The Effective Field Theory of nonsingular cosmology

  • Yong Cai
  • Youping Wan
  • Hai-Guang Li
  • Taotao Qiu
  • Yun-Song Piao
Open Access
Regular Article - Theoretical Physics


In this paper, we explore the nonsingular cosmology within the framework of the Effective Field Theory (EFT) of cosmological perturbations. Due to the recently proved no-go theorem, any nonsingular cosmological models based on the cubic Galileon suffer from pathologies. We show how the EFT could help us clarify the origin of the no-go theorem, and offer us solutions to break the no-go. Particularly, we point out that the gradient instability can be removed by using some spatial derivative operators in EFT. Based on the EFT description, we obtain a realistic healthy nonsingular cosmological model, and show the perturbation spectrum can be consistent with the observations.


Spacetime Singularities Classical Theories of Gravity Cosmology of Theories beyond the SM 


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]
    A. Borde and A. Vilenkin, Eternal inflation and the initial singularity, Phys. Rev. Lett. 72 (1994) 3305 [gr-qc/9312022] [INSPIRE].
  2. [2]
    A. Borde, A.H. Guth and A. Vilenkin, Inflationary space-times are incompletein past directions, Phys. Rev. Lett. 90 (2003) 151301 [gr-qc/0110012] [INSPIRE].
  3. [3]
    Y.-S. Piao, B. Feng and X.-m. Zhang, Suppressing CMB quadrupole with a bounce from contracting phase to inflation, Phys. Rev. D 69 (2004) 103520 [hep-th/0310206] [INSPIRE].ADSGoogle Scholar
  4. [4]
    Y.-S. Piao, A possible explanation to low CMB quadrupole, Phys. Rev. D 71 (2005) 087301 [astro-ph/0502343] [INSPIRE].
  5. [5]
    Y.-S. Piao, S. Tsujikawa and X.-m. Zhang, Inflation in string inspired cosmology and suppression of CMB low multipoles, Class. Quant. Grav. 21 (2004) 4455 [hep-th/0312139] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    Z.-G. Liu, Z.-K. Guo and Y.-S. Piao, Obtaining the CMB anomalies with a bounce from the contracting phase to inflation, Phys. Rev. D 88 (2013) 063539 [arXiv:1304.6527] [INSPIRE].ADSGoogle Scholar
  7. [7]
    T. Qiu and Y.-T. Wang, G-Bounce Inflation: Towards Nonsingular Inflation Cosmology with Galileon Field, JHEP 04 (2015) 130 [arXiv:1501.03568] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    Z.-G. Liu, H. Li and Y.-S. Piao, Preinflationary genesis with CMB B-mode polarization, Phys. Rev. D 90 (2014) 083521 [arXiv:1405.1188] [INSPIRE].ADSGoogle Scholar
  9. [9]
    D. Pirtskhalava, L. Santoni, E. Trincherini and P. Uttayarat, Inflation from Minkowski Space, JHEP 12 (2014) 151 [arXiv:1410.0882] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    T. Kobayashi, M. Yamaguchi and J. Yokoyama, Galilean Creation of the Inflationary Universe, JCAP 07 (2015) 017 [arXiv:1504.05710] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    Y.-F. Cai, T. Qiu, Y.-S. Piao, M. Li and X. Zhang, Bouncing universe with quintom matter, JHEP 10 (2007) 071 [arXiv:0704.1090] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    Y.-F. Cai, T.-t. Qiu, R. Brandenberger and X.-m. Zhang, A Nonsingular Cosmology with a Scale-Invariant Spectrum of Cosmological Perturbations from Lee-Wick Theory, Phys. Rev. D 80 (2009) 023511 [arXiv:0810.4677] [INSPIRE].ADSGoogle Scholar
  13. [13]
    P. Creminelli, A. Nicolis and E. Trincherini, Galilean Genesis: An alternative to inflation, JCAP 11 (2010) 021 [arXiv:1007.0027] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    K. Hinterbichler, A. Joyce, J. Khoury and G.E.J. Miller, Dirac-Born-Infeld Genesis: An Improved Violation of the Null Energy Condition, Phys. Rev. Lett. 110 (2013) 241303 [arXiv:1212.3607] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    Y.-F. Cai, Y. Wan and X. Zhang, Cosmology of the Spinor Emergent Universe and Scale-invariant Perturbations, Phys. Lett. B 731 (2014) 217 [arXiv:1312.0740] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    Y.-S. Piao and E. Zhou, Nearly scale invariant spectrum of adiabatic fluctuations may be from a very slowly expanding phase of the universe, Phys. Rev. D 68 (2003) 083515 [hep-th/0308080] [INSPIRE].ADSGoogle Scholar
  17. [17]
    Z.-G. Liu, J. Zhang and Y.-S. Piao, A Galileon Design of Slow Expansion, Phys. Rev. D 84 (2011) 063508 [arXiv:1105.5713] [INSPIRE].ADSGoogle Scholar
  18. [18]
    Y. Cai and Y.-S. Piao, The slow expansion with nonminimal derivative coupling and its conformal dual, JHEP 03 (2016) 134 [arXiv:1601.07031] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    Z.-G. Liu and Y.-S. Piao, A Galileon Design of Slow Expansion: Emergent universe, Phys. Lett. B 718 (2013) 734 [arXiv:1207.2568] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    V.A. Rubakov, The Null Energy Condition and its violation, Phys. Usp. 57 (2014) 128 [arXiv:1401.4024] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    S.M. Carroll, M. Hoffman and M. Trodden, Can the dark energy equation-of-state parameter w be less than −1?, Phys. Rev. D 68 (2003) 023509 [astro-ph/0301273] [INSPIRE].
  22. [22]
    J.M. Cline, S. Jeon and G.D. Moore, The Phantom menaced: Constraints on low-energy effective ghosts, Phys. Rev. D 70 (2004) 043543 [hep-ph/0311312] [INSPIRE].
  23. [23]
    A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    C. Deffayet, X. Gao, D.A. Steer and G. Zahariade, From k-essence to generalised Galileons, Phys. Rev. D 84 (2011) 064039 [arXiv:1103.3260] [INSPIRE].ADSGoogle Scholar
  26. [26]
    J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Healthy theories beyond Horndeski, Phys. Rev. Lett. 114 (2015) 211101 [arXiv:1404.6495] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    T. Qiu, J. Evslin, Y.-F. Cai, M. Li and X. Zhang, Bouncing Galileon Cosmologies, JCAP 10 (2011) 036 [arXiv:1108.0593] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    T. Qiu, X. Gao and E.N. Saridakis, Towards anisotropy-free and nonsingular bounce cosmology with scale-invariant perturbations, Phys. Rev. D 88 (2013) 043525 [arXiv:1303.2372] [INSPIRE].ADSGoogle Scholar
  29. [29]
    Y.-F. Cai, D.A. Easson and R. Brandenberger, Towards a Nonsingular Bouncing Cosmology, JCAP 08 (2012) 020 [arXiv:1206.2382] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    Y. Wan, T. Qiu, F.P. Huang, Y.-F. Cai, H. Li and X. Zhang, Bounce Inflation Cosmology with Standard Model Higgs Boson, JCAP 12 (2015) 019 [arXiv:1509.08772] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    L. Battarra, M. Koehn, J.-L. Lehners and B.A. Ovrut, Cosmological Perturbations Through a Non-Singular Ghost-Condensate/Galileon Bounce, JCAP 07 (2014) 007 [arXiv:1404.5067] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    M. Koehn, J.-L. Lehners and B. Ovrut, Nonsingular bouncing cosmology: Consistency of the effective description, Phys. Rev. D 93 (2016) 103501 [arXiv:1512.03807] [INSPIRE].ADSGoogle Scholar
  33. [33]
    M. Libanov, S. Mironov and V. Rubakov, Generalized Galileons: instabilities of bouncing and Genesis cosmologies and modified Genesis, JCAP 08 (2016) 037 [arXiv:1605.05992] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    R. Kolevatov and S. Mironov, Cosmological bounces and Lorentzian wormholes in Galileon theories with an extra scalar field, Phys. Rev. D 94 (2016) 123516 [arXiv:1607.04099] [INSPIRE].ADSGoogle Scholar
  35. [35]
    T. Kobayashi, Generic instabilities of nonsingular cosmologies in Horndeski theory: A no-go theorem, Phys. Rev. D 94 (2016) 043511 [arXiv:1606.05831] [INSPIRE].ADSGoogle Scholar
  36. [36]
    A. Ijjas and P.J. Steinhardt, Fully stable cosmological solutions with a non-singular classical bounce, Phys. Lett. B 764 (2017) 289 [arXiv:1609.01253] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    A. Ijjas and P.J. Steinhardt, Classically stable nonsingular cosmological bounces, Phys. Rev. Lett. 117 (2016) 121304 [arXiv:1606.08880] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    C. Cheung, P. Creminelli, A.L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    S. Weinberg, Effective Field Theory for Inflation, Phys. Rev. D 77 (2008) 123541 [arXiv:0804.4291] [INSPIRE].ADSMathSciNetGoogle Scholar
  40. [40]
    G. Gubitosi, F. Piazza and F. Vernizzi, The Effective Field Theory of Dark Energy, JCAP 02 (2013) 032 [arXiv:1210.0201] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Essential Building Blocks of Dark Energy, JCAP 08 (2013) 025 [arXiv:1304.4840] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    F. Piazza and F. Vernizzi, Effective Field Theory of Cosmological Perturbations, Class. Quant. Grav. 30 (2013) 214007 [arXiv:1307.4350] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    R. Kase and S. Tsujikawa, Effective field theory approach to modified gravity including Horndeski theory and Hořava-Lifshitz gravity, Int. J. Mod. Phys. D 23 (2015) 1443008 [arXiv:1409.1984] [INSPIRE].ADSzbMATHGoogle Scholar
  44. [44]
    X. Gao, Unifying framework for scalar-tensor theories of gravity, Phys. Rev. D 90 (2014) 081501 [arXiv:1406.0822] [INSPIRE].ADSGoogle Scholar
  45. [45]
    X. Gao, Hamiltonian analysis of spatially covariant gravity, Phys. Rev. D 90 (2014) 104033 [arXiv:1409.6708] [INSPIRE].ADSGoogle Scholar
  46. [46]
    P. Creminelli, D. Pirtskhalava, L. Santoni and E. Trincherini, Stability of Geodesically Complete Cosmologies, JCAP 11 (2016) 047 [arXiv:1610.04207] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    D.A. Easson, I. Sawicki and A. Vikman, G-Bounce, JCAP 11 (2011) 021 [arXiv:1109.1047] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, The ekpyrotic universe: Collidingbranes and the origin of the hot big bang, Phys. Rev. D 64 (2001) 123522 [hep-th/0103239] [INSPIRE].ADSGoogle Scholar
  49. [49]
    J.-L. Lehners, P. McFadden, N. Turok and P.J. Steinhardt, Generating ekpyrotic curvature perturbations before the big bang, Phys. Rev. D 76 (2007) 103501 [hep-th/0702153] [INSPIRE].ADSMathSciNetGoogle Scholar
  50. [50]
    E.I. Buchbinder, J. Khoury and B.A. Ovrut, New ekpyrotic cosmology, Phys. Rev. D 76 (2007) 123503 [hep-th/0702154] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  51. [51]
    H.-G. Li, Y. Cai and Y.-S. Piao, Towards the bounce inflationary gravitational wave, Eur. Phys. J. C 76 (2016) 699 [arXiv:1605.09586] [INSPIRE].CrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.School of PhysicsUniversity of Chinese Academy of SciencesBeijingP.R. China
  2. 2.CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of ChinaChinese Academy of SciencesHefeiP.R. China
  3. 3.Institute of AstrophysicsCentral China Normal UniversityWuhanP.R. China
  4. 4.Key Laboratory of Quark and Lepton Physics (MOE)Central China Normal UniversityWuhanP.R. China
  5. 5.Institute of Theoretical PhysicsChinese Academy of SciencesBeijingP.R. China

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