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The open effective field theory of inflation

  • Regular Article - Theoretical Physics
  • Open access
  • Published: 31 October 2024
  • Volume 2024, article number 248 (2024)
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Journal of High Energy Physics Aims and scope Submit manuscript
The open effective field theory of inflation
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  • Santiago Agüí Salcedo  ORCID: orcid.org/0009-0003-5390-63671,
  • Thomas Colas  ORCID: orcid.org/0000-0003-3913-80341 &
  • Enrico Pajer  ORCID: orcid.org/0000-0002-7921-44791 
  • 3886 Accesses

  • 40 Citations

  • 18 Altmetric

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A preprint version of the article is available at arXiv.

Abstract

In our quest to understand the generation of cosmological perturbations, we face two serious obstacles: we do not have direct information about the environment experienced by primordial perturbations during inflation, and our observables are practically limited to correlators of massless fields, heavier fields and derivatives decaying exponentially in the number of e-foldings. The flexible and general framework of open systems has been developed precisely to face similar challenges. Building on previous work, we develop a Schwinger-Keldysh path integral description for an open effective field theory of inflation, describing the possibly dissipative and non-unitary evolution of the Goldstone boson of time translations interacting with an unspecified environment, under the key assumption of locality in space and time. Working in the decoupling limit, we study the linear and interacting theory in de Sitter and derive predictions for the power spectrum and bispectrum that depend on a finite number of effective couplings organised in a derivative expansion. The smoking gun of interactions with the environment is an enhanced but finite bispectrum close to the folded kinematical limit. We demonstrate the generality of our approach by matching our open effective theory to an explicit model. Our construction provides a standard model to simultaneously study phenomenological predictions as well as quantum information aspects of the inflationary dynamics.

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References

  1. A.A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30 (1979) 682 [INSPIRE].

  2. BICEP/Keck collaboration, The Latest Constraints on Inflationary B-modes from the BICEP/Keck Telescopes, in the proceedings of the 56th Rencontres de Moriond on Cosmology, La Thuile, Italy, January 23–30 (2022) [arXiv:2203.16556] [INSPIRE].

  3. C. Cheung et al., The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  4. B. Finelli, G. Goon, E. Pajer and L. Santoni, The Effective Theory of Shift-Symmetric Cosmologies, JCAP 05 (2018) 060 [arXiv:1802.01580] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  5. N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].

  6. X. Tong, Y. Wang and Y. Zhu, Cutting rule for cosmological collider signals: a bulk evolution perspective, JHEP 03 (2022) 181 [arXiv:2112.03448] [INSPIRE].

    Article  ADS  Google Scholar 

  7. H. Lee, D. Baumann and G.L. Pimentel, Non-Gaussianity as a Particle Detector, JHEP 12 (2016) 040 [arXiv:1607.03735] [INSPIRE].

    Article  ADS  Google Scholar 

  8. X. Chen, Y. Wang and Z.-Z. Xianyu, Neutrino Signatures in Primordial Non-Gaussianities, JHEP 09 (2018) 022 [arXiv:1805.02656] [INSPIRE].

    Article  ADS  Google Scholar 

  9. L.-T. Wang and Z.-Z. Xianyu, In Search of Large Signals at the Cosmological Collider, JHEP 02 (2020) 044 [arXiv:1910.12876] [INSPIRE].

    Article  ADS  Google Scholar 

  10. A. Bodas, S. Kumar and R. Sundrum, The Scalar Chemical Potential in Cosmological Collider Physics, JHEP 02 (2021) 079 [arXiv:2010.04727] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  11. X. Tong and Z.-Z. Xianyu, Large spin-2 signals at the cosmological collider, JHEP 10 (2022) 194 [arXiv:2203.06349] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  12. C.M. Sou, X. Tong and Y. Wang, Chemical-potential-assisted particle production in FRW spacetimes, JHEP 06 (2021) 129 [arXiv:2104.08772] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  13. C. McCulloch, E. Pajer and X. Tong, A cosmological tachyon collider: enhancing the long-short scale coupling, JHEP 05 (2024) 262 [arXiv:2401.11009] [INSPIRE].

    Article  ADS  Google Scholar 

  14. A. Berera and L.-Z. Fang, Thermally induced density perturbations in the inflation era, Phys. Rev. Lett. 74 (1995) 1912 [astro-ph/9501024] [INSPIRE].

  15. A. Berera, Warm inflation, Phys. Rev. Lett. 75 (1995) 3218 [astro-ph/9509049] [INSPIRE].

  16. A. Berera, The Warm Inflation Story, Universe 9 (2023) 272 [arXiv:2305.10879] [INSPIRE].

    Article  ADS  Google Scholar 

  17. M.M. Anber and L. Sorbo, Naturally inflating on steep potentials through electromagnetic dissipation, Phys. Rev. D 81 (2010) 043534 [arXiv:0908.4089] [INSPIRE].

    Article  ADS  Google Scholar 

  18. D. Lopez Nacir, R.A. Porto, L. Senatore and M. Zaldarriaga, Dissipative effects in the Effective Field Theory of Inflation, JHEP 01 (2012) 075 [arXiv:1109.4192] [INSPIRE].

    Article  Google Scholar 

  19. M. Crossley, P. Glorioso and H. Liu, Effective field theory of dissipative fluids, JHEP 09 (2017) 095 [arXiv:1511.03646] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  20. P. Glorioso, M. Crossley and H. Liu, Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current, JHEP 09 (2017) 096 [arXiv:1701.07817] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  21. H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics, PoS TASI2017 (2018) 008 [arXiv:1805.09331] [INSPIRE].

  22. M. Hongo, S. Kim, T. Noumi and A. Ota, Effective field theory of time-translational symmetry breaking in nonequilibrium open system, JHEP 02 (2019) 131 [arXiv:1805.06240] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  23. M. Hongo, S. Kim, T. Noumi and A. Ota, Effective Lagrangian for Nambu-Goldstone modes in nonequilibrium open systems, Phys. Rev. D 103 (2021) 056020 [arXiv:1907.08609] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  24. Y. Donath and E. Pajer, The in-out formalism for in-in correlators, JHEP 07 (2024) 064 [arXiv:2402.05999] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  25. N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [INSPIRE].

  26. C.P. Burgess, R. Holman and G. Tasinato, Open EFTs, IR effects & late-time resummations: systematic corrections in stochastic inflation, JHEP 01 (2016) 153 [arXiv:1512.00169] [INSPIRE].

  27. C.P. Burgess et al., Cosmic purity lost: perturbative and resummed late-time inflationary decoherence, JCAP 08 (2024) 042 [arXiv:2403.12240] [INSPIRE].

    Article  Google Scholar 

  28. A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].

  29. A.A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the De Sitter background, Phys. Rev. D 50 (1994) 6357 [astro-ph/9407016] [INSPIRE].

  30. V. Gorbenko and L. Senatore, λϕ4 in dS, arXiv:1911.00022 [INSPIRE].

  31. S. Céspedes, A.-C. Davis and D.-G. Wang, On the IR divergences in de Sitter space: loops, resummation and the semi-classical wavefunction, JHEP 04 (2024) 004 [arXiv:2311.17990] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  32. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2007) [https://doi.org/10.1093/acprof:oso/9780199213900.001.0001] [INSPIRE].

  33. R.P. Feynman and F.L. Vernon Jr., The theory of a general quantum system interacting with a linear dissipative system, Annals Phys. 24 (1963) 118 [INSPIRE].

  34. P. Glorioso and H. Liu, The second law of thermodynamics from symmetry and unitarity, arXiv:1612.07705 [INSPIRE].

  35. C.O. Akyuz, G. Goon and R. Penco, The Schwinger-Keldysh coset construction, JHEP 06 (2024) 004 [arXiv:2306.17232] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  36. A. Berera, I.G. Moss and R.O. Ramos, Warm Inflation and its Microphysical Basis, Rept. Prog. Phys. 72 (2009) 026901 [arXiv:0808.1855] [INSPIRE].

    Article  ADS  Google Scholar 

  37. G. Ballesteros, A. Pérez Rodríguez and M. Pierre, Monomial warm inflation revisited, JCAP 03 (2024) 003 [arXiv:2304.05978] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  38. G. Montefalcone, V. Aragam, L. Visinelli and K. Freese, WarmSPy: a numerical study of cosmological perturbations in warm inflation, JCAP 01 (2024) 032 [arXiv:2306.16190] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  39. X. Chen, Y. Wang and Z.-Z. Xianyu, Schwinger-Keldysh Diagrammatics for Primordial Perturbations, JCAP 12 (2017) 006 [arXiv:1703.10166] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  40. D. Green and R.A. Porto, Signals of a Quantum Universe, Phys. Rev. Lett. 124 (2020) 251302 [arXiv:2001.09149] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  41. R. Holman and A.J. Tolley, Enhanced Non-Gaussianity from Excited Initial States, JCAP 05 (2008) 001 [arXiv:0710.1302] [INSPIRE].

    ADS  Google Scholar 

  42. X. Chen, M.-x. Huang, S. Kachru and G. Shiu, Observational signatures and non-Gaussianities of general single field inflation, JCAP 01 (2007) 002 [hep-th/0605045].

  43. P.D. Meerburg, J.P. van der Schaar and P.S. Corasaniti, Signatures of Initial State Modifications on Bispectrum Statistics, JCAP 05 (2009) 018 [arXiv:0901.4044] [INSPIRE].

    Article  ADS  Google Scholar 

  44. I. Agullo and L. Parker, Non-gaussianities and the Stimulated creation of quanta in the inflationary universe, Phys. Rev. D 83 (2011) 063526 [arXiv:1010.5766] [INSPIRE].

    Article  ADS  Google Scholar 

  45. N. Agarwal, R. Holman, A.J. Tolley and J. Lin, Effective field theory and non-Gaussianity from general inflationary states, JHEP 05 (2013) 085 [arXiv:1212.1172] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  46. A. Albrecht, N. Bolis and R. Holman, Cosmological Consequences of Initial State Entanglement, JHEP 11 (2014) 093 [arXiv:1408.6859] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  47. J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].

  48. P. Creminelli and M. Zaldarriaga, Single field consistency relation for the 3-point function, JCAP 10 (2004) 006 [astro-ph/0407059] [INSPIRE].

  49. C. Cheung, A.L. Fitzpatrick, J. Kaplan and L. Senatore, On the consistency relation of the 3-point function in single field inflation, JCAP 02 (2008) 021 [arXiv:0709.0295] [INSPIRE].

    Article  ADS  Google Scholar 

  50. P. Creminelli, J. Noreña and M. Simonović, Conformal consistency relations for single-field inflation, JCAP 07 (2012) 052 [arXiv:1203.4595] [INSPIRE].

    Article  ADS  Google Scholar 

  51. K. Hinterbichler, L. Hui and J. Khoury, Conformal Symmetries of Adiabatic Modes in Cosmology, JCAP 08 (2012) 017 [arXiv:1203.6351] [INSPIRE].

    Article  ADS  Google Scholar 

  52. V. Assassi, D. Baumann and D. Green, On Soft Limits of Inflationary Correlation Functions, JCAP 11 (2012) 047 [arXiv:1204.4207] [INSPIRE].

    Article  ADS  Google Scholar 

  53. E. Pajer and S. Jazayeri, Systematics of Adiabatic Modes: Flat Universes, JCAP 03 (2018) 013 [arXiv:1710.02177] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  54. G. Avis, S. Jazayeri, E. Pajer and J. Supeł, Spatial Curvature at the Sound Horizon, JCAP 02 (2020) 034 [arXiv:1911.04454] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  55. P. Creminelli, S. Kumar, B. Salehian and L. Santoni, Dissipative inflation via scalar production, JCAP 08 (2023) 076 [arXiv:2305.07695] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  56. D. Anninos, T. Anous, D.Z. Freedman and G. Konstantinidis, Late-time Structure of the Bunch-Davies De Sitter Wavefunction, JCAP 11 (2015) 048 [arXiv:1406.5490] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  57. N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, JHEP 04 (2020) 105 [arXiv:1811.00024] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  58. E. Pajer, D. Stefanyszyn and J. Supeł, The Boostless Bootstrap: Amplitudes without Lorentz boosts, JHEP 12 (2020) 198 [Erratum ibid. 04 (2022) 023] [arXiv:2007.00027] [INSPIRE].

  59. H. Goodhew, S. Jazayeri and E. Pajer, The Cosmological Optical Theorem, JCAP 04 (2021) 021 [arXiv:2009.02898] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  60. S. Céspedes, A.-C. Davis and S. Melville, On the time evolution of cosmological correlators, JHEP 02 (2021) 012 [arXiv:2009.07874] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  61. H. Goodhew, S. Jazayeri, M.H.G. Lee and E. Pajer, Cutting cosmological correlators, JCAP 08 (2021) 003 [arXiv:2104.06587] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  62. S. Melville and E. Pajer, Cosmological Cutting Rules, JHEP 05 (2021) 249 [arXiv:2103.09832] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  63. S.A. Salcedo, M.H.G. Lee, S. Melville and E. Pajer, The Analytic Wavefunction, JHEP 06 (2023) 020 [arXiv:2212.08009] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  64. D. Baumann et al., Linking the singularities of cosmological correlators, JHEP 09 (2022) 010 [arXiv:2106.05294] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  65. S. Albayrak, P. Benincasa and C. Duaso Pueyo, Perturbative unitarity and the wavefunction of the Universe, SciPost Phys. 16 (2024) 157 [arXiv:2305.19686] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  66. D. Stefanyszyn, X. Tong and Y. Zhu, Cosmological correlators through the looking glass: reality, parity, and factorisation, JHEP 05 (2024) 196 [arXiv:2309.07769] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  67. C. Sleight and M. Taronna, From dS to AdS and back, JHEP 12 (2021) 074 [arXiv:2109.02725] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  68. M. Celoria, P. Creminelli, G. Tambalo and V. Yingcharoenrat, Beyond perturbation theory in inflation, JCAP 06 (2021) 051 [arXiv:2103.09244] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  69. P. Creminelli, S. Renaux-Petel, G. Tambalo and V. Yingcharoenrat, Non-perturbative wavefunction of the universe in inflation with (resonant) features, JHEP 03 (2024) 010 [arXiv:2401.10212] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  70. T. Chakraborty et al., The Hilbert space of de Sitter quantum gravity, JHEP 01 (2024) 132 [arXiv:2303.16315] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  71. D. Baumann et al., Snowmass White Paper: The Cosmological Bootstrap, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2203.08121] [INSPIRE].

  72. P. Benincasa, Amplitudes meet Cosmology: A (Scalar) Primer, arXiv:2203.15330 [https://doi.org/10.1142/S0217751X22300101] [INSPIRE].

  73. D. Boyanovsky, Effective Field Theory out of Equilibrium: Brownian quantum fields, New J. Phys. 17 (2015) 063017 [arXiv:1503.00156] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  74. J. Oppenheim and Z. Weller-Davies, Covariant path integrals for quantum fields back-reacting on classical space-time, arXiv:2302.07283 [INSPIRE].

  75. A. Kamenev, Field Theory of Non-Equilibrium Systems, Cambridge University Press (2011) [https://doi.org/10.1017/cbo9781139003667].

  76. A.A. Radovskaya and A.G. Semenov, Semiclassical approximation meets Keldysh-Schwinger diagrammatic technique: scalar φ4, Eur. Phys. J. C 81 (2021) 704 [arXiv:2003.06395] [INSPIRE].

    Article  ADS  Google Scholar 

  77. T. Colas, J. Grain and V. Vennin, Benchmarking the cosmological master equations, Eur. Phys. J. C 82 (2022) 1085 [arXiv:2209.01929] [INSPIRE].

    Article  ADS  Google Scholar 

  78. J.-T. Hsiang and B.-L. Hu, Fluctuation-dissipation relation for a quantum Brownian oscillator in a parametrically squeezed thermal field, Annals Phys. 433 (2021) 168594 [arXiv:2107.13343] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  79. P. Creminelli, M.A. Luty, A. Nicolis and L. Senatore, Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies, JHEP 12 (2006) 080 [hep-th/0606090] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  80. H.-P. Breuer, E.-M. Laine, J. Piilo and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys. 88 (2016) 021002 [INSPIRE].

  81. S. Prudhoe and S. Shandera, Classifying the non-time-local and entangling dynamics of an open qubit system, JHEP 02 (2023) 007 [arXiv:2201.07080] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  82. S. Shandera, N. Agarwal and A. Kamal, Open quantum cosmological system, Phys. Rev. D 98 (2018) 083535 [arXiv:1708.00493] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  83. T. Colas, J. Grain and V. Vennin, Quantum recoherence in the early universe, EPL 142 (2023) 69002 [arXiv:2212.09486] [INSPIRE].

    Article  ADS  Google Scholar 

  84. T. Colas, C. de Rham and G. Kaplanek, Decoherence out of fire: purity loss in expanding and contracting universes, JCAP 05 (2024) 025 [arXiv:2401.02832] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  85. M. Braglia and L. Pinol, No time to derive: unraveling total time derivatives in in-in perturbation theory, JHEP 08 (2024) 068 [arXiv:2403.14558] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  86. H. Collins, R. Holman and A. Ross, Effective field theory in time-dependent settings, JHEP 02 (2013) 108 [arXiv:1208.3255] [INSPIRE].

    Article  ADS  Google Scholar 

  87. L. Senatore and M. Zaldarriaga, On Loops in Inflation, JHEP 12 (2010) 008 [arXiv:0912.2734] [INSPIRE].

    Article  ADS  Google Scholar 

  88. M.H.G. Lee, C. McCulloch and E. Pajer, Leading loops in cosmological correlators, JHEP 11 (2023) 038 [arXiv:2305.11228] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  89. E.A. Calzetta and B.-L.B. Hu, Nonequilibrium Quantum Field Theory, Cambridge University Press (2008) [https://doi.org/10.1017/cbo9780511535123].

  90. A. Kamenev and A. Levchenko, Keldysh technique and nonlinear sigma-model: Basic principles and applications, Adv. Phys. 58 (2009) 197 [arXiv:0901.3586] [INSPIRE].

    Article  ADS  Google Scholar 

  91. D. Babich, P. Creminelli and M. Zaldarriaga, The shape of non-Gaussianities, JCAP 08 (2004) 009 [astro-ph/0405356] [INSPIRE].

  92. Planck collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, Astron. Astrophys. 641 (2020) A9 [arXiv:1905.05697] [INSPIRE].

  93. SPHEREx collaboration, Cosmology with the SPHEREX All-Sky Spectral Survey, arXiv:1412.4872 [INSPIRE].

  94. G. Cabass et al., Constraining single-field inflation with MegaMapper, Phys. Lett. B 841 (2023) 137912 [arXiv:2211.14899] [INSPIRE].

    Article  Google Scholar 

  95. W. Sohn, J.R. Fergusson and E.P.S. Shellard, High-resolution CMB bispectrum estimator with flexible modal bases, Phys. Rev. D 108 (2023) 063504 [arXiv:2305.14646] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  96. T. Colas, Open Effective Field Theories for primordial cosmology: dissipation, decoherence and late-time resummation of cosmological inhomogeneities, Ph.D. thesis, Institut d’astrophysique spatiale, France, AstroParticule et Cosmologie, APC, Paris, France (2023) [INSPIRE].

  97. R.L. Stratonovich, On a Method of Calculating Quantum Distribution Functions, Sov. Phys. Dokl. 2 (1957) 416.

  98. J. Hubbard, Calculation of partition functions, Phys. Rev. Lett. 3 (1959) 77 [INSPIRE].

  99. Planck collaboration, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641 (2020) A10 [arXiv:1807.06211] [INSPIRE].

  100. G. D’Amico et al., The Cosmological Analysis of the SDSS/BOSS data from the Effective Field Theory of Large-Scale Structure, JCAP 05 (2020) 005 [arXiv:1909.05271] [INSPIRE].

    Article  ADS  Google Scholar 

  101. T. Colas et al., Efficient Cosmological Analysis of the SDSS/BOSS data from the Effective Field Theory of Large-Scale Structure, JCAP 06 (2020) 001 [arXiv:1909.07951] [INSPIRE].

    ADS  Google Scholar 

  102. C.P. Burgess and G. Kaplanek, Gravity, Horizons and Open EFTs, arXiv:2212.09157 [https://doi.org/10.1007/978-981-19-3079-9_7-1] [INSPIRE].

  103. A. Ota, Fluctuation-dissipation relation in cosmic microwave background, JCAP 05 (2024) 062 [arXiv:2402.07623] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  104. R. Kubo, The fluctuation-dissipation theorem, Rept. Prog. Phys. 29 (1966) 255 [INSPIRE].

  105. P. Adshead, C.P. Burgess, R. Holman and S. Shandera, Power-counting during single-field slow-roll inflation, JCAP 02 (2018) 016 [arXiv:1708.07443] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  106. T. Grall and S. Melville, Inflation in motion: unitarity constraints in effective field theories with (spontaneously) broken Lorentz symmetry, JCAP 09 (2020) 017 [arXiv:2005.02366] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  107. D. Green, K. Gupta and Y. Huang, A Goldstone boson equivalence for inflation, JHEP 09 (2024) 117 [arXiv:2403.05274] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  108. R. Flauger et al., Oscillations in the CMB from Axion Monodromy Inflation, JCAP 06 (2010) 009 [arXiv:0907.2916] [INSPIRE].

    Article  ADS  Google Scholar 

  109. R. Flauger and E. Pajer, Resonant Non-Gaussianity, JCAP 01 (2011) 017 [arXiv:1002.0833] [INSPIRE].

    Article  ADS  Google Scholar 

  110. S.R. Behbahani, A. Dymarsky, M. Mirbabayi and L. Senatore, (Small) Resonant non-Gaussianities: Signatures of a Discrete Shift Symmetry in the Effective Field Theory of Inflation, JCAP 12 (2012) 036 [arXiv:1111.3373] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  111. C. Duaso Pueyo and E. Pajer, A cosmological bootstrap for resonant non-Gaussianity, JHEP 03 (2024) 098 [arXiv:2311.01395] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  112. A. Matsumura, Reduced dynamics with Poincaré symmetry in an open quantum system, Phys. Rev. A 108 (2023) 042217 [arXiv:2301.01451] [INSPIRE].

    Article  ADS  Google Scholar 

  113. K. Kashiwagi and A. Matsumura, Markovian quantum master equation with Poincaré symmetry, Phys. Rev. A 109 (2024) 052214 [arXiv:2312.04069] [INSPIRE].

    Article  ADS  Google Scholar 

  114. A.I. Lotkov et al., Conformal symmetry in quasifree Markovian open quantum systems, Phys. Rev. B 108 (2023) 064312 [arXiv:2305.01629] [INSPIRE].

    Article  ADS  Google Scholar 

  115. T. Liu, X. Tong, Y. Wang and Z.-Z. Xianyu, Probing P and CP Violations on the Cosmological Collider, JHEP 04 (2020) 189 [arXiv:1909.01819] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  116. G. Cabass, S. Jazayeri, E. Pajer and D. Stefanyszyn, Parity violation in the scalar trispectrum: no-go theorems and yes-go examples, JHEP 02 (2023) 021 [arXiv:2210.02907] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  117. M. Peloso and L. Sorbo, Instability in axion inflation with strong backreaction from gauge modes, JCAP 01 (2023) 038 [arXiv:2209.08131] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  118. S. Agüí Salcedo and S. Melville, The cosmological tree theorem, JHEP 12 (2023) 076 [arXiv:2308.00680] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  119. M. Dias, J. Frazer, D.J. Mulryne and D. Seery, Numerical evaluation of the bispectrum in multiple field inflation — the transport approach with code, JCAP 12 (2016) 033 [arXiv:1609.00379] [INSPIRE].

    Article  ADS  Google Scholar 

  120. J.W. Ronayne and D.J. Mulryne, Numerically evaluating the bispectrum in curved field-space— with PyTransport 2.0, JCAP 01 (2018) 023 [arXiv:1708.07130] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  121. D. Werth, L. Pinol and S. Renaux-Petel, Cosmological Flow of Primordial Correlators, Phys. Rev. Lett. 133 (2024) 141002 [arXiv:2302.00655] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  122. L. Pinol, S. Renaux-Petel and D. Werth, The Cosmological Flow: A Systematic Approach to Primordial Correlators, arXiv:2312.06559 [INSPIRE].

  123. D. Werth, L. Pinol and S. Renaux-Petel, CosmoFlow: Python Package for Cosmological Correlators, Class. Quant. Grav. 41 (2024) 175015 [arXiv:2402.03693] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  124. J. Oppenheim, C. Sparaciari, B. Šoda and Z. Weller-Davies, Objective trajectories in hybrid classical-quantum dynamics, Quantum 7 (2023) 891 [arXiv:2011.06009] [INSPIRE].

    Article  Google Scholar 

  125. J. Oppenheim, C. Sparaciari, B. Šoda and Z. Weller-Davies, Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity, Nature Commun. 14 (2023) 7910 [arXiv:2203.01982] [INSPIRE].

    Article  ADS  Google Scholar 

  126. J. Oppenheim, C. Sparaciari, B. Šoda and Z. Weller-Davies, The two classes of hybrid classical-quantum dynamics, arXiv:2203.01332 [INSPIRE].

  127. I. Layton and J. Oppenheim, The Classical-Quantum Limit, PRX Quantum 5 (2024) 020331 [arXiv:2310.18271] [INSPIRE].

    Article  ADS  Google Scholar 

  128. J. Oppenheim, A. Russo and Z. Weller-Davies, Diffeomorphism invariant classical-quantum path integrals for Nordström gravity, Phys. Rev. D 110 (2024) 024007 [arXiv:2401.05514] [INSPIRE].

    Article  Google Scholar 

  129. A. Grudka et al., Renormalisation of postquantum-classical gravity, arXiv:2402.17844 [INSPIRE].

  130. L. Parker, Particle creation in expanding universes, Phys. Rev. Lett. 21 (1968) 562 [INSPIRE].

  131. L. Parker, Quantized fields and particle creation in expanding universes. 1, Phys. Rev. 183 (1969) 1057 [INSPIRE].

  132. Y.B. Zeldovich and A.A. Starobinsky, Particle production and vacuum polarization in an anisotropic gravitational field, Zh. Eksp. Teor. Fiz. 61 (1971) 2161 [INSPIRE].

  133. N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge, U.K. (1982) [https://doi.org/10.1017/CBO9780511622632] [INSPIRE].

  134. J. Yokoyama and A.D. Linde, Is warm inflation possible?, Phys. Rev. D 60 (1999) 083509 [hep-ph/9809409] [INSPIRE].

  135. M. Bastero-Gil, A. Berera and R.O. Ramos, Shear viscous effects on the primordial power spectrum from warm inflation, JCAP 07 (2011) 030 [arXiv:1106.0701] [INSPIRE].

    Article  ADS  Google Scholar 

  136. S.P. de Alwis, Cosmological fluctuations: Comparing Quantum and Classical Statistical and Stringy Effects, arXiv:1504.05211 [INSPIRE].

  137. M. Mirbabayi and A. Gruzinov, Shapes of non-Gaussianity in warm inflation, JCAP 02 (2023) 012 [arXiv:2205.13227] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  138. K.V. Berghaus, M. Forslund and M.V. Guevarra, Minimal warm inflation with a heavy QCD axion, arXiv:2402.13535 [INSPIRE].

  139. A. Tinwala, A. Narang, S. Mohanty and S. Panda, Open EFT treatment of Inflation with Thermal Initial Conditions, arXiv:2402.18494 [INSPIRE].

  140. W. Cheng et al., Exploring the impact of the dissipation coefficient in warm Higgs inflation, Phys. Rev. D 109 (2024) 083509 [arXiv:2401.11794] [INSPIRE].

    Article  ADS  Google Scholar 

  141. D. Boyanovsky et al., Dissipation via particle production in scalar field theories, Phys. Rev. D 51 (1995) 4419 [hep-ph/9408214] [INSPIRE].

  142. D. Boyanovsky, D. Cormier, H.J. de Vega and R. Holman, Out-of-equilibrium dynamics of an inflationary phase transition, Phys. Rev. D 55 (1997) 3373 [hep-ph/9610396] [INSPIRE].

  143. D. Boyanovsky, R. Holman and S.P. Kumar, Inflaton decay in De Sitter space-time, Phys. Rev. D 56 (1997) 1958 [hep-ph/9606208] [INSPIRE].

  144. D. Green, B. Horn, L. Senatore and E. Silverstein, Trapped Inflation, Phys. Rev. D 80 (2009) 063533 [arXiv:0902.1006] [INSPIRE].

    Article  ADS  Google Scholar 

  145. N. Barnaby, E. Pajer and M. Peloso, Gauge Field Production in Axion Inflation: Consequences for Monodromy, non-Gaussianity in the CMB, and Gravitational Waves at Interferometers, Phys. Rev. D 85 (2012) 023525 [arXiv:1110.3327] [INSPIRE].

    Article  ADS  Google Scholar 

  146. M. Putti, N. Bartolo, S. Bhattacharya and M. Peloso, CMB spectral distortions from enhanced primordial perturbations: the role of spectator axions, JCAP 08 (2024) 016 [arXiv:2403.08594] [INSPIRE].

    Article  Google Scholar 

  147. V. Briaud et al., Revisiting the stochastic QCD axion window: departure from equilibrium during inflation, JCAP 05 (2024) 085 [arXiv:2312.08231] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  148. Y. Nambu and M. Sasaki, Stochastic Stage of an Inflationary Universe Model, Phys. Lett. B 205 (1988) 441 [INSPIRE].

  149. Y. Nambu and M. Sasaki, Stochastic approach to chaotic inflation and the distribution of universes, Phys. Lett. B 219 (1989) 240 [INSPIRE].

  150. H.E. Kandrup, Stochastic inflation as a time dependent random walk, Phys. Rev. D 39 (1989) 2245 [INSPIRE].

  151. K.-I. Nakao, Y. Nambu and M. Sasaki, Stochastic Dynamics of New Inflation, Prog. Theor. Phys. 80 (1988) 1041 [INSPIRE].

  152. Y. Nambu, Stochastic Dynamics of an Inflationary Model and Initial Distribution of Universes, Prog. Theor. Phys. 81 (1989) 1037 [INSPIRE].

  153. D.S. Salopek and J.R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D 42 (1990) 3936 [INSPIRE].

  154. S. Mollerach, S. Matarrese, A. Ortolan and F. Lucchin, Stochastic inflation in a simple two field model, Phys. Rev. D 44 (1991) 1670 [INSPIRE].

  155. R.H. Brandenberger, V.F. Mukhanov and T. Prokopec, Entropy of a classical stochastic field and cosmological perturbations, Phys. Rev. Lett. 69 (1992) 3606 [astro-ph/9206005] [INSPIRE].

  156. F. Finelli et al., Generation of fluctuations during inflation: Comparison of stochastic and field-theoretic approaches, Phys. Rev. D 79 (2009) 044007 [arXiv:0808.1786] [INSPIRE].

    Article  ADS  Google Scholar 

  157. B. Garbrecht, G. Rigopoulos and Y. Zhu, Infrared correlations in de Sitter space: Field theoretic versus stochastic approach, Phys. Rev. D 89 (2014) 063506 [arXiv:1310.0367] [INSPIRE].

    Article  ADS  Google Scholar 

  158. T. Fujita, M. Kawasaki and Y. Tada, Non-perturbative approach for curvature perturbations in stochastic δN formalism, JCAP 10 (2014) 030 [arXiv:1405.2187] [INSPIRE].

    Article  ADS  Google Scholar 

  159. V. Vennin and A.A. Starobinsky, Correlation Functions in Stochastic Inflation, Eur. Phys. J. C 75 (2015) 413 [arXiv:1506.04732] [INSPIRE].

    Article  ADS  Google Scholar 

  160. J. Grain and V. Vennin, Stochastic inflation in phase space: Is slow roll a stochastic attractor?, JCAP 05 (2017) 045 [arXiv:1703.00447] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  161. L. Pinol, S. Renaux-Petel and Y. Tada, A manifestly covariant theory of multifield stochastic inflation in phase space: solving the discretisation ambiguity in stochastic inflation, JCAP 04 (2021) 048 [arXiv:2008.07497] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  162. Y.L. Launay, G.I. Rigopoulos and E.P.S. Shellard, Stochastic inflation in general relativity, Phys. Rev. D 109 (2024) 123523 [arXiv:2401.08530] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  163. D. Koks, A. Matacz and B.L. Hu, Entropy and uncertainty of squeezed quantum open systems, Phys. Rev. D 55 (1997) 5917 [Erratum ibid. 56 (1997) 5281] [quant-ph/9612016] [INSPIRE].

  164. C.P. Burgess, R. Holman and D. Hoover, Decoherence of inflationary primordial fluctuations, Phys. Rev. D 77 (2008) 063534 [astro-ph/0601646] [INSPIRE].

  165. C. Anastopoulos and B.L. Hu, A Master Equation for Gravitational Decoherence: Probing the Textures of Spacetime, Class. Quant. Grav. 30 (2013) 165007 [arXiv:1305.5231] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  166. M. Fukuma, Y. Sakatani and S. Sugishita, Master equation for the Unruh-DeWitt detector and the universal relaxation time in de Sitter space, Phys. Rev. D 89 (2014) 064024 [arXiv:1305.0256] [INSPIRE].

    Article  ADS  Google Scholar 

  167. C.P. Burgess, R. Holman, G. Tasinato and M. Williams, EFT Beyond the Horizon: Stochastic Inflation and How Primordial Quantum Fluctuations Go Classical, JHEP 03 (2015) 090 [arXiv:1408.5002] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  168. D. Boyanovsky, Effective field theory during inflation. II. Stochastic dynamics and power spectrum suppression, Phys. Rev. D 93 (2016) 043501 [arXiv:1511.06649] [INSPIRE].

  169. D. Boyanovsky, Effective field theory during inflation: Reduced density matrix and its quantum master equation, Phys. Rev. D 92 (2015) 023527 [arXiv:1506.07395] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  170. E. Nelson, Quantum Decoherence During Inflation from Gravitational Nonlinearities, JCAP 03 (2016) 022 [arXiv:1601.03734] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  171. T.J. Hollowood and J.I. McDonald, Decoherence, discord and the quantum master equation for cosmological perturbations, Phys. Rev. D 95 (2017) 103521 [arXiv:1701.02235] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  172. D. Boyanovsky, Information loss in effective field theory: entanglement and thermal entropies, Phys. Rev. D 97 (2018) 065008 [arXiv:1801.06840] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  173. D. Boyanovsky, Imprint of entanglement entropy in the power spectrum of inflationary fluctuations, Phys. Rev. D 98 (2018) 023515 [arXiv:1804.07967] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  174. J. Martin and V. Vennin, Observational constraints on quantum decoherence during inflation, JCAP 05 (2018) 063 [arXiv:1801.09949] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  175. H. Bohra et al., Relating the curvature of De Sitter Universe to Open Quantum Lamb Shift Spectroscopy, Eur. Phys. J. C 81 (2021) 196 [arXiv:1905.07403] [INSPIRE].

    Article  ADS  Google Scholar 

  176. S. Akhtar et al., Open Quantum Entanglement: A study of two atomic system in static patch of de Sitter space, Eur. Phys. J. C 80 (2020) 748 [arXiv:1908.09929] [INSPIRE].

    Article  ADS  Google Scholar 

  177. G. Kaplanek and C.P. Burgess, Hot Accelerated Qubits: Decoherence, Thermalization, Secular Growth and Reliable Late-time Predictions, JHEP 03 (2020) 008 [arXiv:1912.12951] [INSPIRE].

  178. S. Brahma, O. Alaryani and R. Brandenberger, Entanglement entropy of cosmological perturbations, Phys. Rev. D 102 (2020) 043529 [arXiv:2005.09688] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  179. G. Kaplanek and C.P. Burgess, Qubits on the Horizon: Decoherence and Thermalization near Black Holes, JHEP 01 (2021) 098 [arXiv:2007.05984] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  180. M. Rai and D. Boyanovsky, Origin of entropy of gravitationally produced dark matter: The entanglement entropy, Phys. Rev. D 102 (2020) 063532 [arXiv:2007.09196] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  181. C.P. Burgess, R. Holman and G. Kaplanek, Quantum Hotspots: Mean Fields, Open EFTs, Nonlocality and Decoherence Near Black Holes, Fortsch. Phys. 70 (2022) 2200019 [arXiv:2106.10804] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  182. G. Kaplanek, C.P. Burgess and R. Holman, Qubit heating near a hotspot, JHEP 08 (2021) 132 [arXiv:2106.10803] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  183. S. Brahma, A. Berera and J. Calderón-Figueroa, Universal signature of quantum entanglement across cosmological distances, Class. Quant. Grav. 39 (2022) 245002 [arXiv:2107.06910] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  184. S. Banerjee et al., Thermalization in quenched open quantum cosmology, Nucl. Phys. B 996 (2023) 116368 [arXiv:2104.10692] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  185. S. Brahma, A. Berera and J. Calderón-Figueroa, Quantum corrections to the primordial tensor spectrum: open EFTs & Markovian decoupling of UV modes, JHEP 08 (2022) 225 [arXiv:2206.05797] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  186. G. Kaplanek and E. Tjoa, Effective master equations for two accelerated qubits, Phys. Rev. A 107 (2023) 012208 [arXiv:2207.13750] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  187. G. Kaplanek, Some Applications of Open Effective Field Theories to Gravitating Quantum Systems, Ph.D. thesis, McMaster University, Hamilton, Ontario, Canada (2022) [INSPIRE].

  188. A. Daddi Hammou and N. Bartolo, Cosmic decoherence: primordial power spectra and non-Gaussianities, JCAP 04 (2023) 055 [arXiv:2211.07598] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  189. C.P. Burgess et al., Minimal decoherence from inflation, JCAP 07 (2023) 022 [arXiv:2211.11046] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  190. S. Cao and D. Boyanovsky, Nonequilibrium dynamics of axionlike particles: The quantum master equation, Phys. Rev. D 107 (2023) 063518 [arXiv:2212.05161] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  191. S. Brahma, J. Calderón-Figueroa, M. Hassan and X. Mi, Momentum-space entanglement entropy in de Sitter spacetime, Phys. Rev. D 108 (2023) 043522 [arXiv:2302.13894] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  192. M. Sharifian et al., Open quantum system approach to the gravitational decoherence of spin-1/2 particles, Phys. Rev. D 109 (2024) 043510 [arXiv:2309.07236] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  193. R. Alicki, G. Barenboim and A. Jenkins, The irreversible relaxation of inflation, arXiv:2307.04803 [INSPIRE].

  194. R. Alicki, G. Barenboim and A. Jenkins, Quantum thermodynamics of de Sitter space, Phys. Rev. D 108 (2023) 123530 [arXiv:2307.04800] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  195. S. Ning, C.M. Sou and Y. Wang, On the decoherence of primordial gravitons, JHEP 06 (2023) 101 [arXiv:2305.08071] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  196. A. Bhattacharyya et al., The early universe as an open quantum system: complexity and decoherence, JHEP 05 (2024) 058 [arXiv:2401.12134] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  197. B.J. Carr and S.W. Hawking, Black holes in the early Universe, Mon. Not. Roy. Astron. Soc. 168 (1974) 399 [INSPIRE].

  198. J. García-Bellido and E. Ruiz Morales, Primordial black holes from single field models of inflation, Phys. Dark Univ. 18 (2017) 47 [arXiv:1702.03901] [INSPIRE].

  199. C. Germani and T. Prokopec, On primordial black holes from an inflection point, Phys. Dark Univ. 18 (2017) 6 [arXiv:1706.04226] [INSPIRE].

    Article  Google Scholar 

  200. C. Pattison, V. Vennin, H. Assadullahi and D. Wands, Quantum diffusion during inflation and primordial black holes, JCAP 10 (2017) 046 [arXiv:1707.00537] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  201. J.M. Ezquiaga and J. García-Bellido, Quantum diffusion beyond slow-roll: implications for primordial black-hole production, JCAP 08 (2018) 018 [arXiv:1805.06731] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  202. J.M. Ezquiaga, J. García-Bellido and V. Vennin, The exponential tail of inflationary fluctuations: consequences for primordial black holes, JCAP 03 (2020) 029 [arXiv:1912.05399] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  203. A. Kalaja et al., From Primordial Black Holes Abundance to Primordial Curvature Power Spectrum (and back), JCAP 10 (2019) 031 [arXiv:1908.03596] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  204. V. Vennin, Stochastic inflation and primordial black holes, Ph.D. thesis, Université Paris-Saclay, 91190 Saint-Aubin, France (2020) [arXiv:2009.08715] [INSPIRE].

  205. G. Ballesteros, S. Céspedes and L. Santoni, Large power spectrum and primordial black holes in the effective theory of inflation, JHEP 01 (2022) 074 [arXiv:2109.00567] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  206. C. Animali and V. Vennin, Primordial black holes from stochastic tunnelling, JCAP 02 (2023) 043 [arXiv:2210.03812] [INSPIRE].

  207. A.D. Gow et al., Non-perturbative non-Gaussianity and primordial black holes, EPL 142 (2023) 49001 [arXiv:2211.08348] [INSPIRE].

    Article  ADS  Google Scholar 

  208. J.M. Ezquiaga, J. García-Bellido and V. Vennin, Massive Galaxy Clusters Like El Gordo Hint at Primordial Quantum Diffusion, Phys. Rev. Lett. 130 (2023) 121003 [arXiv:2207.06317] [INSPIRE].

    Article  ADS  Google Scholar 

  209. G. Ballesteros et al., Primordial black holes and gravitational waves from dissipation during inflation, JCAP 12 (2022) 006 [arXiv:2208.14978] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  210. LISA Cosmology Working Group collaboration, Primordial black holes and their gravitational-wave signatures, arXiv:2310.19857 [INSPIRE].

  211. V. Briaud and V. Vennin, Uphill inflation, JCAP 06 (2023) 029 [arXiv:2301.09336] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  212. V. Vennin and D. Wands, Quantum diffusion and large primordial perturbations from inflation, arXiv:2402.12672 [INSPIRE].

  213. C. Animali and V. Vennin, Clustering of primordial black holes from quantum diffusion during inflation, JCAP 08 (2024) 026 [arXiv:2402.08642] [INSPIRE].

  214. S. Choudhury, A. Karde, P. Padiyar and M. Sami, Primordial Black Holes from Effective Field Theory of Stochastic Single Field Inflation at NNNLO, arXiv:2403.13484 [INSPIRE].

  215. J. Serreau and R. Parentani, Nonperturbative resummation of de Sitter infrared logarithms in the large-N limit, Phys. Rev. D 87 (2013) 085012 [arXiv:1302.3262] [INSPIRE].

    Article  ADS  Google Scholar 

  216. A.Y. Kamenshchik, A.A. Starobinsky and T. Vardanyan, Massive scalar field in de Sitter spacetime: a two-loop calculation and a comparison with the stochastic approach, Eur. Phys. J. C 82 (2022) 345 [arXiv:2109.05625] [INSPIRE].

    Article  ADS  Google Scholar 

  217. S.P. Miao, N.C. Tsamis and R.P. Woodard, Summing inflationary logarithms in nonlinear sigma models, JHEP 03 (2022) 069 [arXiv:2110.08715] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  218. T. Cohen, D. Green, A. Premkumar and A. Ridgway, Stochastic Inflation at NNLO, JHEP 09 (2021) 159 [arXiv:2106.09728] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  219. T. Cohen, D. Green and A. Premkumar, A tail of eternal inflation, SciPost Phys. 14 (2023) 109 [arXiv:2111.09332] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  220. T. Cohen, D. Green and A. Premkumar, Large deviations in the early Universe, Phys. Rev. D 107 (2023) 083501 [arXiv:2212.02535] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  221. R.P. Woodard and B. Yesilyurt, Unfinished business in a nonlinear sigma model on de Sitter background, JHEP 06 (2023) 206 [arXiv:2302.11528] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  222. G. Kaplanek and C.P. Burgess, Hot Cosmic Qubits: Late-Time de Sitter Evolution and Critical Slowing Down, JHEP 02 (2020) 053 [arXiv:1912.12955] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  223. S. Chaykov, N. Agarwal, S. Bahrami and R. Holman, Loop corrections in Minkowski spacetime away from equilibrium. Part I. Late-time resummations, JHEP 02 (2023) 093 [arXiv:2206.11288] [INSPIRE].

  224. G. Gubitosi, F. Piazza and F. Vernizzi, The Effective Field Theory of Dark Energy, JCAP 02 (2013) 032 [arXiv:1210.0201] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  225. F. Piazza and F. Vernizzi, Effective Field Theory of Cosmological Perturbations, Class. Quant. Grav. 30 (2013) 214007 [arXiv:1307.4350] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  226. J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Essential Building Blocks of Dark Energy, JCAP 08 (2013) 025 [arXiv:1304.4840] [INSPIRE].

    Article  ADS  Google Scholar 

  227. M. Raveri, B. Hu, N. Frusciante and A. Silvestri, Effective Field Theory of Cosmic Acceleration: constraining dark energy with CMB data, Phys. Rev. D 90 (2014) 043513 [arXiv:1405.1022] [INSPIRE].

    Article  ADS  Google Scholar 

  228. N. Frusciante, G. Papadomanolakis and A. Silvestri, An extended action for the effective field theory of dark energy: a stability analysis and a complete guide to the mapping at the basis of EFTCAMB, JCAP 07 (2016) 018 [arXiv:1601.04064] [INSPIRE].

    Article  ADS  Google Scholar 

  229. D. Baumann, A. Nicolis, L. Senatore and M. Zaldarriaga, Cosmological Non-Linearities as an Effective Fluid, JCAP 07 (2012) 051 [arXiv:1004.2488] [INSPIRE].

    Article  ADS  Google Scholar 

  230. J.J.M. Carrasco, M.P. Hertzberg and L. Senatore, The Effective Field Theory of Cosmological Large Scale Structures, JHEP 09 (2012) 082 [arXiv:1206.2926] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  231. S. Peirone, M. Martinelli, M. Raveri and A. Silvestri, Impact of theoretical priors in cosmological analyses: the case of single field quintessence, Phys. Rev. D 96 (2017) 063524 [arXiv:1702.06526] [INSPIRE].

    Article  ADS  Google Scholar 

  232. C. de Rham and S. Melville, Gravitational Rainbows: LIGO and Dark Energy at its Cutoff, Phys. Rev. Lett. 121 (2018) 221101 [arXiv:1806.09417] [INSPIRE].

    Article  ADS  Google Scholar 

  233. D. de Boe et al., Phenomenology of Horndeski gravity under positivity bounds, JCAP 08 (2024) 029 [arXiv:2403.13096] [INSPIRE].

    Article  ADS  Google Scholar 

  234. LIGO Scientific and Virgo collaborations, GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119 (2017) 161101 [arXiv:1710.05832] [INSPIRE].

  235. LIGO Scientific et al. collaborations, Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, Astrophys. J. Lett. 848 (2017) L13 [arXiv:1710.05834] [INSPIRE].

  236. DESI collaboration, DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations, arXiv:2404.03002 [INSPIRE].

  237. S. Giardino, V. Faraoni and A. Giusti, First-order thermodynamics of scalar-tensor cosmology, JCAP 04 (2022) 053 [arXiv:2202.07393] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  238. M. Miranda, S. Giardino, A. Giusti and L. Heisenberg, First-order thermodynamics of Horndeski cosmology, Phys. Rev. D 109 (2024) 124033 [arXiv:2401.10351] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  239. J. García-Bellido and L. Espinosa-Portalés, Cosmic acceleration from first principles, Phys. Dark Univ. 34 (2021) 100892 [arXiv:2106.16014] [INSPIRE].

    Article  Google Scholar 

  240. L. Espinosa-Portalés and J. García-Bellido, Covariant formulation of non-equilibrium thermodynamics in General Relativity, Phys. Dark Univ. 34 (2021) 100893 [arXiv:2106.16012] [INSPIRE].

    Article  Google Scholar 

  241. R. Arjona, L. Espinosa-Portalés, J. García-Bellido and S. Nesseris, A GREAT model comparison against the cosmological constant, Phys. Dark Univ. 36 (2022) 101029 [arXiv:2111.13083] [INSPIRE].

    Article  Google Scholar 

  242. S.J. Landau, M. Benetti, A. Pérez and D. Sudarsky, Cosmological constraints on unimodular gravity models with diffusion, Phys. Rev. D 108 (2023) 043524 [arXiv:2211.07424] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  243. J. Oppenheim and A. Russo, Anomalous contribution to galactic rotation curves due to stochastic spacetime, arXiv:2402.19459 [INSPIRE].

  244. D. Campo and R. Parentani, Inflationary spectra and violations of Bell inequalities, Phys. Rev. D 74 (2006) 025001 [astro-ph/0505376] [INSPIRE].

  245. L. Lello, D. Boyanovsky and R. Holman, Entanglement entropy in particle decay, JHEP 11 (2013) 116 [arXiv:1304.6110] [INSPIRE].

    Article  ADS  Google Scholar 

  246. L. Lello, D. Boyanovsky and R. Holman, Superhorizon entanglement entropy from particle decay in inflation, JHEP 04 (2014) 055 [arXiv:1305.2441] [INSPIRE].

    Article  ADS  Google Scholar 

  247. J. Maldacena, A model with cosmological Bell inequalities, Fortsch. Phys. 64 (2016) 10 [arXiv:1508.01082] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  248. S. Choudhury, S. Panda and R. Singh, Bell violation in the Sky, Eur. Phys. J. C 77 (2017) 60 [arXiv:1607.00237] [INSPIRE].

    Article  ADS  Google Scholar 

  249. J. Martin and V. Vennin, Bell inequalities for continuous-variable systems in generic squeezed states, Phys. Rev. A 93 (2016) 062117 [arXiv:1605.02944] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  250. J. Martin and V. Vennin, Obstructions to Bell CMB Experiments, Phys. Rev. D 96 (2017) 063501 [arXiv:1706.05001] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  251. N. Bolis, A. Albrecht and R. Holman, Non-Gaussianity from Entanglement During Inflation, JCAP 07 (2019) 021 [arXiv:1902.07567] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  252. K. Ando and V. Vennin, Bipartite temporal Bell inequalities for two-mode squeezed states, Phys. Rev. A 102 (2020) 052213 [arXiv:2007.00458] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  253. L. Espinosa-Portalés and V. Vennin, Real-space Bell inequalities in de Sitter, JCAP 07 (2022) 037 [arXiv:2203.03505] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  254. A. Adil et al., Entanglement masquerading in the CMB, JCAP 06 (2023) 024 [arXiv:2211.11079] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  255. P. Tejerina-Pérez, D. Bertacca and R. Jimenez, An Entangled Universe, arXiv:2403.15742 [INSPIRE].

  256. T. Colas, J. Grain and V. Vennin, Four-mode squeezed states: two-field quantum systems and the symplectic group Sp(4, ℝ), Eur. Phys. J. C 82 (2022) 6 [arXiv:2104.14942] [INSPIRE].

  257. C. Cheung, T. He and A. Sivaramakrishnan, Entropy growth in perturbative scattering, Phys. Rev. D 108 (2023) 045013 [arXiv:2304.13052] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  258. R. Aoude, G. Elor, G.N. Remmen and O. Sumensari, Positivity in Amplitudes from Quantum Entanglement, arXiv:2402.16956 [INSPIRE].

  259. S. Melville and G.L. Pimentel, A de Sitter S-matrix for the masses, arXiv:2309.07092 [INSPIRE].

  260. S. Melville and G.L. Pimentel, A de Sitter S-matrix from amputated cosmological correlators, JHEP 08 (2024) 211 [arXiv:2404.05712] [INSPIRE].

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We thank C. P. Burgess, S. Cespedes, P. Creminelli, J. Grain, R. Holman, G. Kaplanek, S. Melville, A. Nicolis, R. Penco, L. Santoni, B. Salehian, D. Stefanyszyn, V. Vennin and G. Villa for the insightful discussions. T.C. warmly thanks D. Comelli, F. Piazza, A. Tolley and F. Vernizzi for inspiring this research direction. This work has been supported by STFC consolidated grant ST/X001113/1, ST/T000694/1, ST/X000664/1 and EP/V048422/1. S.A.S. is supported by a Harding Distinguished Postgraduate Scholarship.

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  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K.

    Santiago Agüí Salcedo, Thomas Colas & Enrico Pajer

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  1. Santiago Agüí Salcedo
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  2. Thomas Colas
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Correspondence to Thomas Colas.

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Salcedo, S.A., Colas, T. & Pajer, E. The open effective field theory of inflation. J. High Energ. Phys. 2024, 248 (2024). https://doi.org/10.1007/JHEP10(2024)248

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  • Received: 03 May 2024

  • Accepted: 03 October 2024

  • Published: 31 October 2024

  • Version of record: 31 October 2024

  • DOI: https://doi.org/10.1007/JHEP10(2024)248

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Keywords

  • Cosmological models
  • Effective Field Theories
  • de Sitter space
  • Non-Equilibrium Field Theory

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