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
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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)