Abstract
Calculating the quantum evolution of a de Sitter universe on superhorizon scales is notoriously difficult. To address this challenge, we introduce the Soft de Sitter Effective Theory (SdSET). This framework holds for superhorizon modes whose comoving momentum is far below the UV scale, which is set by the inverse comoving horizon. The SdSET is formulated using the same approach that yields the Heavy Quark Effective Theory. The degrees of freedom that capture the long wavelength dynamics are identified with the growing and decaying solutions to the equations of motion. The operator expansion is organized using a power counting scheme, and loops can be regulated while respecting the low energy symmetries. For massive quantum fields in a fixed de Sitter background, power counting implies that all interactions beyond the horizon are irrelevant. Alternatively, if the fields are very light, the leading interactions are at most marginal, and resumming the associated logarithms using (dynamical) renormalization group techniques yields the evolution equation for canonical stochastic inflation. The SdSET is also applicable to models where gravity is dynamical, including inflation. In this case, diffeomorphism invariance ensures that all interactions are irrelevant, trivially implying the all-orders conservation of adiabatic density fluctuations and gravitational waves. We briefly touch on the application to slow-roll eternal inflation by identifying novel relevant operators. This work serves to demystify many aspects of perturbation theory outside the horizon, and has a variety of applications to problems of cosmological interest.
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References
R. Bousso, Holography in general space-times, JHEP 06 (1999) 028 [hep-th/9906022] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, in Strings 2001: International Conference, (2001) [hep-th/0106109] [INSPIRE].
P.O. Mazur and E. Mottola, Weyl cohomology and the effective action for conformal anomalies, Phys. Rev. D 64 (2001) 104022 [hep-th/0106151] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
M. Alishahiha, A. Karch, E. Silverstein and D. Tong, The dS/dS correspondence, AIP Conf. Proc. 743 (2004) 393 [hep-th/0407125] [INSPIRE].
B. Freivogel, Making predictions in the multiverse, Class. Quant. Grav. 28 (2011) 204007 [arXiv:1105.0244] [INSPIRE].
D.S. Salopek and J.R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D 42 (1990) 3936 [INSPIRE].
C. Cheung, P. Creminelli, A. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].
L. Senatore and M. Zaldarriaga, On Loops in Inflation, JHEP 12 (2010) 008 [arXiv:0912.2734] [INSPIRE].
L. Senatore and M. Zaldarriaga, The constancy of ζ in single-clock Inflation at all loops, JHEP 09 (2013) 148 [arXiv:1210.6048] [INSPIRE].
V. Assassi, D. Baumann and D. Green, Symmetries and Loops in Inflation, JHEP 02 (2013) 151 [arXiv:1210.7792] [INSPIRE].
L.H. Ford, Quantum Instability of de Sitter Space-time, Phys. Rev. D 31 (1985) 710 [INSPIRE].
I. Antoniadis, J. Iliopoulos and T.N. Tomaras, Quantum Instability of de Sitter Space, Phys. Rev. Lett. 56 (1986) 1319 [INSPIRE].
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].
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].
N.C. Tsamis and R.P. Woodard, Strong infrared effects in quantum gravity, Annals Phys. 238 (1995) 1.
N.C. Tsamis and R.P. Woodard, The Quantum gravitational back reaction on inflation, Annals Phys. 253 (1997) 1 [hep-ph/9602316] [INSPIRE].
N.C. Tsamis and R.P. Woodard, Matter contributions to the expansion rate of the universe, Phys. Lett. B 426 (1998) 21 [hep-ph/9710466] [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations. II. Can these corrections become large?, Phys. Rev. D 74 (2006) 023508 [hep-th/0605244] [INSPIRE].
D. Seery, Infrared effects in inflationary correlation functions, Class. Quant. Grav. 27 (2010) 124005 [arXiv:1005.1649] [INSPIRE].
C.P. Burgess, R. Holman, L. Leblond and S. Shandera, Breakdown of Semiclassical Methods in de Sitter Space, JCAP 10 (2010) 017 [arXiv:1005.3551] [INSPIRE].
A. Rajaraman, On the proper treatment of massless fields in Euclidean de Sitter space, Phys. Rev. D 82 (2010) 123522 [arXiv:1008.1271] [INSPIRE].
D. Marolf and I.A. Morrison, The IR stability of de Sitter: Loop corrections to scalar propagators, Phys. Rev. D 82 (2010) 105032 [arXiv:1006.0035] [INSPIRE].
D. Marolf and I.A. Morrison, The IR stability of de Sitter QFT: Physical initial conditions, Gen. Rel. Grav. 43 (2011) 3497 [arXiv:1104.4343] [INSPIRE].
D. Marolf, I.A. Morrison and M. Srednicki, Perturbative S-matrix for massive scalar fields in global de Sitter space, Class. Quant. Grav. 30 (2013) 155023 [arXiv:1209.6039] [INSPIRE].
M. Beneke and P. Moch, On “dynamical mass” generation in Euclidean de Sitter space, Phys. Rev. D 87 (2013) 064018 [arXiv:1212.3058] [INSPIRE].
E.T. Akhmedov, Lecture notes on interacting quantum fields in de Sitter space, Int. J. Mod. Phys. D 23 (2014) 1430001 [arXiv:1309.2557] [INSPIRE].
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].
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].
E.T. Akhmedov, U. Moschella, K.E. Pavlenko and F.K. Popov, Infrared dynamics of massive scalars from the complementary series in de Sitter space, Phys. Rev. D 96 (2017) 025002 [arXiv:1701.07226] [INSPIRE].
B.-L. Hu, Infrared Behavior of Quantum Fields in Inflationary Cosmology — Issues and Approaches: an overview, arXiv:1812.11851 [INSPIRE].
E.T. Akhmedov, U. Moschella and F.K. Popov, Characters of different secular effects in various patches of de Sitter space, Phys. Rev. D 99 (2019) 086009 [arXiv:1901.07293] [INSPIRE].
V. Gorbenko and L. Senatore, λϕ4 in dS, arXiv:1911.00022 [INSPIRE].
M. Baumgart and R. Sundrum, de Sitter Diagrammar and the Resummation of Time, JHEP 07 (2020) 119 [arXiv:1912.09502] [INSPIRE].
M. Mirbabayi, Infrared dynamics of a light scalar field in de Sitter, arXiv:1911.00564 [INSPIRE].
D. Green and A. Premkumar, Dynamical RG and Critical Phenomena in de Sitter Space, JHEP 04 (2020) 064 [arXiv:2001.05974] [INSPIRE].
W. Hu, D.N. Spergel and M.J. White, Distinguishing causal seeds from inflation, Phys. Rev. D 55 (1997) 3288 [astro-ph/9605193] [INSPIRE].
D.N. Spergel and M. Zaldarriaga, CMB polarization as a direct test of inflation, Phys. Rev. Lett. 79 (1997) 2180 [astro-ph/9705182] [INSPIRE].
S. Dodelson, Coherent phase argument for inflation, AIP Conf. Proc. 689 (2003) 184 [hep-ph/0309057] [INSPIRE].
N. Dalal, O. Dore, D. Huterer and A. Shirokov, The imprints of primordial non-Gaussianities on large-scale structure: scale dependent bias and abundance of virialized objects, Phys. Rev. D 77 (2008) 123514 [arXiv:0710.4560] [INSPIRE].
M. Alvarez et al., Testing Inflation with Large Scale Structure: Connecting Hopes with Reality, arXiv:1412.4671 [INSPIRE].
S. Weinberg, Effective Gauge Theories, Phys. Lett. B 91 (1980) 51 [INSPIRE].
H. Georgi, Effective field theory, Ann. Rev. Nucl. Part. Sci. 43 (1993) 209 [INSPIRE].
F. Tanaka, Coherent Representation of Dynamical Renormalization Group in Bose Systems, Prog. Theor. Phys. 54 (1975) 289 [INSPIRE].
D. Boyanovsky, H.J. de Vega, R. Holman and M. Simionato, Dynamical renormalization group resummation of finite temperature infrared divergences, Phys. Rev. D 60 (1999) 065003 [hep-ph/9809346] [INSPIRE].
D. Boyanovsky and H.J. de Vega, Dynamical renormalization group approach to relaxation in quantum field theory, Annals Phys. 307 (2003) 335 [hep-ph/0302055] [INSPIRE].
J. Glimm and A.M. Jaffe, Positivity of the 𝜙4 in Three-dimensions Hamiltonian, Fortsch. Phys. 21 (1973) 327 [INSPIRE].
T. Appelquist and J. Carazzone, Infrared Singularities and Massive Fields, Phys. Rev. D 11 (1975) 2856 [INSPIRE].
S. Weinberg, High-energy behavior in quantum field theory, Phys. Rev. 118 (1960) 838 [INSPIRE].
J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B 231 (1984) 269 [INSPIRE].
J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP 09 (2011) 045 [arXiv:1104.2846] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
A. Strominger, Inflation and the dS/CFT correspondence, JHEP 11 (2001) 049 [hep-th/0110087] [INSPIRE].
P. McFadden and K. Skenderis, Holography for Cosmology, Phys. Rev. D 81 (2010) 021301 [arXiv:0907.5542] [INSPIRE].
P. Creminelli, Conformal invariance of scalar perturbations in inflation, Phys. Rev. D 85 (2012) 041302 [arXiv:1108.0874] [INSPIRE].
N. Isgur and M.B. Wise, Weak Decays of Heavy Mesons in the Static Quark Approximation, Phys. Lett. B 232 (1989) 113 [INSPIRE].
E. Eichten and B.R. Hill, An Effective Field Theory for the Calculation of Matrix Elements Involving Heavy Quarks, Phys. Lett. B 234 (1990) 511 [INSPIRE].
H. Georgi, An Effective Field Theory for Heavy Quarks at Low-energies, Phys. Lett. B 240 (1990) 447 [INSPIRE].
B. Grinstein, The Static Quark Effective Theory, Nucl. Phys. B 339 (1990) 253 [INSPIRE].
D.I. Podolsky, Dynamical renormalization group methods in theory of eternal inflation, Grav. Cosmol. 15 (2009) 69 [arXiv:0809.2453] [INSPIRE].
M. Dias, R.H. Ribeiro and D. Seery, The δN formula is the dynamical renormalization group, JCAP 10 (2013) 062 [arXiv:1210.7800] [INSPIRE].
M. Mirbabayi and M. Simonović, Effective Theory of Squeezed Correlation Functions, JCAP 03 (2016) 056 [arXiv:1507.04755] [INSPIRE].
H. Georgi, Heavy quark effective field theory, in TASI 91, pp. 0589–630 (1991) [INSPIRE].
M. Neubert, Heavy quark symmetry, Phys. Rept. 245 (1994) 259 [hep-ph/9306320] [INSPIRE].
M.A. Shifman, Lectures on heavy quarks in quantum chromodynamics, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and Beyond, pp. 409–514 (1995) [hep-ph/9510377] [INSPIRE].
M.B. Wise, Heavy quark physics: Course, in Les Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, pp. 1051–1089 (1997) [hep-ph/9805468] [INSPIRE].
A.V. Manohar and M.B. Wise, Heavy Quark Physics, vol. 10, Cambridge University Press (2000).
A.V. Manohar, Effective field theories, in 10th Lake Louise Winter Institute: Quarks and Colliders, pp. 274–315, 6, 1995 [hep-ph/9508245] [INSPIRE].
D.B. Kaplan, Effective field theories, in 7th Summer School in Nuclear Physics Symmetries, 6, 1995 [nucl-th/9506035] [INSPIRE].
I.Z. Rothstein, TASI lectures on effective field theories, in TASI 2002, (2003) [hep-ph/0308266] [INSPIRE].
D.B. Kaplan, Five lectures on effective field theory, in 7th Summer School in Nuclear Physics Symmetries, (2005) [nucl-th/0510023] [INSPIRE].
A.A. Petrov and A.E. Blechman, Effective Field Theories, WSP (2016) [DOI] [INSPIRE].
A.V. Manohar, Introduction to Effective Field Theories, Les Houches Lect. Notes 108 (2020) [arXiv:1804.05863] [INSPIRE].
T. Cohen, As Scales Become Separated: Lectures on Effective Field Theory, PoS TASI2018 (2019) 011 [arXiv:1903.03622] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1 [INSPIRE].
L.P. Grishchuk and Y. Sidorov, Squeezed quantum states of relic gravitons and primordial density fluctuations, Phys. Rev. D 42 (1990) 3413 [INSPIRE].
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].
I. Mata, S. Raju and S. Trivedi, CMB from CFT, JHEP 07 (2013) 015 [arXiv:1211.5482] [INSPIRE].
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].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds, arXiv:1910.14051 [INSPIRE].
C. Sleight, A Mellin Space Approach to Cosmological Correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].
N. Arkani-Hamed and P. Benincasa, On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes, arXiv:1811.01125 [INSPIRE].
P. Benincasa, From the flat-space S-matrix to the Wavefunction of the Universe, arXiv:1811.02515 [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization, arXiv:2005.04234 [INSPIRE].
M.E. Luke, A.V. Manohar and I.Z. Rothstein, Renormalization group scaling in nonrelativistic QCD, Phys. Rev. D 61 (2000) 074025 [hep-ph/9910209] [INSPIRE].
D. Green and E. Pajer, On the Symmetries of Cosmological Perturbations, JCAP 09 (2020) 032 [arXiv:2004.09587] [INSPIRE].
M.E. Luke and A.V. Manohar, Reparametrization invariance constraints on heavy particle effective field theories, Phys. Lett. B 286 (1992) 348 [hep-ph/9205228] [INSPIRE].
S. Weinberg, Effective Field Theory for Inflation, Phys. Rev. D 77 (2008) 123541 [arXiv:0804.4291] [INSPIRE].
D. Green, M. Lewandowski, L. Senatore, E. Silverstein and M. Zaldarriaga, Anomalous Dimensions and Non-Gaussianity, JHEP 10 (2013) 171 [arXiv:1301.2630] [INSPIRE].
S. Raju, New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators, Phys. Rev. D 85 (2012) 126009 [arXiv:1201.6449] [INSPIRE].
D. Green and R.A. Porto, Signals of a Quantum Universe, Phys. Rev. Lett. 124 (2020) 251302 [arXiv:2001.09149] [INSPIRE].
F. Bernardeau, S. Colombi, E. Gaztanaga and R. Scoccimarro, Large scale structure of the universe and cosmological perturbation theory, Phys. Rept. 367 (2002) 1 [astro-ph/0112551] [INSPIRE].
D. Boyanovsky and H.J. de Vega, Particle decay in inflationary cosmology, Phys. Rev. D 70 (2004) 063508 [astro-ph/0406287] [INSPIRE].
P. McDonald, Dark matter clustering: a simple renormalization group approach, Phys. Rev. D 75 (2007) 043514 [astro-ph/0606028] [INSPIRE].
C.P. Burgess, L. Leblond, R. Holman and S. Shandera, Super-Hubble de Sitter Fluctuations and the Dynamical RG, JCAP 03 (2010) 033 [arXiv:0912.1608] [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, Dynamical Structure and Definition of Energy in General Relativity, Phys. Rev. 116 (1959) 1322 [INSPIRE].
K. Hinterbichler, L. Hui and J. Khoury, Conformal Symmetries of Adiabatic Modes in Cosmology, JCAP 08 (2012) 017 [arXiv:1203.6351] [INSPIRE].
K. Hinterbichler, L. Hui and J. Khoury, An Infinite Set of Ward Identities for Adiabatic Modes in Cosmology, JCAP 01 (2014) 039 [arXiv:1304.5527] [INSPIRE].
W.H. Kinney, Horizon crossing and inflation with large eta, Phys. Rev. D 72 (2005) 023515 [gr-qc/0503017] [INSPIRE].
M.H. Namjoo, H. Firouzjahi and M. Sasaki, Violation of non-Gaussianity consistency relation in a single field inflationary model, EPL 101 (2013) 39001 [arXiv:1210.3692] [INSPIRE].
J. Martin, H. Motohashi and T. Suyama, Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation, Phys. Rev. D 87 (2013) 023514 [arXiv:1211.0083] [INSPIRE].
P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore and M. Zaldarriaga, The Phase Transition to Slow-roll Eternal Inflation, JHEP 09 (2008) 036 [arXiv:0802.1067] [INSPIRE].
S. Dubovsky, L. Senatore and G. Villadoro, The Volume of the Universe after Inflation and de Sitter Entropy, JHEP 04 (2009) 118 [arXiv:0812.2246] [INSPIRE].
M. Lewandowski and A. Perko, Leading slow roll corrections to the volume of the universe and the entropy bound, JHEP 12 (2014) 060 [arXiv:1309.6705] [INSPIRE].
G. Barenboim, W.-I. Park and W.H. Kinney, Eternal Hilltop Inflation, JCAP 05 (2016) 030 [arXiv:1601.08140] [INSPIRE].
M.A.G. Garcia, M.A. Amin, S.G. Carlsten and D. Green, Stochastic Particle Production in a de Sitter Background, JCAP 05 (2019) 012 [arXiv:1902.09598] [INSPIRE].
M.A.G. Garcia, M.A. Amin and D. Green, Curvature Perturbations From Stochastic Particle Production During Inflation, JCAP 06 (2020) 039 [arXiv:2001.09158] [INSPIRE].
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Cohen, T., Green, D. Soft de Sitter Effective Theory. J. High Energ. Phys. 2020, 41 (2020). https://doi.org/10.1007/JHEP12(2020)041
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DOI: https://doi.org/10.1007/JHEP12(2020)041