Abstract
We identify the effective field theory describing the physics of super-Hubble scales and show it to be a special case of a class of effective field theories appropriate to open systems. Open systems are those that allow information to be exchanged between the degrees of freedom of interest and those that are integrated out, such as would be appropriate for particles moving through a fluid. Strictly speaking they cannot in general be described by an effective lagrangian; rather the appropriate ‘low-energy’ limit is instead a Lindblad equation describing the time-evolution of the density matrix of the slow degrees of freedom. We derive the equation relevant to super-Hubble modes of quantum fields in de Sitter (and near-de Sitter) spacetimes and derive two of its implications. We show that the evolution of the diagonal density-matrix elements quickly approach the Fokker-Planck equation of Starobinsky’s stochastic inflationary picture. This allows us both to identify the leading corrections and provide an alternative first-principles derivation of this picture’s stochastic noise and drift. (As applications we show that the noise for massless fields is independent of the details of the window function used, and also compute how the noise changes for systems with a sub-luminal speed of sound, c s < 1.) We then argue that the presence of interactions drive the off-diagonal density-matrix elements to zero in the field
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References
C.P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel. 7 (2004) 5 [gr-qc/0311082] [INSPIRE].
W.D. Goldberger, Les Houches lectures on effective field theories and gravitational radiation, hep-ph/0701129 [INSPIRE].
J.F. Donoghue, The effective field theory treatment of quantum gravity, AIP Conf. Proc. 1483 (2012) 73 [arXiv:1209.3511] [INSPIRE].
C.P. Burgess, J.M. Cline, F. Lemieux and R. Holman, Are inflationary predictions sensitive to very high-energy physics?, JHEP 02 (2003) 048 [hep-th/0210233] [INSPIRE].
C.P. Burgess, J.M. Cline and R. Holman, Effective field theories and inflation, JCAP 10 (2003) 004 [hep-th/0306079] [INSPIRE].
S. Weinberg, Effective Field Theory for Inflation, Phys. Rev. D 77 (2008) 123541 [arXiv:0804.4291] [INSPIRE].
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].
A. Achucarro, J.-O. Gong, S. Hardeman, G.A. Palma and S.P. Patil, Effective theories of single field inflation when heavy fields matter, JHEP 05 (2012) 066 [arXiv:1201.6342] [INSPIRE].
A. Avgoustidis, S. Cremonini, A.-C. Davis, R.H. Ribeiro, K. Turzynski and S. Watson, Decoupling Survives Inflation: A Critical Look at Effective Field Theory Violations During Inflation, JCAP 06 (2012) 025 [arXiv:1203.0016] [INSPIRE].
C.P. Burgess, M.W. Horbatsch and S.P. Patil, Inflating in a Trough: Single-Field Effective Theory from Multiple-Field Curved Valleys, JHEP 01 (2013) 133 [arXiv:1209.5701] [INSPIRE].
C. Cheung, P. Creminelli, A.L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].
L. Senatore and M. Zaldarriaga, The Effective Field Theory of Multifield Inflation, JHEP 04 (2012) 024 [arXiv:1009.2093] [INSPIRE].
N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, Large non-Gaussianities in the Effective Field Theory Approach to Single-Field Inflation: the Bispectrum, JCAP 08 (2010) 008 [arXiv:1004.0893] [INSPIRE].
N. Bartolo, M. Fasiello, S. Matarrese and A. Riotto, Large non-Gaussianities in the Effective Field Theory Approach to Single-Field Inflation: the Trispectrum, JCAP 09 (2010) 035 [arXiv:1006.5411] [INSPIRE].
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].
E. Dimastrogiovanni, M. Fasiello and A.J. Tolley, Low-Energy Effective Field Theory for Chromo-Natural Inflation, JCAP 02 (2013) 046 [arXiv:1211.1396] [INSPIRE].
V.F. Mukhanov and G.V. Chibisov, Quantum Fluctuation and Nonsingular Universe (in Russian), JETP Lett. 33 (1981) 532 [Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 549] [INSPIRE].
A.H. Guth and S.-Y. Pi, Fluctuations in the New Inflationary Universe, Phys. Rev. Lett. 49 (1982) 1110 [INSPIRE].
A.A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B 117 (1982) 175 [INSPIRE].
S.W. Hawking, The Development of Irregularities in a Single Bubble Inflationary Universe, Phys. Lett. B 115 (1982) 295 [INSPIRE].
V.N. Lukash, ??, Pisma Zh. Eksp. Teor. Fiz. 31 (1980) 631.
V.N. Lukash, Production of phonons in an isotropic universe, Sov. Phys. JETP 52 (1980) 807 [Zh. Eksp. Teor. Fiz. 79 (1980) 1601] [INSPIRE].
W.H. Press, Spontaneous Production of the Zel’dovich Spectrum of Cosmological Fluctuations, Phys. Scr. 21 (1980) 702 [INSPIRE].
K. Sato, First Order Phase Transition of a Vacuum and Expansion of the Universe, Mon. Not. Roy. Astron. Soc. 195 (1981) 467 [INSPIRE].
M.S. Sloth, On the one loop corrections to inflation and the CMB anisotropies, Nucl. Phys. B 748 (2006) 149 [astro-ph/0604488] [INSPIRE].
M.S. Sloth, On the one loop corrections to inflation. II. The Consistency relation, Nucl. Phys. B 775 (2007) 78 [hep-th/0612138] [INSPIRE].
A. Bilandzic and T. Prokopec, Quantum radiative corrections to slow-roll inflation, Phys. Rev. D 76 (2007) 103507 [arXiv:0704.1905] [INSPIRE].
M. van der Meulen and J. Smit, Classical approximation to quantum cosmological correlations, JCAP 11 (2007) 023 [arXiv:0707.0842] [INSPIRE].
G. Petri, A Diagrammatic Approach to Scalar Field Correlators during Inflation, arXiv:0810.3330 [INSPIRE].
D.H. Lyth, The curvature perturbation in a box, JCAP 12 (2007) 016 [arXiv:0707.0361] [INSPIRE].
K. Enqvist, S. Nurmi, D. Podolsky and G.I. Rigopoulos, On the divergences of inflationary superhorizon perturbations, JCAP 04 (2008) 025 [arXiv:0802.0395] [INSPIRE].
N. Bartolo, S. Matarrese, M. Pietroni, A. Riotto and D. Seery, On the Physical Significance of Infra-red Corrections to Inflationary Observables, JCAP 01 (2008) 015 [arXiv:0711.4263] [INSPIRE].
A. Riotto and M.S. Sloth, On Resumming Inflationary Perturbations beyond One-loop, JCAP 04 (2008) 030 [arXiv:0801.1845] [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].
P. Adshead, R. Easther and E.A. Lim, Cosmology With Many Light Scalar Fields: Stochastic Inflation and Loop Corrections, Phys. Rev. D 79 (2009) 063504 [arXiv:0809.4008] [INSPIRE].
Y. Urakawa and T. Tanaka, Influence on observation from IR divergence during inflation. II. Multi field inflation, Prog. Theor. Phys. 122 (2009) 1207 [arXiv:0904.4415] [INSPIRE].
Y. Urakawa and T. Tanaka, Influence on Observation from IR Divergence during Inflation. I. Single Field Inflation, Prog. Theor. Phys. 122 (2009) 779 [arXiv:0902.3209] [INSPIRE].
S.B. Giddings and M.S. Sloth, Semiclassical relations and IR effects in de Sitter and slow-roll space-times, JCAP 01 (2011) 023 [arXiv:1005.1056] [INSPIRE].
C.T. Byrnes, M. Gerstenlauer, A. Hebecker, S. Nurmi and G. Tasinato, Inflationary Infrared Divergences: Geometry of the Reheating Surface versus δN Formalism, JCAP 08 (2010) 006 [arXiv:1005.3307] [INSPIRE].
S.B. Giddings and M.S. Sloth, Cosmological observables, IR growth of fluctuations and scale-dependent anisotropies, Phys. Rev. D 84 (2011) 063528 [arXiv:1104.0002] [INSPIRE].
M. Gerstenlauer, A. Hebecker and G. Tasinato, Inflationary Correlation Functions without Infrared Divergences, JCAP 06 (2011) 021 [arXiv:1102.0560] [INSPIRE].
D. Seery, A parton picture of de Sitter space during slow-roll inflation, JCAP 05 (2009) 021 [arXiv:0903.2788] [INSPIRE].
N. Afshordi and R.H. Brandenberger, Super Hubble nonlinear perturbations during inflation, Phys. Rev. D 63 (2001) 123505 [gr-qc/0011075] [INSPIRE].
B. Losic and W.G. Unruh, Cosmological Perturbation Theory in Slow-Roll Spacetimes, Phys. Rev. Lett. 101 (2008) 111101 [arXiv:0804.4296] [INSPIRE].
T.M. Janssen, S.P. Miao, T. Prokopec and R.P. Woodard, Infrared Propagator Corrections for Constant Deceleration, Class. Quant. Grav. 25 (2008) 245013 [arXiv:0808.2449] [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].
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].
L.H. Ford, Quantum Instability of de Sitter Space-time, Phys. Rev. D 31 (1985) 710 [INSPIRE].
V. Muller, H.J. Schmidt and A.A. Starobinsky, The Stability of the de Sitter Space-time in Fourth Order Gravity, Phys. Lett. B 202 (1988) 198 [INSPIRE].
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, in Field Theory, Quantum Gravity and Strings. Proceedings of a Seminar Series Held at DAPHE, Observatoire de Meudon, and LPTHE, Université Pierre et Marie Curie, Paris, Between October 1984 and October 1985, H.j. De Vega and N. Sánchez eds., Springer Berlin Heidelberg (1986), pp. 107-126 [ISBN: 978-3-540-16452-4, 978-3-540-39789-2] [Lect. Notes Phys. 246 (1986) 107] [INSPIRE].
I. Antoniadis and E. Mottola, Graviton Fluctuations in de Sitter Space, J. Math. Phys. 32 (1991) 1037 [INSPIRE].
M. Sasaki, H. Suzuki, K. Yamamoto and J. Yokoyama, Superexpansionary divergence: Breakdown of perturbative quantum field theory in space-time with accelerated expansion, Class. Quant. Grav. 10 (1993) L55 [INSPIRE].
A.D. Dolgov, M.B. Einhorn and V.I. Zakharov, On Infrared effects in de Sitter background, Phys. Rev. D 52 (1995) 717 [gr-qc/9403056] [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].
M.-a. Sakagami, Evolution From Pure States Into Mixed States in de Sitter Space, Prog. Theor. Phys. 79 (1988) 442 [INSPIRE].
L.P. Grishchuk and Y.V. Sidorov, On the Quantum State of Relic Gravitons, Class. Quant. Grav. 6 (1989) L161 [INSPIRE].
R.H. Brandenberger, R. Laflamme and M. Mijic, Classical Perturbations From Decoherence of Quantum Fluctuations in the Inflationary Universe, Mod. Phys. Lett. A 5 (1990) 2311 [INSPIRE].
E. Calzetta and B.L. Hu, Quantum fluctuations, decoherence of the mean field and structure formation in the early universe, Phys. Rev. D 52 (1995) 6770 [gr-qc/9505046] [INSPIRE].
J. Lesgourgues, D. Polarski and A.A. Starobinsky, Quantum to classical transition of cosmological perturbations for nonvacuum initial states, Nucl. Phys. B 497 (1997) 479 [gr-qc/9611019] [INSPIRE].
C. Kiefer, D. Polarski and A.A. Starobinsky, Quantum to classical transition for fluctuations in the early universe, Int. J. Mod. Phys. D 7 (1998) 455 [gr-qc/9802003] [INSPIRE].
C. Kiefer and D. Polarski, Emergence of classicality for primordial fluctuations: Concepts and analogies, Annalen Phys. 7 (1998) 137 [gr-qc/9805014] [INSPIRE].
F.C. Lombardo and D. Lopez Nacir, Decoherence during inflation: The Generation of classical inhomogeneities, Phys. Rev. D 72 (2005) 063506 [gr-qc/0506051] [INSPIRE].
J.W. Sharman and G.D. Moore, Decoherence due to the Horizon after Inflation, JCAP 11 (2007) 020 [arXiv:0708.3353] [INSPIRE].
A.H. Guth and S.-Y. Pi, The Quantum Mechanics of the Scalar Field in the New Inflationary Universe, Phys. Rev. D 32 (1985) 1899 [INSPIRE].
D. Polarski and A.A. Starobinsky, Semiclassicality and decoherence of cosmological perturbations, Class. Quant. Grav. 13 (1996) 377 [gr-qc/9504030] [INSPIRE].
L.P. Grishchuk and Y.V. Sidorov, Squeezed quantum states of relic gravitons and primordial density fluctuations, Phys. Rev. D 42 (1990) 3413 [INSPIRE].
A. Albrecht, P. Ferreira, M. Joyce and T. Prokopec, Inflation and squeezed quantum states, Phys. Rev. D 50 (1994) 4807 [astro-ph/9303001] [INSPIRE].
S. Habib, Stochastic inflation: The Quantum phase space approach, Phys. Rev. D 46 (1992) 2408 [gr-qc/9208006] [INSPIRE].
J. Weenink and T. Prokopec, On decoherence of cosmological perturbations and stochastic inflation, arXiv:1108.3994 [INSPIRE].
L. Perreault Levasseur, Lagrangian formulation of stochastic inflation: Langevin equations, one-loop corrections and a proposed recursive approach, Phys. Rev. D 88 (2013) 083537 [arXiv:1304.6408] [INSPIRE].
C.P. Burgess, R. Holman, G. Tasinato and M. Williams, Open EFTs: Effective Field Theories Without Effective Lagrangians, in preparation.
C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Atom Photon Interactions, Wiley, New York (1992).
V.F. Sears, Neutron Optics, Oxford University Press (1989).
H. Haken, The Semiclassical and Quantum Theory of the Laser, in Quantum Optics: Proceedings of the Tenth Session of the Scottish Universities Summer School in Physics, 1969, S.M. Kay and A. Maitland eds., Academic Press (1970).
C.P. Burgess, R. Holman and D. Hoover, Decoherence of inflationary primordial fluctuations, Phys. Rev. D 77 (2008) 063534 [astro-ph/0601646] [INSPIRE].
C. Kiefer, I. Lohmar, D. Polarski and A.A. Starobinsky, Pointer states for primordial fluctuations in inflationary cosmology, Class. Quant. Grav. 24 (2007) 1699 [astro-ph/0610700] [INSPIRE].
C.P. Burgess and D. Michaud, Neutrino propagation in a fluctuating sun, Annals Phys. 256 (1997) 1 [hep-ph/9606295] [INSPIRE].
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].
S. Chaturvedy and F. Shibata, ??, Z. Phys. B 35 (1979) 297.
J.R. Anglin and W.H. Zurek, Decoherence of quantum fields: Pointer states and predictability, Phys. Rev. D 53 (1996) 7327 [quant-ph/9510021] [INSPIRE].
N.C. Tsamis and R.P. Woodard, Stochastic quantum gravitational inflation, Nucl. Phys. B 724 (2005) 295 [gr-qc/0505115] [INSPIRE].
M. Sasaki, Large Scale Quantum Fluctuations in the Inflationary Universe, Prog. Theor. Phys. 76 (1986) 1036 [INSPIRE].
V.F. Mukhanov, Quantum Theory of Gauge Invariant Cosmological Perturbations, Sov. Phys. JETP 67 (1988) 1297 [Zh. Eksp. Teor. Fiz. 94N7 (1988) 1] [INSPIRE].
V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203 [INSPIRE].
R.H. Brandenberger, Lectures on the theory of cosmological perturbations, Lect. Notes Phys. 646 (2004) 127 [hep-th/0306071] [INSPIRE].
C. Schomblond and P. Spindel, ??, Ann. Inst. Henri Poincaré 25A (1976) 67.
T.S. Bunch and P.C.W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117 [INSPIRE].
P. Candelas and D.J. Raine, General Relativistic Quantum Field Theory-An Exactly Soluble Model, Phys. Rev. D 12 (1975) 965 [INSPIRE].
C.P. Burgess and C.A. Lütken, Propagators and Effective Potentials in Anti-de Sitter Space, Phys. Lett. B 153 (1985) 137 [INSPIRE].
L.H. Ford and A. Vilenkin, Global Symmetry Breaking in Two-dimensional Flat Space-time and in de Sitter Space-time, Phys. Rev. D 33 (1986) 2833 [INSPIRE].
D. Boyanovsky, H.J. de Vega and R. Holman, Nonequilibrium evolution of scalar fields in FRW cosmologies I, Phys. Rev. D 49 (1994) 2769 [hep-ph/9310319] [INSPIRE].
F. Finelli, G. Marozzi, A.A. Starobinsky, G.P. Vacca and G. Venturi, Stochastic growth of quantum fluctuations during slow-roll inflation, Phys. Rev. D 82 (2010) 064020 [arXiv:1003.1327] [INSPIRE].
P. Bamert, C.P. Burgess and D. Michaud, Neutrino propagation through helioseismic waves, Nucl. Phys. B 513 (1998) 319 [hep-ph/9707542] [INSPIRE].
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Burgess, C.P., Holman, R., Tasinato, G. et al. EFT beyond the horizon: stochastic inflation and how primordial quantum fluctuations go classical. J. High Energ. Phys. 2015, 90 (2015). https://doi.org/10.1007/JHEP03(2015)090
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DOI: https://doi.org/10.1007/JHEP03(2015)090