linde and D. Roest, Superconformal inflationary α-attractors, JHEP 11 (2013) 198 [arXiv:1311.0472] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Alexey S. Koshelev
    • 1
    • 2
    Email author
  • K. Sravan Kumar
    • 1
  • Alexei A. Starobinsky
    • 3
    • 4
  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.L.D. Landau Institute for Theoretical Physics RASMoscowRussian Federation
  4. 4.Kazan Federal UniversityKazanRussian Federation

Personalised recommendations