Gravitational Theories with Stable (anti-)de Sitter Backgrounds

  • Tirthabir BiswasEmail author
  • Alexey S. Koshelev
  • Anupam Mazumdar
Part of the Fundamental Theories of Physics book series (FTPH, volume 183)


In this article we will construct the most general torsion-free parity-invariant covariant theory of gravity that is free from ghost-like and tachyonic instabilities around constant curvature space-times in four dimensions. Specifically, this includes the Minkowski, de Sitter and anti-de Sitter backgrounds. We will first argue in details how starting from a general covariant action for the metric one arrives at an “equivalent” action that at most contains terms that are quadratic in curvatures but nevertheless is sufficient for the purpose of studying stability of the original action. We will then briefly discuss how such a “quadratic curvature action” can be decomposed in a covariant formalism into separate sectors involving the tensor, vector and scalar modes of the metric tensor; most of the details of the analysis however, will be presented in an accompanying paper. We will find that only the transverse and trace-less spin-2 graviton with its two helicity states and possibly a spin-0 Brans-Dicke type scalar degree of freedom are left to propagate in 4 dimensions. This will also enable us to arrive at the consistency conditions required to make the theory perturbatively stable around constant curvature backgrounds.


Covariant Derivative Weyl Tensor Riemann Tensor Scalar Mode Equivalent Action 
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.



We would like to thank Spyridon Talaganis for discussions. TB would like to thank Carl for his insightful comments on the general subject matter of IDG theories. AM is supported by the STFC grant ST/J000418/1. AK is supported by the FCT Portugal fellowship SFRH/BPD/105212/2014.


  1. 1.
    M.J.G. Veltman, Quantum theory of gravitation. Conf. Proc. C 7507281, 265 (1975)MathSciNetGoogle Scholar
  2. 2.
    B.S. DeWitt, Quantum theory of gravity. 1. The canonical theory. Phys. Rev. 160, 1113 (1967). B.S. DeWitt, Quantum theory of gravity. 2. The manifestly covariant theory. Phys. Rev. 162, 1195 (1967). B.S. DeWitt, Quantum theory of gravity. 3. Applications of the covariant theory. Phys. Rev. 162, 1239 (1967)Google Scholar
  3. 3.
    B.S. DeWitt, G. Esposito, An introduction to quantum gravity. Int. J. Geom. Meth. Mod. Phys. 5, 101 (2008), arXiv:0711.2445 [hep-th]
  4. 4.
    J. Polchinski, String Theory: Superstring Theory and Beyond, vol. 2 (Cambridge, UK: Univ. Pr 1998) , p. 531Google Scholar
  5. 5.
    A. Ashtekar, Introduction to loop quantum gravity and cosmology. Lect. Notes Phys. 863, 31 (2013)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    J. Henson, in The Causal Set Approach to Quantum Gravity, ed. by D. Oriti, Approaches to Quantum Gravity, pp. 393–413, arXiv:gr-qc/0601121 (for a review)
  7. 7.
    S. Weinberg, in Ultraviolet Divergences in Quantum Theories of Gravitation. eds. by S.W. Hawking (Cambridge Univ. (UK)); W. Israel (Alberta Univ., Edmonton (Canada). Theoretical Physics Inst.), pp. 790–831; ISBN 0 521 22285 0; 1979; pp. 790–831; University Press; CambridgeGoogle Scholar
  8. 8.
    E. Witten, Noncommutative geometry and string field theory. Nucl. Phys. B 268, 253 (1986)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Smailagic, E. Spallucci, Lorentz invariance, unitarity in UV-finite of QFT on noncommutative spacetime. J. Phys. A 37, 1 (2004) [Erratum-ibid. A 37, 7169 (2004)] arXiv:hep-th/0406174
  10. 10.
    P.G.O. Freund, M. Olson, Nonarchimedean strings. Phys. Lett. B 199, 186 (1987). P.G.O. Freund, E. Witten, Adelic string amplitudes. Phys. Lett. B 199, 191 (1987). L. Brekke, P.G.O. Freund, M. Olson, E. Witten, Nonarchimedean string dynamics. Nucl. Phys. B 302, 365 (1988). P.H. Frampton, Y. Okada, Effective scalar field theory of \(P^-\)adic string. Phys. Rev. D 37, 3077 (1988)Google Scholar
  11. 11.
    B. Dragovich, Zeta strings, arXiv:hep-th/0703008
  12. 12.
    M.R. Douglas, S.H. Shenker, Strings in less than one-dimension. Nucl. Phys. B 335, 635 (1990). D.J. Gross, A.A. Migdal, Nonperturbative solution of the ising model on a random surface. Phys. Rev. Lett. 64, 717 (1990). E. Brezin, V.A. Kazakov, Exactly solvable field theories of closed strings. Phys. Lett. B 236, 144 (1990). D. Ghoshal, p-adic string theories provide lattice discretization to the ordinary string worldsheet. Phys. Rev. Lett. 97, 151601 (2006)Google Scholar
  13. 13.
    T. Biswas, M. Grisaru, W. Siegel, Linear regge trajectories from worldsheet lattice parton field theory. Nucl. Phys. B 708, 317 (2005), arXiv:hep-th/0409089
  14. 14.
    W. Siegel, Introduction to string field theory, arXiv:hep-th/0107094
  15. 15.
    W. Siegel, Stringy gravity at short distances, arXiv:hep-th/0309093
  16. 16.
    A.A. Tseytlin, On singularities of spherically symmetric backgrounds in string theory. Phys. Lett. B 363, 223 (1995), arXiv:hep-th/9509050
  17. 17.
    T. Biswas, A. Mazumdar, W. Siegel, Bouncing universes in string-inspired gravity. JCAP 0603, 009 (2006), arXiv:hep-th/0508194
  18. 18.
    E. Tomboulis, Renormalizability and asymptotic freedom in quantum gravity. Phys. Lett. B 97, 77 (1980). E.T. Tomboulis, in Renormalization and Asymptotic Freedom in Quantum Gravity, ed. by S.M. Christensen. Quantum Theory of Gravity, pp. 251–266 and Preprint - TOMBOULIS, E.T. (REC.MAR.83) p. 27. E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, arXiv:hep-th/9702146. E. T. Tomboulis, arXiv:1507.00981 [hep-th]
  19. 19.
    L. Modesto, Super-renormalizable quantum gravity, Phys. Rev. D 86, 044005 (2012), arXiv:1107.2403 [hep-th]
  20. 20.
    A.O. Barvinsky, Y.V. Gusev, New representation of the nonlocal ghost-free gravity theory, arXiv:1209.3062 [hep-th]. A.O. Barvinsky, aspects of nonlocality in quantum field theory, quantum gravity and cosmology, arXiv:1209.3062 [hep-th]
  21. 21.
    J.W. Moffat, Ultraviolet complete quantum gravity. Eur. Phys. J. Plus 126, 43 (2011), arXiv:1008.2482 [gr-qc]
  22. 22.
    K. Krasnov, Renormalizable non-metric quantum gravity? arXiv:hep-th/0611182. K. Krasnov, Non-metric gravity i: field equations. Class. Quant. Grav. 25, 025001 (2008), arXiv:gr-qc/0703002
  23. 23.
    D.A. Eliezer, R.P. Woodard, Nucl. Phys. B 325, 389 (1989)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    K.S. Stelle, Renormalization of higher derivative quantum gravity. Phys. Rev. D 16, 953 (1977)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    T. Biswas, E. Gerwick, T. Koivisto, A. Mazumdar, Towards singularity and ghost free theories of gravity. Phys. Rev. Lett. 108, 031101 (2012), arXiv:1110.5249 [gr-qc]
  26. 26.
    T. Biswas, T. Koivisto, A. Mazumdar, Nonlocal theories of gravity: the flat space propagator, arXiv:1302.0532 [gr-qc]
  27. 27.
    S. Talaganis, T. Biswas, A. Mazumdar, Class. Quant. Grav. 32(21), 215017 (2015). doi: 10.1088/0264-9381/32/21/215017, arXiv:1412.3467 [hep-th]
  28. 28.
    T. Biswas, T. Koivisto, A. Mazumdar, Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity. JCAP 1011, 008 (2010), arXiv:1005.0590 [hep-th]
  29. 29.
    T. Biswas, A.S. Koshelev, A. Mazumdar, S.Y. Vernov, Stable bounce and inflation in non-local higher derivative cosmology. JCAP 1208, 024 (2012), arXiv:1206.6374 [astro-ph.CO]
  30. 30.
    V.P. Frolov, A. Zelnikov, T. de Paula Netto, JHEP 1506, 107 (2015). doi: 10.1007/JHEP06(2015)107, arXiv:1504.00412 [hep-th]. V.P. Frolov, Phys. Rev. Lett. 115(5), 051102 (2015). doi: 10.1103/PhysRevLett.115.051102, arXiv:1505.00492 [hep-th]. V.P. Frolov, A. Zelnikov, arXiv:1509.03336 [hep-th]
  31. 31.
    T. Biswas, A. Conroy, A.S. Koshelev, A. Mazumdar, Class. Quant. Grav. 31, 015022 (2014) [Class. Quant. Grav. 31, 159501 (2014)]. doi: 10.1088/0264-9381/31/1/015022,  10.1088/0264-9381/31/15/159501, arXiv:1308.2319 [hep-th]
  32. 32.
    T. Biswas, S. Talaganis, Mod. Phys. Lett. A 30, no. 03n04, 1540009 (2015). doi: 10.1142/S021773231540009X, arXiv:1412.4256 [gr-qc]
  33. 33.
    T. Biswas, A. Mazumdar, Class. Quant. Grav. 31, 025019 (2014). doi: 10.1088/0264-9381/31/2/025019, arXiv:1304.3648 [hep-th]
  34. 34.
    D. Chialva, A. Mazumdar, Mod. Phys. Lett. A 30, no. 03n04, 1540008 (2015). doi: 10.1142/S0217732315400088, arXiv:1405.0513 [hep-th]
  35. 35.
    B. Craps, T. De Jonckheere, A.S. Koshelev, Cosmological perturbations in non-local higher-derivative gravity, arXiv:1407.4982 [hep-th]
  36. 36.
    A. Conroy, A. Mazumdar, A. Teimouri, Phys. Rev. Lett. 114(20), 201101 (2015). doi: 10.1103/PhysRevLett.114.201101, arXiv:1503.05568 [hep-th]. A. Conroy, A. Mazumdar, S. Talaganis, A. Teimouri, arXiv:1509.01247 [hep-th]
  37. 37.
    T. Biswas, A.S. Koshelev, A. Mazumdar, Analysis of stability of gravitational theories around (anti-)deSitter backgrounds, (in preparation)Google Scholar
  38. 38.
    G. Magnano, L.M. Sokolowski, Phys. Rev. D 50, 5039 (1994). doi: 10.1103/PhysRevD.50.5039, arXiv:gr-qc/9312008
  39. 39.
    R.P. Woodard, Nonlocal Models of Cosmic Acceleration. Found. Phys. (2014) 44(2), 213–233 (2014). doi: 10.1007/s10701-014-9780-6 e-Print: arXiv:1401.0254 [astro-ph.CO]
  40. 40.
    T. Chiba, JCAP 0503, 008 (2005), arXiv:gr-qc/0502070
  41. 41.
    A. Nunez, S. Solganik, Phys. Lett. B 608, 189–193 (2005), arXiv:hep-th/0411102
  42. 42.
    B. Allen, Phys. Rev. D 34, 3670 (1986). doi: 10.1103/PhysRevD.34.3670. P.J. Mora, N.C. Tsamis, R.P. Woodard, J. Math. Phys. 53, 122502 (2012). doi: 10.1063/1.4764882, arXiv:1205.4468 [gr-qc]
  43. 43.
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis, L. Rastelli, Nucl. Phys. B 562, 330 (1999). doi: 10.1016/S0550-3213(99)00524-6, arXiv:hep-th/9902042
  44. 44.
    A.A. Starobinsky, Phys. Lett. B 91, 99 (1980). doi: 10.1016/0370-2693(80)90670-X ADSCrossRefGoogle Scholar
  45. 45.
    T. Biswas, J.A.R. Cembranos, J.I. Kapusta, JHEP 1010, 048 (2010), arXiv:1005.0430 [hep-th]
  46. 46.
    T. Biswas, J. Kapusta, A. Reddy, JHEP 1212, 008 (2012), arXiv:1201.1580 [hep-th]

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Tirthabir Biswas
    • 1
    Email author
  • Alexey S. Koshelev
    • 2
  • Anupam Mazumdar
    • 3
  1. 1.Loyola UniversityNew OrleansUSA
  2. 2.Departamento de Física and Centro de Matemática e AplicaçõesUniversidade da Beira InteriorCovilhãPortugal
  3. 3.Consortium for Fundamental PhysicsLancaster UniversityLancasterUK

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