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Revisiting non-Gaussianity in multifield inflation with curved field space

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Abstract

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.

A preprint version of the article is available at ArXiv.

References

  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].

  3. [3]

    X. Chen, Primordial non-Gaussianities from inflation models, Adv. Astron.2010 (2010) 638979 [arXiv:1002.1416] [INSPIRE].

  4. [4]

    Y. Wang, Inflation, cosmic perturbations and non-gaussianities, Commun. Theor. Phys.62 (2014) 109 [arXiv:1303.1523] [INSPIRE].

  5. [5]

    S. Renaux-Petel, Primordial non-Gaussianities after Planck 2015: an introductory review, Comptes Rendus Physique16 (2015) 969 [arXiv:1508.06740] [INSPIRE].

  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].

  10. [10]

    A. Achucarro et al., Features of heavy physics in the CMB power spectrum, JCAP01 (2011) 030 [arXiv:1010.3693] [INSPIRE].

  11. [11]

    A. Achucarro et al., Effective theories of single field inflation when heavy fields matter, JHEP05 (2012) 066 [arXiv:1201.6342] [INSPIRE].

  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].

  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].

  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].

  16. [16]

    H. Lee, D. Baumann and G.L. Pimentel, Non-Gaussianity as a particle detector, JHEP12 (2016) 040 [arXiv:1607.03735] [INSPIRE].

  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].

  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].

  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].

  21. [21]

    C. Cheung et al., The effective field theory of inflation, JHEP03 (2008) 014 [arXiv:0709.0293] [INSPIRE].

  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].

  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].

  25. [25]

    P. Christodoulidis, D. Roest and E.I. Sfakianakis, Angular inflation in multi-field α-attractors, JCAP11 (2019) 002 [arXiv:1803.09841] [INSPIRE].

  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].

  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].

  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].

  29. [29]

    T. Bjorkmo, Rapid-turn inflationary attractors, Phys. Rev. Lett.122 (2019) 251301 [arXiv:1902.10529] [INSPIRE].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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.

  64. [64]

    D. Seery and J.E. Lidsey, Primordial non-Gaussianities from multiple-field inflation, JCAP09 (2005) 011 [astro-ph/0506056] [INSPIRE].

  65. [65]

    D. Langlois and S. Renaux-Petel, Perturbations in generalized multi-field inflation, JCAP04 (2008) 017 [arXiv:0801.1085] [INSPIRE].

  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].

  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].

  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].

  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].

  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.

  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.

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

  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].

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Correspondence to Sébastien Renaux-Petel.

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ArXiv ePrint: 1907.10403

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Garcia-Saenz, S., Pinol, L. & Renaux-Petel, S. Revisiting non-Gaussianity in multifield inflation with curved field space. J. High Energ. Phys. 2020, 73 (2020). https://doi.org/10.1007/JHEP01(2020)073

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Keywords

  • Cosmology of Theories beyond the SM
  • Effective Field Theories
  • Sigma Models