Journal of High Energy Physics

, 2012:83 | Cite as

f(R) gravities, Killing spinor equations, “BPS” domain walls and cosmology



We derive the condition on f(R) gravities that admit Killing spinor equations and construct explicit such examples. The Killing spinor equations can be used to reduce the fourth-order differential equations of motion to the first order for both the domain wall and FLRW cosmological solutions. We obtain exact “BPS” domain walls that describe the smooth Randall-Sundrum II, AdS wormholes and the RG flow from IR to UV. We also obtain exact smooth cosmological solutions that describe the evolution from an inflationary starting point with a larger cosmological constant to an ever-expanding universe with a smaller cosmological constant. In addition, We find exact smooth solutions of pre-big bang models, bouncing or crunching universes. An important feature is that the scalar curvature R of all these metrics is varying rather than a constant. Another intriguing feature is that there are two different f(R) gravities that give rise to the same “BPS” solution. We also study linearized f(R) gravities in (A)dS vacua.


Classical Theories of Gravity Cosmology of Theories beyond the SM AdS-CFT Correspondence 


  1. [1]
    A.S. Eddington, The mathematical theory of relativity, Cambridge University Press, Cambridge U.K. (1923).MATHGoogle Scholar
  2. [2]
    H. Weyl, A new extension of relativity theory. (in german), Annalen Phys. 59 (1919) 101 [INSPIRE].ADSMATHCrossRefGoogle Scholar
  3. [3]
    P.G. Bergmann, Comments on the scalar tensor theory, Int. J. Theor. Phys. 1 (1968) 25 [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    T.V. Ruzmaikina and A.A. Ruzmaikin, Quadratic corrections to the Lagrangian density of the gravitational field and the singularity, Zh. Eksp. Teor. Fiz. 57 (1969) 680 [Sov. Phys. JETP 30 (1970) 372].Google Scholar
  5. [5]
    B.N. Breizman, V.T. Gurovich and V.P. Sokolov, On the possibility of setting up regular cosmological solutions, Zh. Eksp. Teor. Fiz. 59 (1970) 288 [Sov. Phys. JETP 32 (1971) 155].Google Scholar
  6. [6]
    H.A. Buchdahl, Non-linear Lagrangians and cosmological theory, Mon. Not. R. Astron. Soc. 150 (1970) 1.ADSGoogle Scholar
  7. [7]
    J. O’ Hanlon, Intermediate-range gravity - a generally covariant model, Phys. Rev. Lett. 29 (1972) 137 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    P. Teyssandier and P. Tourrenc, The Cauchy problem for the R + R 2 theories of gravity without torsion, J. Math. Phys. 24 (1983) 2793 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  9. [9]
    C. Brans and R. Dicke, Machs principle and a relativistic theory of gravitation, Phys. Rev. 124 (1961) 925 [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].ADSGoogle Scholar
  11. [11]
    S. Nojiri and S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, eConf C 0602061 (2006) 06 [hep-th/0601213] [INSPIRE].Google Scholar
  12. [12]
    T.P. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  13. [13]
    A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].Google Scholar
  14. [14]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133 ] [hep-th/9711200] [INSPIRE].
  15. [15]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  16. [16]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  17. [17]
    A. de la Cruz-Dombriz, A. Dobado and A. Maroto, Black holes in f(R) theories, Phys. Rev. D 80 (2009) 124011 [Erratum ibid. D 83 (2011) 029903] [arXiv:0907.3872] [INSPIRE].
  18. [18]
    A. Larranaga, A rotating charged black hole solution in f(R) gravity, arXiv:1108.6325 [INSPIRE].
  19. [19]
    J. Cembranos, A. de la Cruz-Dombriz and P. Romero, Kerr-Newman black holes in f(R) theories, arXiv:1109.4519 [INSPIRE].
  20. [20]
    D. Freedman, C. Núñez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE].ADSGoogle Scholar
  21. [21]
    H. Lü, C. Pope and Z.-L. Wang, Pseudo-supersymmetry, consistent sphere reduction and Killing spinors for the bosonic string, Phys. Lett. B 702 (2011) 442 [arXiv:1105.6114] [INSPIRE].ADSGoogle Scholar
  22. [22]
    H. Lü and Z.-L. Wang, Killing spinors for the bosonic string, arXiv:1106.1664 [INSPIRE].
  23. [23]
    H. Liu, H. Lü and Z.-L. Wang, Killing spinors for the bosonic string and the Kaluza-Klein theory with scalar potentials, arXiv:1106.4566 [INSPIRE].
  24. [24]
    R.-G. Cai, Y. Liu and Y.-W. Sun, A Lifshitz black hole in four dimensional R 2 gravity, JHEP 10 (2009) 080 [arXiv:0909.2807] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Analytic Lifshitz black holes in higher dimensions, JHEP 04 (2010) 030 [arXiv:1001.2361] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    S. Nojiri and S.D. Odintsov, Unified cosmic history in modified gravity: from f(R) theory to Lorentz non-invariant models, Phys. Rept. 505 (2011) 59 [arXiv:1011.0544] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  28. [28]
    Y. Zhong, Y.-X. Liu and K. Yang, Tensor perturbations of f(R)-branes, Phys. Lett. B 699 (2011) 398 [arXiv:1010.3478] [INSPIRE].ADSGoogle Scholar
  29. [29]
    Y.-X. Liu, Y. Zhong, Z.-H. Zhao and H.-T. Li, Domain wall brane in squared curvature gravity, JHEP 06 (2011) 135 [arXiv:1104.3188] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    M. Gremm, Four-dimensional gravity on a thick domain wall, Phys. Lett. B 478 (2000) 434 [hep-th/9912060] [INSPIRE].ADSGoogle Scholar
  31. [31]
    G. Dotti, J. Oliva and R. Troncoso, Static wormhole solution for higher-dimensional gravity in vacuum, Phys. Rev. D 75 (2007) 024002 [hep-th/0607062] [INSPIRE].ADSGoogle Scholar
  32. [32]
    G. Dotti, J. Oliva and R. Troncoso, Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions: black holes, wormholes and spacetime horns, Phys. Rev. D 76 (2007) 064038 [arXiv:0706.1830] [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    M. Ali, F. Ruiz, C. Saint-Victor and J.F. Vazquez-Poritz, Strings on AdS wormholes, Phys. Rev. D 80 (2009) 046002 [arXiv:0905.4766] [INSPIRE].MathSciNetADSGoogle Scholar
  34. [34]
    G. Galloway, K. Schleich, D. Witt and E. Woolgar, The AdS/CFT correspondence conjecture and topological censorship, Phys. Lett. B 505 (2001) 255 [hep-th/9912119] [INSPIRE].MathSciNetADSGoogle Scholar
  35. [35]
    H. Lü and J. Mei, Ricci-flat and charged wormholes in five dimensions, Phys. Lett. B 666 (2008) 511 [arXiv:0806.3111] [INSPIRE].ADSGoogle Scholar
  36. [36]
    A. Bergman, H. Lü, J. Mei and C. Pope, AdS wormholes, Nucl. Phys. B 810 (2009) 300 [arXiv:0808.2481] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    H. Lü, J. Mei and Z.-L. Wang, GL(n,R) wormholes and waves in diverse dimensions, Class. Quant. Grav. 26 (2009) 085020 [arXiv:0901.0003] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    Z.-L. Wang and H. Lü, Most general spherically symmetric M2-branes and type IIB strings, Phys. Rev. D 80 (2009) 066008 [arXiv:0906.3439] [INSPIRE].ADSGoogle Scholar
  39. [39]
    D. Freedman, S. Gubser, K. Pilch and N. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].MathSciNetMATHGoogle Scholar
  40. [40]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    J. Grover, J.B. Gutowski, C.A. Herdeiro and W. Sabra, HKT geometry and de Sitter supergravity, Nucl. Phys. B 809 (2009) 406 [arXiv:0806.2626] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    J. Grover, J.B. Gutowski, C.A. Herdeiro, P. Meessen, A. Palomo-Lozano, et al., Gauduchon-Tod structures, Sim holonomy and de Sitter supergravity, JHEP 07 (2009) 069 [arXiv:0905.3047] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    Y.S. Myung, Graviton and scalar propagations on AdS 4 space in f(R) gravities, Eur. Phys. J. C 71 (2011) 1550 [arXiv:1012.2153] [INSPIRE].ADSGoogle Scholar
  44. [44]
    W. Li, W. Song and A. Strominger, Chiral gravity in three dimensions, JHEP 04 (2008) 082 [arXiv:0801.4566] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    H. Lü and C. Pope, Critical gravity in four dimensions, Phys. Rev. Lett. 106 (2011) 181302 [arXiv:1101.1971] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    S. Deser, H. Liu, H. Lü, C. Pope, T.C. Sisman, et al., Critical points of d-dimensional extended gravities, Phys. Rev. D 83 (2011) 061502 [arXiv:1101.4009] [INSPIRE].ADSGoogle Scholar
  47. [47]
    H. Lü, C. Pope and Z.-L. Wang, Pseudo-supergravity extension of the bosonic string, Nucl. Phys. B 854 (2012) 293 [arXiv:1106.5794] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    H.-S. Liu, H. Lü, Z.-L. Wang, H. Lü and Z.-L. Wang, Gauged Kaluza-Klein AdS pseudo-supergravity, Phys. Lett. B 703 (2011) 524 [arXiv:1107.2659] [INSPIRE].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Zheijiang Institute of Modern Physics, Department of PhysicsZhejiang UniversityHangzhouChina
  2. 2.China Economics and Management AcademyCentral University of Finance and EconomicsBeijingChina
  3. 3.Institute for Advanced StudyShenzhen UniversityShenzhenChina
  4. 4.School of PhysicsKorea Institute for Advanced StudySeoulKorea

Personalised recommendations