EFT beyond the horizon: stochastic inflation and how primordial quantum fluctuations go classical

Journal of High Energy Physics, Mar 2015

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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EFT beyond the horizon: stochastic inflation and how primordial quantum fluctuations go classical

Received: September EFT beyond the horizon: stochastic inflation and how primordial quantum fluctuations go classical 0 Department of Physics, Swansea University 1 Waterloo, ON , N2L 2Y5 Canada 2 Hamilton , ON, L8S 4M1 Canada 3 Physics Department, Carnegie Mellon University 4 ICG, University of Portsmouth 5 Perimeter Institute for Theoretical Physics 6 Division PH -TH , CERN 7 Physics & Astronomy, McMaster University 8 Open Access , c The Authors 9 Vivian Tower , Swansea, SA2 8PP United Kingdom 10 Pittsburgh , PA, 15213 U.S.A 11 Gen`eve 23 , CH-1211 Suisse 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, cs < 1.) We then argue that the presence of interactions drive the off-diagonal density-matrix elements to zero in the field primordial; quantum; fluctuations; go; classical - basis. This shows why the field basis is generally the pointer basis for the process that decoheres primordial quantum fluctuations while they are outside the horizon, thus allowing them to re-enter later as classical field fluctuations, as assumed when analyzing CMB data. The decoherence process is very efficient, occurring after several Hubble times even for interactions as weak as gravitational-strength. Crucially, the details of the interactions largely control only the decoherence time and not the nature of the final late-time stochastic state, much as interactions can control the equilibration time for thermal systems but are largely irrelevant to the properties of the resulting equilibrium state. ArXiv ePrint: 1408.5002 1 Introduction Open EFTs 2 3 A hierarchy of scales for open systems The Lindblad equation Scalar fields on de Sitter space Time dependence Make some noise! Getting the drift Mass-dependent noise 4 Interactions and decoherence Generalization to FRW Geometries Solutions Decoherence Rates Summary and other possible applications Summary of the argument Future directions: IR resummations, secular behaviour and black holes A Solving for time dependence Gaussian facts Introduction The advent of precision CMB cosmology reveals the Universe to be a somewhat lumpy place whose present crags and wrinkles partly reflect an earlier accelerated lifestyle. Cosmologists infer properties of this earlier epoch much as one might try to guess about past excesses by gazing on the features of one long past the sowing of wild oats. In particular, evidence continues to build that the right explanation for present-epoch super-Hubble correlations lies with quantum fluctuations generated during a much-earlier epoch of accelerated expansion. A common feature of such explanations is that quantum fluctuations pass to super-Hubble scales in the remote past and then re-enter as classical fluctuations after spending a lengthy period frozen beyond the Hubble pale. This kind of picture raises two related and oft-considered issues: 1. What effective theory describes long-wavelength physics in the super-Hubble regime? 2. Why do quantum fluctuations re-enter the Hubble scale as classical distributions? The first of these issues starts with the observation that for most physical systems the long-wavelength limit is usually most efficiently described by a Wilsonian effective field theory (EFT), obtained by integrating out shorter-wavelength modes.1 Since super-Hubble modes have the longest wavelengths of all, one is led to ask what field theory provides its effective description.2 Such an effective description might allow a cleaner understanding of the various thorny infrared issues faced by quantum fields on de Sitter space [1724, 4954]. The second issue asks why fluctuations that are initially described (at horizon exit) interpretable (at horizon re-entry) in terms of an ensemble average of classic (...truncated)


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C. P. Burgess, R. Holman, G. Tasinato, M. Williams. EFT beyond the horizon: stochastic inflation and how primordial quantum fluctuations go classical, Journal of High Energy Physics, 2015, pp. 90, Volume 2015, Issue 3, DOI: 10.1007/JHEP03(2015)090