Revisiting non-Gaussianity in multifield inflation with curved field space

  • Sebastian Garcia-Saenz
  • Lucas Pinol
  • Sébastien Renaux-PetelEmail author
Open Access
Regular Article - Theoretical Physics


Recent studies of inflation with multiple scalar fields have highlighted the importance of non-canonical kinetic terms in novel types of inflationary solutions. This motivates a thorough analysis of non-Gaussianities in this context, which we revisit here by studying the primordial bispectrum in a general two-field model. Our main result is the complete cubic action for inflationary fluctuations written in comoving gauge, i.e. in terms of the curvature perturbation and the entropic mode. Although full expressions for the cubic action have already been derived in terms of fields fluctuations in the flat gauge, their applicability is mostly restricted to numerical evaluations. Our form of the action is instead amenable to several analytical approximations, as our calculation in terms of the directly observable quantity makes manifest the scaling of every operator in terms of the slow-roll parameters, what is essentially a generalization of Maldacena’s single-field result to non-canonical two-field models. As an important application we derive the single-field effective field theory that is valid when the entropic mode is heavy and may be integrated out, underlining the observable effects that derive from a curved field space.


Cosmology of Theories beyond the SM Effective Field Theories Sigma Models 


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]
    Planck collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, arXiv:1905.05697 [INSPIRE].
  2. [2]
    D. Wands, Local non-Gaussianity from inflation, Class. Quant. Grav.27 (2010) 124002 [arXiv:1004.0818] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    X. Chen, Primordial non-Gaussianities from inflation models, Adv. Astron.2010 (2010) 638979 [arXiv:1002.1416] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    Y. Wang, Inflation, cosmic perturbations and non-gaussianities, Commun. Theor. Phys.62 (2014) 109 [arXiv:1303.1523] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    S. Renaux-Petel, Primordial non-Gaussianities after Planck 2015: an introductory review, Comptes Rendus Physique16 (2015) 969 [arXiv:1508.06740] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    P.D. Meerburg et al., Primordial non-Gaussianity, arXiv:1903.04409 [INSPIRE].
  7. [7]
    X. Chen and Y. Wang, Large non-Gaussianities with intermediate shapes from quasi-single field inflation, Phys. Rev.D 81 (2010) 063511 [arXiv:0909.0496] [INSPIRE].
  8. [8]
    A.J. Tolley and M. Wyman, The Gelaton scenario: equilateral non-Gaussianity from multi-field dynamics, Phys. Rev.D 81 (2010) 043502 [arXiv:0910.1853] [INSPIRE].
  9. [9]
    S. Cremonini, Z. Lalak and K. Turzynski, Strongly coupled perturbations in two-field inflationary models, JCAP03 (2011) 016 [arXiv:1010.3021] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A. Achucarro et al., Features of heavy physics in the CMB power spectrum, JCAP01 (2011) 030 [arXiv:1010.3693] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A. Achucarro et al., Effective theories of single field inflation when heavy fields matter, JHEP05 (2012) 066 [arXiv:1201.6342] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    L. McAllister, S. Renaux-Petel and G. Xu, A statistical approach to multifield inflation: many-field perturbations beyond slow roll, JCAP10 (2012) 046 [arXiv:1207.0317] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    C.P. Burgess, M.W. Horbatsch and S. Patil, Inflating in a trough: single-field effective theory from multiple-field curved valleys, JHEP01 (2013) 133 [arXiv:1209.5701] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    N. Arkani-Hamed and J. Maldacena, Cosmological collider physics, arXiv:1503.08043 [INSPIRE].
  15. [15]
    R. Flauger, M. Mirbabayi, L. Senatore and E. Silverstein, Productive Interactions: heavy particles and non-Gaussianity, JCAP10 (2017) 058 [arXiv:1606.00513] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    H. Lee, D. Baumann and G.L. Pimentel, Non-Gaussianity as a particle detector, JHEP12 (2016) 040 [arXiv:1607.03735] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  17. [17]
    X. Chen, Y. Wang and Z.-Z. Xianyu, Standard model background of the cosmological collider, Phys. Rev. Lett.118 (2017) 261302 [arXiv:1610.06597] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    X. Chen, A. Loeb and Z.-Z. Xianyu, Unique fingerprints of alternatives to inflation in the primordial power spectrum, Phys. Rev. Lett.122 (2019) 121301 [arXiv:1809.02603] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The cosmological bootstrap: inflationary correlators from symmetries and singularities, arXiv:1811.00024 [INSPIRE].
  20. [20]
    P. Creminelli, M.A. Luty, A. Nicolis and L. Senatore, Starting the universe: stable violation of the null energy condition and non-standard cosmologies, JHEP12 (2006) 080 [hep-th/0606090] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    C. Cheung et al., The effective field theory of inflation, JHEP03 (2008) 014 [arXiv:0709.0293] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    S. Renaux-Petel and K. Turzyński, Geometrical destabilization of inflation, Phys. Rev. Lett.117 (2016) 141301 [arXiv:1510.01281] [INSPIRE].
  23. [23]
    A.R. Brown, Hyperbolic inflation, Phys. Rev. Lett.121 (2018) 251601 [arXiv:1705.03023] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S. Mizuno and S. Mukohyama, Primordial perturbations from inflation with a hyperbolic field-space, Phys. Rev.D 96 (2017) 103533 [arXiv:1707.05125] [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    P. Christodoulidis, D. Roest and E.I. Sfakianakis, Angular inflation in multi-field α-attractors, JCAP11 (2019) 002 [arXiv:1803.09841] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    S. Garcia-Saenz, S. Renaux-Petel and J. Ronayne, Primordial fluctuations and non-Gaussianities in sidetracked inflation, JCAP07 (2018) 057 [arXiv:1804.11279] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    T. Bjorkmo and M.C.D. Marsh, Hyperinflation generalised: from its attractor mechanism to its tension with the ‘swampland conditions’, JHEP04 (2019) 172 [arXiv:1901.08603] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    J. Fumagalli et al., Hyper non-Gaussianities in inflation with strongly non-geodesic motion, Phys. Rev. Lett.123 (2019) 201302 [arXiv:1902.03221] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    T. Bjorkmo, Rapid-turn inflationary attractors, Phys. Rev. Lett.122 (2019) 251301 [arXiv:1902.10529] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    P. Christodoulidis, D. Roest and E.I. Sfakianakis, Attractors, bifurcations and curvature in multi-field inflation, arXiv:1903.03513 [INSPIRE].
  31. [31]
    P. Christodoulidis, D. Roest and E.I. Sfakianakis, Scaling attractors in multi-field inflation, JCAP12 (2019) 059 [arXiv:1903.06116] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    V. Aragam, S. Paban and R. Rosati, Multi-field inflation in high-slope potentials, arXiv:1905.07495 [INSPIRE].
  33. [33]
    A. Hetz and G.A. Palma, Sound speed of primordial fluctuations in supergravity inflation, Phys. Rev. Lett.117 (2016) 101301 [arXiv:1601.05457] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    A. Achúcarro, V. Atal, C. Germani and G.A. Palma, Cumulative effects in inflation with ultra-light entropy modes, JCAP02 (2017) 013 [arXiv:1607.08609] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  35. [35]
    S. Renaux-Petel, K. Turzyński and V. Vennin, Geometrical destabilization, premature end of inflation and Bayesian model selection, JCAP11 (2017) 006 [arXiv:1706.01835] [INSPIRE].
  36. [36]
    A. Achúcarro et al., Universality of multi-field α-attractors, JCAP04 (2018) 028 [arXiv:1711.09478] [INSPIRE].
  37. [37]
    A. Linde et al., Hypernatural inflation, JCAP07 (2018) 035 [arXiv:1803.09911] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    X. Chen et al., Landscape tomography through primordial non-Gaussianity, Phys. Rev.D 98 (2018) 083528 [arXiv:1804.07315] [INSPIRE].
  39. [39]
    A. Achúcarro and G.A. Palma, The string swampland constraints require multi-field inflation, JCAP02 (2019) 041 [arXiv:1807.04390] [INSPIRE].
  40. [40]
    A. Achúcarro, S. Céspedes, A.-C. Davis and G.A. Palma, Constraints on holographic multifield inflation and models based on the Hamilton-Jacobi formalism, Phys. Rev. Lett.122 (2019) 191301 [arXiv:1809.05341] [INSPIRE].
  41. [41]
    A. Achúcarro et al., Shift-symmetric orbital inflation: single field or multi-field?, arXiv:1901.03657 [INSPIRE].
  42. [42]
    O. Grocholski et al., On backreaction effects in geometrical destabilisation of inflation, JCAP05 (2019) 008 [arXiv:1901.10468] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    M. Cicoli, V. Guidetti and F.G. Pedro, Geometrical destabilisation of ultra-light axions in string inflation, JCAP05 (2019) 046 [arXiv:1903.01497] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    S. Mizuno, S. Mukohyama, S. Pi and Y.-L. Zhang, Hyperbolic field space and swampland conjecture for DBI scalar, JCAP09 (2019) 072 [arXiv:1905.10950] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    G. Panagopoulos and E. Silverstein, Primordial black holes from non-Gaussian tails, arXiv:1906.02827 [INSPIRE].
  46. [46]
    R. Bravo, G.A. Palma and S. Riquelme, A tip for landscape riders: multi-field inflation can fulfill the swampland distance conjecture, arXiv:1906.05772 [INSPIRE].
  47. [47]
    A. Achúcarro and Y. Welling, Orbital inflation: inflating along an angular isometry of field space, arXiv:1907.02020 [INSPIRE].
  48. [48]
    J. Elliston, D. Seery and R. Tavakol, The inflationary bispectrum with curved field-space, JCAP11 (2012) 060 [arXiv:1208.6011] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    D.J. Mulryne and J.W. Ronayne, PyTransport: a Python package for the calculation of inflationary correlation functions, arXiv:1609.00381 [INSPIRE].
  50. [50]
    M. Dias, J. Frazer, D.J. Mulryne and D. Seery, Numerical evaluation of the bispectrum in multiple field inflation — The transport approach with code, JCAP12 (2016) 033 [arXiv:1609.00379] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    D. Seery, CppTransport: a platform to automate calculation of inflationary correlation functions, arXiv:1609.00380 [INSPIRE].
  52. [52]
    J.W. Ronayne and D.J. Mulryne, Numerically evaluating the bispectrum in curved field-space — with PyTransport 2.0, JCAP01 (2018) 023 [arXiv:1708.07130] [INSPIRE].
  53. [53]
    S. Butchers and D. Seery, Numerical evaluation of inflationary 3-point functions on curved field space — with the transport method & CppTransport, JCAP07 (2018) 031 [arXiv:1803.10563] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP05 (2003) 013 [astro-ph/0210603] [INSPIRE].
  55. [55]
    C. Armendariz-Picon, T. Damour and V.F. Mukhanov, k-inflation, Phys. Lett.B 458 (1999) 209 [hep-th/9904075] [INSPIRE].
  56. [56]
    J. Garriga and V.F. Mukhanov, Perturbations in k-inflation, Phys. Lett.B 458 (1999) 219 [hep-th/9904176] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  57. [57]
    D. Seery and J.E. Lidsey, Primordial non-Gaussianities in single field inflation, JCAP06 (2005) 003 [astro-ph/0503692] [INSPIRE].
  58. [58]
    X. Chen, M.-x. Huang, S. Kachru and G. Shiu, Observational signatures and non-Gaussianities of general single field inflation, JCAP01 (2007) 002 [hep-th/0605045] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    S. Groot Nibbelink and B.J.W. van Tent, Density perturbations arising from multiple field slow roll inflation, hep-ph/0011325 [INSPIRE].
  60. [60]
    S. Groot Nibbelink and B.J.W. van Tent, Scalar perturbations during multiple field slow-roll inflation, Class. Quant. Grav.19 (2002) 613 [hep-ph/0107272] [INSPIRE].
  61. [61]
    J.-O. Gong and T. Tanaka, A covariant approach to general field space metric in multi-field inflation, JCAP03 (2011) 015 [Erratum ibid.02 (2012) E01] [arXiv:1101.4809] [INSPIRE].
  62. [62]
    R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, Gen. Rel. Grav.40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
  63. [63]
    D.S. Salopek, Nonlinear evolution of long-wavelength metric fluctuations in inflationary models, Phys. Rev.D 42 (1990) 3936.ADSMathSciNetGoogle Scholar
  64. [64]
    D. Seery and J.E. Lidsey, Primordial non-Gaussianities from multiple-field inflation, JCAP09 (2005) 011 [astro-ph/0506056] [INSPIRE].CrossRefGoogle Scholar
  65. [65]
    D. Langlois and S. Renaux-Petel, Perturbations in generalized multi-field inflation, JCAP04 (2008) 017 [arXiv:0801.1085] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    D. Langlois, S. Renaux-Petel, D.A. Steer and T. Tanaka, Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation, Phys. Rev.D 78 (2008) 063523 [arXiv:0806.0336] [INSPIRE].
  67. [67]
    E. Tzavara and B. van Tent, Gauge-invariant perturbations at second order in two-field inflation, JCAP08 (2012) 023 [arXiv:1111.5838] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    E. Tzavara, S. Mizuno and B. van Tent, Covariant second-order perturbations in generalized two-field inflation, JCAP07 (2014) 027 [arXiv:1312.6139] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    G.I. Rigopoulos, E.P.S. Shellard and B.J.W. van Tent, Non-linear perturbations in multiple-field inflation, Phys. Rev.D 73 (2006) 083521 [astro-ph/0504508] [INSPIRE].
  70. [70]
    D. Langlois and F. Vernizzi, Nonlinear perturbations of cosmological scalar fields, JCAP02 (2007) 017 [astro-ph/0610064] [INSPIRE].CrossRefGoogle Scholar
  71. [71]
    S. Renaux-Petel and G. Tasinato, Nonlinear perturbations of cosmological scalar fields with non-standard kinetic terms, JCAP01 (2009) 012 [arXiv:0810.2405] [INSPIRE].ADSGoogle Scholar
  72. [72]
    J.-L. Lehners and S. Renaux-Petel, Multifield cosmological perturbations at third order and the Ekpyrotic trispectrum, Phys. Rev.D 80 (2009) 063503 [arXiv:0906.0530] [INSPIRE].
  73. [73]
    H. Collins, Primordial non-Gaussianities from inflation, arXiv:1101.1308 [INSPIRE].
  74. [74]
    R.D. Jordan, Effective field equations for expectation values, Phys. Rev.D 33 (1986) 444.ADSMathSciNetGoogle Scholar
  75. [75]
    E. Calzetta and B.L. Hu, Closed-time-path functional formalism in curved spacetime: Application to cosmological back-reaction problems, Phys. Rev.D 35 (1987) 495.ADSMathSciNetGoogle Scholar
  76. [76]
    S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev.D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
  77. [77]
    F. Arroja and T. Tanaka, A note on the role of the boundary terms for the non-Gaussianity in general k-inflation, JCAP05 (2011) 005 [arXiv:1103.1102] [INSPIRE].ADSCrossRefGoogle Scholar
  78. [78]
    C. Burrage, R.H. Ribeiro and D. Seery, Large slow-roll corrections to the bispectrum of noncanonical inflation, JCAP07 (2011) 032 [arXiv:1103.4126] [INSPIRE].ADSCrossRefGoogle Scholar
  79. [79]
    G. Rigopoulos, Gauge invariance and non-Gaussianity in inflation, Phys. Rev.D 84 (2011) 021301 [arXiv:1104.0292] [INSPIRE].
  80. [80]
    D. Baumann and D. Green, Equilateral non-gaussianity and new physics on the horizon, JCAP09 (2011) 014 [arXiv:1102.5343] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    G. Shiu and J. Xu, Effective field theory and decoupling in multi-field inflation: an illustrative case study, Phys. Rev.D 84 (2011) 103509 [arXiv:1108.0981] [INSPIRE].ADSGoogle Scholar
  82. [82]
    S. Cespedes, V. Atal and G.A. Palma, On the importance of heavy fields during inflation, JCAP05 (2012) 008 [arXiv:1201.4848] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    A. Avgoustidis et al., Decoupling survives inflation: a critical look at effective field theory violations during inflation, JCAP06 (2012) 025 [arXiv:1203.0016] [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    A. Achucarro et al., Heavy fields, reduced speeds of sound and decoupling during inflation, Phys. Rev.D 86 (2012) 121301 [arXiv:1205.0710] [INSPIRE].ADSGoogle Scholar
  85. [85]
    R. Gwyn, G.A. Palma, M. Sakellariadou and S. Sypsas, Effective field theory of weakly coupled inflationary models, JCAP04 (2013) 004 [arXiv:1210.3020] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  86. [86]
    S. Céspedes and G.A. Palma, Cosmic inflation in a landscape of heavy-fields, JCAP10 (2013) 051 [arXiv:1303.4703] [INSPIRE].
  87. [87]
    J.-O. Gong, S. Pi and M. Sasaki, Equilateral non-Gaussianity from heavy fields, JCAP11 (2013) 043 [arXiv:1306.3691] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  88. [88]
    R. Gwyn, G.A. Palma, M. Sakellariadou and S. Sypsas, On degenerate models of cosmic inflation, JCAP10 (2014) 005 [arXiv:1406.1947] [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    J.-O. Gong, M.-S. Seo and S. Sypsas, Higher derivatives and power spectrum in effective single field inflation, JCAP03 (2015) 009 [arXiv:1407.8268] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  90. [90]
    S. Garcia-Saenz and S. Renaux-Petel, Flattened non-Gaussianities from the effective field theory of inflation with imaginary speed of sound, JCAP11 (2018) 005 [arXiv:1805.12563] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  91. [91]
    P. Creminelli et al., Limits on non-Gaussianities from wmap data, JCAP05 (2006) 004 [astro-ph/0509029] [INSPIRE].
  92. [92]
    L. Senatore, K.M. Smith and M. Zaldarriaga, Non-Gaussianities in single field inflation and their optimal limits from the WMAP 5-year data, JCAP01 (2010) 028 [arXiv:0905.3746] [INSPIRE].ADSCrossRefGoogle Scholar
  93. [93]
    S. Renaux-Petel, On the redundancy of operators and the bispectrum in the most general second-order scalar-tensor theory, JCAP02 (2012) 020 [arXiv:1107.5020] [INSPIRE].ADSCrossRefGoogle Scholar
  94. [94]
    A.A. Starobinsky, S. Tsujikawa and J. Yokoyama, Cosmological perturbations from multifield inflation in generalized Einstein theories, Nucl. Phys.B 610 (2001) 383 [astro-ph/0107555] [INSPIRE].
  95. [95]
    F. Di Marco, F. Finelli and R. Brandenberger, Adiabatic and isocurvature perturbations for multifield generalized Einstein models, Phys. Rev.D 67 (2003) 063512 [astro-ph/0211276] [INSPIRE].
  96. [96]
    F. Di Marco and F. Finelli, Slow-roll inflation for generalized two-field Lagrangians, Phys. Rev.D 71 (2005) 123502 [astro-ph/0505198] [INSPIRE].
  97. [97]
    Z. Lalak, D. Langlois, S. Pokorski and K. Turzynski, Curvature and isocurvature perturbations in two-field inflation, JCAP07 (2007) 014 [arXiv:0704.0212] [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    L. Pinol, S. Renaux-Petel and Y. Tada, Inflationary stochastic anomalies, Class. Quant. Grav.36 (2019) 07LT01 [arXiv:1806.10126] [INSPIRE].
  99. [99]
    C. Burrage, C. de Rham, D. Seery and A.J. Tolley, Galileon inflation, JCAP01 (2011) 014 [arXiv:1009.2497] [INSPIRE].ADSCrossRefGoogle Scholar
  100. [100]
    G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Shapes of gravity: tensor non-gaussianity and massive spin-2 fields, JHEP10 (2019) 182 [arXiv:1812.07571] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2020

Authors and Affiliations

  • Sebastian Garcia-Saenz
    • 1
  • Lucas Pinol
    • 1
  • Sébastien Renaux-Petel
    • 1
    Email author
  1. 1.Institut d’Astrophysique de Paris, GReCO, UMR 7095 du CNRS et de Sorbonne UniversitéParisFrance

Personalised recommendations