Holography for inflation using conformal perturbation theory

  • Adam Bzowski
  • Paul McFadden
  • Kostas Skenderis


We provide a precise and quantitative holographic description of a class of inflationary slow-roll models. The dual QFT is a deformation of a three-dimensional CFT by a nearly marginal operator, which, in the models we consider, generates an RG flow to a nearby IR fixed point. These models describe hilltop inflation, where the inflaton rolls from a local maximum of the potential in the infinite past (corresponding to the IR fixed point of the dual QFT) to reach a nearby local minimum in the infinite future (corresponding to the UV of the dual QFT). Through purely holographic means, we compute the spectra and bispectra of scalar and tensor cosmological perturbations. The QFT correlators to which these observables map holographically may be calculated using conformal perturbation theory, even when the dual QFT is strongly coupled. Both the spectra and the bispectra may be expressed this way in terms of CFT correlators that are fixed, up to a few constants, by conformal invariance. The form of slow-roll inflationary correlators is thus determined by the perturbative breaking of the de Sitter isometries away from the fixed point. Setting the constants to their values obtained by AdS/CFT at the fixed point, we find exact agreement with known expressions for the slow-roll power spectra and non-Gaussianities.


Gauge-gravity correspondence Cosmology of Theories beyond the SM AdSCFT Correspondence Conformal and W Symmetry 


  1. [1]
    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].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    I. Antoniadis, P.O. Mazur and E. Mottola, Conformal invariance, dark energy and CMB non-Gaussianity, JCAP 09 (2012) 024 [arXiv:1103.4164] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    K. Hinterbichler and J. Khoury, The pseudo-conformal universe: scale invariance from spontaneous breaking of conformal symmetry, JCAP 04 (2012) 023 [arXiv:1106.1428] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    P. Creminelli, Conformal invariance of scalar perturbations in inflation, Phys. Rev. D 85 (2012) 041302 [arXiv:1108.0874] [INSPIRE].ADSGoogle Scholar
  6. [6]
    K. Hinterbichler, A. Joyce and J. Khoury, Non-linear realizations of conformal symmetry and effective field theory for the pseudo-conformal universe, JCAP 06 (2012) 043 [arXiv:1202.6056] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    P. Creminelli, J. Norena and M. Simonovic, Conformal consistency relations for single-field inflation, JCAP 07 (2012) 052 [arXiv:1203.4595] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    K. Hinterbichler, L. Hui and J. Khoury, Conformal symmetries of adiabatic modes in cosmology, JCAP 08 (2012) 017 [arXiv:1203.6351] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    V. Assassi, D. Baumann and D. Green, On soft limits of inflationary correlation functions, JCAP 11 (2012) 047 [arXiv:1204.4207] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A. Kehagias and A. Riotto, Operator product expansion of inflationary correlators and conformal symmetry of de Sitter, Nucl. Phys. B 864 (2012) 492 [arXiv:1205.1523] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    A. Kehagias and A. Riotto, The four-point correlator in multifield inflation, the operator product expansion and the symmetries of de Sitter, Nucl. Phys. B 868 (2013) 577 [arXiv:1210.1918] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    V. Assassi, D. Baumann and D. Green, Symmetries and loops in inflation, JHEP 02 (2013) 151 [arXiv:1210.7792] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    P. McFadden and K. Skenderis, Holography for cosmology, Phys. Rev. D 81 (2010) 021301 [arXiv:0907.5542] [INSPIRE].MathSciNetADSGoogle Scholar
  14. [14]
    P. McFadden and K. Skenderis, The holographic universe, J. Phys. Conf. Ser. 222 (2010) 012007 [arXiv:1001.2007] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    P. McFadden and K. Skenderis, Observational signatures of holographic models of inflation, arXiv:1010.0244 [INSPIRE].
  16. [16]
    P. McFadden and K. Skenderis, Holographic non-Gaussianity, JCAP 05 (2011) 013 [arXiv:1011.0452] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    P. McFadden and K. Skenderis, Cosmological 3-point correlators from holography, JCAP 06 (2011) 030 [arXiv:1104.3894] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    R. Easther, R. Flauger, P. McFadden and K. Skenderis, Constraining holographic inflation with WMAP, JCAP 09 (2011) 030 [arXiv:1104.2040] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Bzowski, P. McFadden and K. Skenderis, Holographic predictions for cosmological 3-point functions, JHEP 03 (2012) 091 [arXiv:1112.1967] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    M. Dias, Cosmology at the boundary of de Sitter using the dS/QFT correspondence, Phys. Rev. D 84 (2011) 023512 [arXiv:1104.0625] [INSPIRE].ADSGoogle Scholar
  21. [21]
    C. Corianò, L. Delle Rose and M. Serino, Three and four point functions of stress energy tensors in D = 3 for the analysis of cosmological non-Gaussianities, JHEP 12 (2012) 090 [arXiv:1210.0136] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    A. Strominger, Inflation and the dS/CFT correspondence, JHEP 11 (2001) 049 [hep-th/0110087] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    F. Larsen, J.P. van der Schaar and R.G. Leigh, de Sitter holography and the cosmic microwave background, JHEP 04 (2002) 047 [hep-th/0202127] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    E. Halyo, Holographic inflation, JHEP 02 (2004) 062 [hep-th/0203235] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    F. Larsen and R. McNees, Inflation and de Sitter holography, JHEP 07 (2003) 051 [hep-th/0307026] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    J.P. van der Schaar, Inflationary perturbations from deformed CFT, JHEP 01 (2004) 070 [hep-th/0307271] [INSPIRE].CrossRefGoogle Scholar
  27. [27]
    F. Larsen and R. McNees, Holography, diffeomorphisms and scaling violations in the CMB, JHEP 07 (2004) 062 [hep-th/0402050] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    D. Seery and J.E. Lidsey, Non-Gaussian inflationary perturbations from the dS/CFT correspondence, JCAP 06 (2006) 001 [astro-ph/0604209] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    E. Witten, Quantum gravity in de Sitter space, hep-th/0106109 [INSPIRE].
  31. [31]
    J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    D. Harlow and D. Stanford, Operator dictionaries and wave functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
  33. [33]
    X. Dong, B. Horn, S. Matsuura, E. Silverstein and G. Torroba, FRW solutions and holography from uplifted AdS/CFT, Phys. Rev. D 85 (2012) 104035 [arXiv:1108.5732] [INSPIRE].ADSGoogle Scholar
  34. [34]
    D. Anninos, T. Hartman and A. Strominger, Higher spin realization of the dS/CFT correspondence, arXiv:1108.5735 [INSPIRE].
  35. [35]
    D. Anninos, S.A. Hartnoll and D.M. Hofman, Static patch solipsism: conformal symmetry of the de Sitter worldline, Class. Quant. Grav. 29 (2012) 075002 [arXiv:1109.4942] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    T. Hertog and J. Hartle, Holographic no-boundary measure, JHEP 05 (2012) 095 [arXiv:1111.6090] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    J.B. Hartle, S. Hawking and T. Hertog, Accelerated expansion from negative λ, arXiv:1205.3807 [INSPIRE].
  38. [38]
    D. Anninos, F. Denef and D. Harlow, The wave function of Vasilievs universe - A few slices thereof, arXiv:1207.5517 [INSPIRE].
  39. [39]
    J.B. Hartle, S. Hawking and T. Hertog, Inflation with negative λ, arXiv:1207.6653 [INSPIRE].
  40. [40]
    A. Castro and A. Maloney, The wave function of quantum de Sitter, JHEP 11 (2012) 096 [arXiv:1209.5757] [INSPIRE].MathSciNetADSGoogle Scholar
  41. [41]
    D. Marolf, I.A. Morrison and M. Srednicki, Perturbative S-matrix for massive scalar fields in global de Sitter space, arXiv:1209.6039 [INSPIRE].
  42. [42]
    A. Ludwig and J.L. Cardy, Perturbative evaluation of the conformal anomaly at new critical points with applications to random systems, Nucl. Phys. B 285 (1987) 687 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    A. Zamolodchikov, Renormalization group and perturbation theory near fixed points in two-dimensional field theory, Sov. J. Nucl. Phys. 46 (1987) 1090 [INSPIRE].MathSciNetGoogle Scholar
  44. [44]
    J. Cardy, Conformal invariance and statistical mechanics, Les Houches lectures (1988).Google Scholar
  45. [45]
    A. Zamolodchikov, Exact solutions of conformal field theory in two-dimensions and critical phenomena, Rev. Math. Phys. 1 (1990) 197.MathSciNetCrossRefGoogle Scholar
  46. [46]
    A. Zamolodchikov, Two point correlation function in scaling Lee-Yang model, Nucl. Phys. B 348 (1991) 619 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  47. [47]
    R. Guida and N. Magnoli, All order IR finite expansion for short distance behavior of massless theories perturbed by a relevant operator, Nucl. Phys. B 471 (1996) 361 [hep-th/9511209] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  48. [48]
    WMAP collaboration, E. Komatsu et al., Seven-year Wilkinson microwave anisotropy probe (WMAP) observations: cosmological interpretation, Astrophys. J. Suppl. 192 (2011) 18 [arXiv:1001.4538] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    K. Schalm, G. Shiu and T. van der Aalst, Consistency condition for inflation from (broken) conformal symmetry, JCAP JCAP03 (2013) 005 [arXiv:1211.2157] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    A.M. Polyakov, Gauge fields and strings, Contemp. Concepts Phys. 3 (1987) 1.MathSciNetADSGoogle Scholar
  51. [51]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    H. Osborn and A. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  53. [53]
    J.L. Cardy, Anisotropic corrections to correlation functions in finite size systems, Nucl. Phys. B 290 (1987) 355 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  54. [54]
    M. Cvetič and H.H. Soleng, Naked singularities in dilatonic domain wall space times, Phys. Rev. D 51 (1995) 5768 [hep-th/9411170] [INSPIRE].ADSGoogle Scholar
  55. [55]
    K. Skenderis and P.K. Townsend, Hidden supersymmetry of domain walls and cosmologies, Phys. Rev. Lett. 96 (2006) 191301 [hep-th/0602260] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    D. Salopek and J. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D 42 (1990) 3936 [INSPIRE].MathSciNetADSGoogle Scholar
  57. [57]
    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
  58. [58]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  59. [59]
    E.D. Stewart and D.H. Lyth, A more accurate analytic calculation of the spectrum of cosmological perturbations produced during inflation, Phys. Lett. B 302 (1993) 171 [gr-qc/9302019] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    J.-O. Gong and E.D. Stewart, The density perturbation power spectrum to second order corrections in the slow roll expansion, Phys. Lett. B 510 (2001) 1 [astro-ph/0101225] [INSPIRE].ADSGoogle Scholar
  61. [61]
    D.Z. Freedman, K. Johnson and J.I. Latorre, Differential regularization and renormalization: a new method of calculation in quantum field theory, Nucl. Phys. B 371 (1992) 353 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  62. [62]
    Y. Brychkov, A. Prudnikov, and O. Marichev, Tables of indefinite integrals, vol. 2, Gordon & Breach Science, New York U.S.A. (1989).Google Scholar
  63. [63]
    E. Boos and A.I. Davydychev, A method of evaluation of vertex type Feynman integrals, Moscow Univ. Phys. Bull. 42N3 (1987) 6.MathSciNetGoogle Scholar
  64. [64]
    C. Anastasiou, E.N. Glover and C. Oleari, Scalar one loop integrals using the negative dimension approach, Nucl. Phys. B 572 (2000) 307 [hep-ph/9907494] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    H. Exton, On the system of partial differential equations associated with Appells function F 4, J. Phys. A 28 (1995) 631.MathSciNetADSGoogle Scholar
  66. [66]
    R. Alkofer, M.Q. Huber and K. Schwenzer, Infrared behavior of three-point functions in Landau gauge Yang-Mills theory, Eur. Phys. J. C 62 (2009) 761 [arXiv:0812.4045] [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    A. Erdélyi, Bateman manuscript project, higher transcendental functions, McGraw-Hill, New York, U.S.A. (1953).Google Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Adam Bzowski
    • 1
    • 3
  • Paul McFadden
    • 2
  • Kostas Skenderis
    • 1
    • 3
    • 4
  1. 1.School of MathematicsUniversity of SouthamptonSouthamptonU.K
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.Korteweg-de Vries Institute for MathematicsAmsterdamThe Netherlands
  4. 4.Institute for Theoretical PhysicsAmsterdamThe Netherlands

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