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Gravitational Theories with Stable (anti-)de Sitter Backgrounds

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At the Frontier of Spacetime

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 183))

Abstract

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.

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Notes

  1. 1.

    Finite higher derivative theories suffer from Ostrogradsky instabilities, see Ref. [23]. However, the Ostrogradsky argument relies on having a highest “momentum” associated with the highest derivative in the theory, in which the energy comes as a linear term, as opposed to quadratic. In a classical theory this would lead to instability and in a quantum theory, this would yield ghosts or extra poles in the propagator. A classic example is Stelle’s 4th derivative theory of gravity [24], which has been argued to be UV finite, but contains massive spin-2 ghost, therefore shows vacuum instabilities.

  2. 2.

    The action of Ref. [25] also provides the UV complete Starobinsky inflation [33–35]. Also, it was noted that the gravitational entropy for this action, for a static spherically symmetric background, gets no contribution from the quadratic curvature part [36].

  3. 3.

    Although, Brans and Dicke formulated their theory by adding a new nonminimally coupled scalar field, as is well known, this scalar degree of freedom can be incorporated within the metric degrees of freedom by replacing \(R\rightarrow F(R)\) in the gravitational action [38]. This is the approach that naturally emerges in our analysis.

References

  1. M.J.G. Veltman, Quantum theory of gravitation. Conf. Proc. C 7507281, 265 (1975)

    MathSciNet  Google Scholar 

  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. B.S. DeWitt, G. Esposito, An introduction to quantum gravity. Int. J. Geom. Meth. Mod. Phys. 5, 101 (2008), arXiv:0711.2445 [hep-th]

    Google Scholar 

  4. J. Polchinski, String Theory: Superstring Theory and Beyond, vol. 2 (Cambridge, UK: Univ. Pr 1998) , p. 531

    Google Scholar 

  5. A. Ashtekar, Introduction to loop quantum gravity and cosmology. Lect. Notes Phys. 863, 31 (2013)

    Article  ADS  MATH  Google Scholar 

  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. 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; Cambridge

    Google Scholar 

  8. E. Witten, Noncommutative geometry and string field theory. Nucl. Phys. B 268, 253 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  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. 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. B. Dragovich, Zeta strings, arXiv:hep-th/0703008

  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. 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

    Google Scholar 

  14. W. Siegel, Introduction to string field theory, arXiv:hep-th/0107094

  15. W. Siegel, Stringy gravity at short distances, arXiv:hep-th/0309093

  16. A.A. Tseytlin, On singularities of spherically symmetric backgrounds in string theory. Phys. Lett. B 363, 223 (1995), arXiv:hep-th/9509050

    Google Scholar 

  17. T. Biswas, A. Mazumdar, W. Siegel, Bouncing universes in string-inspired gravity. JCAP 0603, 009 (2006), arXiv:hep-th/0508194

    Google Scholar 

  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. L. Modesto, Super-renormalizable quantum gravity, Phys. Rev. D 86, 044005 (2012), arXiv:1107.2403 [hep-th]

  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. J.W. Moffat, Ultraviolet complete quantum gravity. Eur. Phys. J. Plus 126, 43 (2011), arXiv:1008.2482 [gr-qc]

  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. D.A. Eliezer, R.P. Woodard, Nucl. Phys. B 325, 389 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  24. K.S. Stelle, Renormalization of higher derivative quantum gravity. Phys. Rev. D 16, 953 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  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. T. Biswas, T. Koivisto, A. Mazumdar, Nonlocal theories of gravity: the flat space propagator, arXiv:1302.0532 [gr-qc]

  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]

    Google Scholar 

  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]

    Google Scholar 

  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]

    Google Scholar 

  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. 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]

    Google Scholar 

  32. T. Biswas, S. Talaganis, Mod. Phys. Lett. A 30, no. 03n04, 1540009 (2015). doi:10.1142/S021773231540009X, arXiv:1412.4256 [gr-qc]

    Google Scholar 

  33. T. Biswas, A. Mazumdar, Class. Quant. Grav. 31, 025019 (2014). doi:10.1088/0264-9381/31/2/025019, arXiv:1304.3648 [hep-th]

    Google Scholar 

  34. D. Chialva, A. Mazumdar, Mod. Phys. Lett. A 30, no. 03n04, 1540008 (2015). doi:10.1142/S0217732315400088, arXiv:1405.0513 [hep-th]

    Google Scholar 

  35. B. Craps, T. De Jonckheere, A.S. Koshelev, Cosmological perturbations in non-local higher-derivative gravity, arXiv:1407.4982 [hep-th]

  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. T. Biswas, A.S. Koshelev, A. Mazumdar, Analysis of stability of gravitational theories around (anti-)deSitter backgrounds, (in preparation)

    Google Scholar 

  38. G. Magnano, L.M. Sokolowski, Phys. Rev. D 50, 5039 (1994). doi:10.1103/PhysRevD.50.5039, arXiv:gr-qc/9312008

    Google Scholar 

  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. T. Chiba, JCAP 0503, 008 (2005), arXiv:gr-qc/0502070

  41. A. Nunez, S. Solganik, Phys. Lett. B 608, 189–193 (2005), arXiv:hep-th/0411102

  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]

    Google Scholar 

  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

    Google Scholar 

  44. A.A. Starobinsky, Phys. Lett. B 91, 99 (1980). doi:10.1016/0370-2693(80)90670-X

    Article  ADS  Google Scholar 

  45. T. Biswas, J.A.R. Cembranos, J.I. Kapusta, JHEP 1010, 048 (2010), arXiv:1005.0430 [hep-th]

  46. T. Biswas, J. Kapusta, A. Reddy, JHEP 1212, 008 (2012), arXiv:1201.1580 [hep-th]

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Acknowledgments

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.

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Authors and Affiliations

  1. Loyola University, 6363 St. Charles Avenue, Box 92, New Orleans, 70118, USA

    Tirthabir Biswas

  2. Departamento de Física and Centro de Matemática e Aplicações, Universidade da Beira Interior, 6200, Covilhã, Portugal

    Alexey S. Koshelev

  3. Consortium for Fundamental Physics, Lancaster University, Lancaster, LA1 4YB, UK

    Anupam Mazumdar

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  1. Tirthabir Biswas
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Correspondence to Tirthabir Biswas .

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  1. PT-SW, DLR Projektträger, Berlin, Germany

    Torsten Asselmeyer-Maluga

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Biswas, T., Koshelev, A.S., Mazumdar, A. (2016). Gravitational Theories with Stable (anti-)de Sitter Backgrounds. In: Asselmeyer-Maluga, T. (eds) At the Frontier of Spacetime. Fundamental Theories of Physics, vol 183. Springer, Cham. https://doi.org/10.1007/978-3-319-31299-6_5

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