Journal of High Energy Physics

, 2010:31 | Cite as

Bi-galileon theory I: motivation and formulation

  • Antonio Padilla
  • Paul M. Saffin
  • Shuang-Yong Zhou
Article

Abstract

We introduce bi-galileon theory, the generalisation of the single galileon model introduced by Nicolis et al. The theory contains two coupled scalar fields and is described by a Lagrangian that is invariant under Galilean shifts in those fields. This paper is the first of two, and focuses on the motivation and formulation of the theory. We show that the boundary effective theory of the cascading cosmology model corresponds to a bi-galileon theory in the decoupling limit, and argue that this is to be expected for co-dimension 2 braneworld models exhibiting infra-red modification of gravity. We then generalise this, by constructing the most general bi-galileon Lagrangian. By coupling one of the galileons to the energy-momentum tensor, we pitch this as a modified gravity theory in which the modifications to General Relativity are encoded in the dynamics of the two galileons. We initiate a study of phenomenology by looking at maximally symmetric vacua and their stability, developing elegant geometric techniques that trivially explain why some of the vacua have to be unstable in certain cases (eg DGP). A detailed study of phenomenology appears in our companion paper.

Keywords

Large Extra Dimensions Cosmology of Theories beyond the SM Classical Theories of Gravity 

References

  1. [1]
    U.J. Le Verrier, Theorie du mouvement de Mercure, Annales de l’ Observatoir lmpirial de Paris 76 (1859) 195.Google Scholar
  2. [2]
    A. Einstein, The field equations of gravitation, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1915 (1915) 844 [SPIRES].Google Scholar
  3. [3]
    A. Einstein, Explanation of the perihelion motion of Mercury from the general theory of relativity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1915 (1915) 831 [SPIRES].Google Scholar
  4. [4]
    A. Einstein, On the general theory of relativity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1915 (1915) 778 [Addendum ibid. 1915 (1915) 799] [SPIRES].Google Scholar
  5. [5]
    V.C. Rubin, N. Thonnard and W.K. Ford, Jr., Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605/R = 4kpc/to UGC 2885/R = 122kpc/, Astrophys. J. 238 (1980) 471 [SPIRES].CrossRefADSGoogle Scholar
  6. [6]
    Supernova Search Team collaboration, A.G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009 [astro-ph/9805201] [SPIRES].CrossRefADSGoogle Scholar
  7. [7]
    Supernova Search Team collaboration, A.G. Riess et al., Type Ia supernova discoveries at z > 1 from the hubble space telescope: evidence for past deceleration and constraints on dark energy evolution, Astrophys. J. 607 (2004) 665 [astro-ph/0402512] [SPIRES].CrossRefADSGoogle Scholar
  8. [8]
    Supernova Cosmology Project collaboration, S. Perlmutter et al., Measurements of Ω and Λ from 42 high-redshift supernovae, Astrophys. J. 517 (1999) 565 [astro-ph/9812133] [SPIRES].CrossRefADSGoogle Scholar
  9. [9]
    WMAP collaboration, D.N. Spergel et al., Wilkinson Microwave Anisotropy Probe (WMAP) three year results: Implications for cosmology, Astrophys. J. Suppl. 170 (2007) 377 [astro-ph/0603449] [SPIRES].CrossRefADSGoogle Scholar
  10. [10]
    G. Bertone, D. Hooper and J. Silk, Particle dark matter: evidence, candidates and constraints, Phys. Rept. 405 (2005) 279 [hep-ph/0404175] [SPIRES].CrossRefADSGoogle Scholar
  11. [11]
    E.J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15 (2006) 1753 [hep-th/0603057] [SPIRES].ADSMathSciNetGoogle Scholar
  12. [12]
    D. Clowe et al., A direct empirical proof of the existence of dark matter, Astrophys. J. 648 (2006) L109 [astro-ph/0608407] [SPIRES].CrossRefADSGoogle Scholar
  13. [13]
    M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211 [SPIRES].ADSMathSciNetGoogle Scholar
  14. [14]
    H. van Dam and M.J.G. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B 22 (1970) 397 [SPIRES].ADSGoogle Scholar
  15. [15]
    V.I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett. 12 (1970) 312 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 447] [SPIRES].ADSGoogle Scholar
  16. [16]
    A.I. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. B 39 (1972) 393 [SPIRES].ADSGoogle Scholar
  17. [17]
    D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D6 (1972) 3368 [SPIRES].ADSGoogle Scholar
  18. [18]
    C. Deffayet and J.-W. Rombouts, Ghosts, strong coupling and accidental symmetries in massive gravity, Phys. Rev. D 72 (2005) 044003 [gr-qc/0505134] [SPIRES].ADSGoogle Scholar
  19. [19]
    N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429 (1998) 263 [hep-ph/9803315] [SPIRES].ADSGoogle Scholar
  20. [20]
    N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, Phenomenology, astrophysics and cosmology of theories with sub-millimeter dimensions and TeV scale quantum gravity, Phys. Rev. D 59 (1999) 086004 [hep-ph/9807344] [SPIRES].ADSGoogle Scholar
  21. [21]
    I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV, Phys. Lett. B 436 (1998) 257 [hep-ph/9804398] [SPIRES].ADSGoogle Scholar
  22. [22]
    L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  23. [23]
    L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  24. [24]
    G.R. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [SPIRES].ADSMathSciNetGoogle Scholar
  25. [25]
    I.I. Kogan, S. Mouslopoulos, A. Papazoglou, G.G. Ross and J. Santiago, A three three-brane universe: New phenomenology for the new millennium?, Nucl. Phys. B 584 (2000) 313 [hep-ph/9912552] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    I.I. Kogan and G.G. Ross, Brane universe and multigravity: modification of gravity at large and small distances, Phys. Lett. B 485 (2000) 255 [hep-th/0003074] [SPIRES].ADSMathSciNetGoogle Scholar
  27. [27]
    R. Gregory, V.A. Rubakov and S.M. Sibiryakov, Opening up extra dimensions at ultra-large scales, Phys. Rev. Lett. 84 (2000) 5928 [hep-th/0002072] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  28. [28]
    R. Gregory, V.A. Rubakov and S.M. Sibiryakov, Gravity and antigravity in a brane world with metastable gravitons, Phys. Lett. B 489 (2000) 203 [hep-th/0003045] [SPIRES].ADSGoogle Scholar
  29. [29]
    A. Padilla, Ghost-free braneworld bigravity, Class. Quant. Grav. 21 (2004) 2899 [hep-th/0402079] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  30. [30]
    A. Padilla, Cosmic acceleration from asymmetric branes, Class. Quant. Grav. 22 (2005) 681 [hep-th/0406157] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  31. [31]
    A. Padilla, Infra-red modification of gravity from asymmetric branes, Class. Quant. Grav. 22 (2005) 1087 [hep-th/0410033] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  32. [32]
    C. Charmousis, R. Gregory and A. Padilla, Stealth acceleration and modified gravity, JCAP 10 (2007) 006 [arXiv:0706.0857] [SPIRES].ADSMathSciNetGoogle Scholar
  33. [33]
    K. Koyama, A. Padilla and F.P. Silva, Ghosts in asymmetric brane gravity and the decoupled stealth limit, JHEP 03 (2009) 134 [arXiv:0901.0713] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  34. [34]
    C. Deffayet, Cosmology on a brane in Minkowski bulk, Phys. Lett. B 502 (2001) 199 [hep-th/0010186] [SPIRES].ADSGoogle Scholar
  35. [35]
    C. Deffayet, G.R. Dvali and G. Gabadadze, Accelerated universe from gravity leaking to extra dimensions, Phys. Rev. D 65 (2002) 044023 [astro-ph/0105068] [SPIRES].ADSMathSciNetGoogle Scholar
  36. [36]
    C. Charmousis, R. Gregory, N. Kaloper and A. Padilla, DGP specteroscopy, JHEP 10 (2006) 066 [hep-th/0604086] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    R. Gregory, N. Kaloper, R.C. Myers and A. Padilla, A new perspective on DGP gravity, JHEP 10 (2007) 069 [arXiv:0707.2666] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  38. [38]
    A. Padilla, A short review of ’DGP Specteroscopy’, J. Phys. A 40 (2007) 6827 [hep-th/0610093] [SPIRES].ADSGoogle Scholar
  39. [39]
    K. Koyama, Are there ghosts in the self-accelerating brane universe?, Phys. Rev. D 72 (2005) 123511 [hep-th/0503191] [SPIRES].ADSGoogle Scholar
  40. [40]
    D. Gorbunov, K. Koyama and S. Sibiryakov, More on ghosts in DGP model, Phys. Rev. D 73 (2006) 044016 [hep-th/0512097] [SPIRES].ADSGoogle Scholar
  41. [41]
    M.A. Luty, M. Porrati and R. Rattazzi, Strong interactions and stability in the DGP model, JHEP 09 (2003) 029 [hep-th/0303116] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  42. [42]
    C. Deffayet, G.R. Dvali, G. Gabadadze and A.I. Vainshtein, Nonperturbative continuity in graviton mass versus perturbative discontinuity, Phys. Rev. D 65 (2002) 044026 [hep-th/0106001] [SPIRES].ADSGoogle Scholar
  43. [43]
    A. Nicolis and R. Rattazzi, Classical and quantum consistency of the DGP model, JHEP 06 (2004) 059 [hep-th/0404159] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  44. [44]
    G. Dvali, Predictive power of strong coupling in theories with large distance modified gravity, New J. Phys. 8 (2006) 326 [hep-th/0610013] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  45. [45]
    A. Nicolis, R. Rattazzi and E. Trincherini, The galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [SPIRES].ADSMathSciNetGoogle Scholar
  46. [46]
    D.B. Fairlie, J. Govaerts and A. Morozov, Universal field equations with covariant solutions D.B. Fairlie, Nucl. Phys. B 373 (1992) 214 [hep-th/9110022] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  47. [47]
    D.B. Fairlie and J. Govaerts, Euler hierarchies and universal equations, J. Math. Phys. 33 (1992) 3543 [hep-th/9204074] [SPIRES].CrossRefMATHADSMathSciNetGoogle Scholar
  48. [48]
    C. Burrage and D. Seery, Revisiting fifth forces in the Galileon model, JCA P 08 (2010) 011 [arXiv:1005.1927] [SPIRES].ADSGoogle Scholar
  49. [49]
    C. Deffayet, G. Esposito-Farese and A. Vikman, Covariant Galileon, Phys. Rev. D 79 (2009) 084003 [arXiv:0901.1314] [SPIRES].ADSGoogle Scholar
  50. [50]
    C. de Rham and A.J. Tolley, DBI and the Galileon reunited, JCAP 05 (2010) 015 [arXiv:1003.5917] [SPIRES].Google Scholar
  51. [51]
    C. de Rham et al., Cascading gravity: extending the Dvali-Gabadadze-Porrati model to higher dimension, Phys. Rev. Lett. 100 (2008) 251603 [arXiv:0711.2072] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  52. [52]
    C. de Rham, S. Hofmann, J. Khoury and A.J. Tolley, Cascading gravity and degravitation, JCAP 02 (2008) 011 [arXiv:0712.2821] [SPIRES].Google Scholar
  53. [53]
    C. de Rham, An introduction to cascading gravity and degravitation, Can. J. Phys. 87 (2009) 201 [arXiv:0810.0269] [SPIRES].CrossRefADSGoogle Scholar
  54. [54]
    C. de Rham, J. Khoury and A.J. Tolley, Flat 3-brane with tension in cascading gravity, Phys. Rev. Lett. 103 (2009) 161601 [arXiv:0907.0473] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  55. [55]
    M. Minamitsuji, Self-accelerating solutions in cascading DGP braneworld, Phys. Lett. B 684 (2010) 92 [arXiv:0806.2390] [SPIRES].ADSMathSciNetGoogle Scholar
  56. [56]
    N. Agarwal, R. Bean, J. Khoury and M. Trodden, Cascading cosmology, Phys. Rev. D 81 (2010) 084020 [arXiv:0912.3798] [SPIRES].ADSGoogle Scholar
  57. [57]
    O. Corradini, K. Koyama and G. Tasinato, Induced gravity on intersecting brane-worlds Part I: Maximally symmetric solutions, Phys. Rev. D 77 (2008) 084006 [arXiv:0712.0385] [SPIRES].ADSMathSciNetGoogle Scholar
  58. [58]
    O. Corradini, K. Koyama and G. Tasinato, Induced gravity on intersecting brane-worlds Part II: Cosmology, Phys. Rev. D 78 (2008) 124002 [arXiv:0803.1850] [SPIRES].ADSMathSciNetGoogle Scholar
  59. [59]
    J.M. Cline, J. Descheneau, M. Giovannini and J. Vinet, Cosmology of codimension-two braneworlds, JHEP 06 (2003) 048 [hep-th/0304147] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  60. [60]
    J. Vinet and J.M. Cline, Can codimension-two branes solve the cosmological constant problem?, Phys. Rev. D 70 (2004) 083514 [hep-th/0406141] [SPIRES].ADSGoogle Scholar
  61. [61]
    Y. Aghababaie, C.P. Burgess, S.L. Parameswaran and F. Quevedo, Towards a naturally small cosmological constant from branes in 6D supergravity, Nucl. Phys. B 680 (2004) 389 [hep-th/0304256] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  62. [62]
    C.P. Burgess, Supersymmetric large extra dimensions and the cosmological constant: An update, Ann. Phys. 313 (2004) 283 [hep-th/0402200] [SPIRES].MathSciNetGoogle Scholar
  63. [63]
    C.P. Burgess, Towards a natural theory of dark energy: Supersymmetric large extra dimensions, AIP Conf. Proc. 743 (2005) 417 [hep-th/0411140] [SPIRES].CrossRefADSGoogle Scholar
  64. [64]
    N. Kaloper and D. Kiley, Charting the landscape of modified gravity, JHEP 05 (2007) 045 [hep-th/0703190] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  65. [65]
    N. Kaloper, Brane induced gravity: Codimension-2, Mod. Phys. Lett. A 23 (2008) 781 [arXiv:0711.3210] [SPIRES].ADSMathSciNetGoogle Scholar
  66. [66]
    C. Charmousis, G. Kofinas and A. Papazoglou, The consistency of codimension-2 braneworlds and their cosmology, JCA P 01 (2010) 022 [arXiv:0907.1640] [SPIRES].ADSGoogle Scholar
  67. [67]
    C. Charmousis and A. Papazoglou, Properties of codimension-2 braneworlds in six-dimensional Lovelock theory, J. Phys. Conf. Ser. 189 (2009) 012007 [arXiv:0902.2174] [SPIRES].CrossRefADSGoogle Scholar
  68. [68]
    C. Charmousis and A. Papazoglou, Self-properties of codimension-2 braneworlds, JHEP 07 (2008) 062 [arXiv:0804.2121] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  69. [69]
    C. Charmousis and R. Zegers, Einstein gravity on an even codimension brane, Phys. Rev. D 72 (2005) 064005 [hep-th/0502171] [SPIRES].ADSMathSciNetGoogle Scholar
  70. [70]
    C. Charmousis and R. Zegers, Matching conditions for a brane of arbitrary codimension, JHEP 08 (2005) 075 [hep-th/0502170] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  71. [71]
    E. Papantonopoulos, A. Papazoglou and V. Zamarias, Induced cosmology on a regularized brane in six-dimensional flux compactification, Nucl. Phys. B 797 (2008) 520 [arXiv:0707.1396] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  72. [72]
    E. Papantonopoulos, A. Papazoglou and V. Zamarias, Regularization of conical singularities in warped six-dimensional compactifications, JHEP 03 (2007) 002 [hep-th/0611311] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  73. [73]
    B. Cuadros-Melgar, E. Papantonopoulos, M. Tsoukalas and V. Zamarias, Black holes on thin 3-branes of codimension-2 and their extension into the bulk, Nucl. Phys. B 810 (2009) 246 [arXiv:0804.4459] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  74. [74]
    B. Cuadros-Melgar, E. Papantonopoulos, M. Tsoukalas and V. Zamarias, BTZ like-string on codimension-2 braneworlds in the thin brane limit, Phys. Rev. Lett. 100 (2008) 221601 [arXiv:0712.3232] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  75. [75]
    N. Arkani-Hamed, S. Dimopoulos, G. Dvali and G. Gabadadze, Non-local modification of gravity and the cosmological constant problem, hep-th/0209227 [SPIRES].
  76. [76]
    G. Dvali, S. Hofmann and J. Khoury, Degravitation of the cosmological constant and graviton width, Phys. Rev. D 76 (2007) 084006 [hep-th/0703027] [SPIRES].ADSMathSciNetGoogle Scholar
  77. [77]
    S.L. Dubovsky and V.A. Rubakov, Brane-induced gravity in more than one extra dimensions: Violation of equivalence principle and ghost, Phys. Rev. D 67 (2003) 104014 [hep-th/0212222] [SPIRES].ADSGoogle Scholar
  78. [78]
    G. Gabadadze and M. Shifman, Softly massive gravity, Phys. Rev. D 69 (2004) 124032 [hep-th/0312289] [SPIRES].ADSGoogle Scholar
  79. [79]
    A. Padilla, P.M. Saffin and S.-Y. Zhou, Bi-galileon theory II: phenomenology, arXiv:1008.3312 [SPIRES].
  80. [80]
    C. Deffayet, S. Deser and G. Esposito-Farese, Arbitrary p-form Galileons, Phys. Rev. D 82 (2010) 061501 [arXiv:1007.5278] [SPIRES].ADSGoogle Scholar
  81. [81]
    C. Deffayet, S. Deser and G. Esposito-Farese, Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors, Phys. Rev. D 80 (2009) 064015 [arXiv:0906.1967] [SPIRES].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  • Antonio Padilla
    • 1
  • Paul M. Saffin
    • 1
  • Shuang-Yong Zhou
    • 1
  1. 1.School of Physics and AstronomyUniversity of NottinghamNottinghamU.K.

Personalised recommendations