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De Sitter Space Without Dynamical Quantum Fluctuations

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Abstract

We argue that, under certain plausible assumptions, de Sitter space settles into a quiescent vacuum in which there are no dynamical quantum fluctuations. Such fluctuations require either an evolving microstate, or time-dependent histories of out-of-equilibrium recording devices, which we argue are absent in stationary states. For a massive scalar field in a fixed de Sitter background, the cosmic no-hair theorem implies that the state of the patch approaches the vacuum, where there are no fluctuations. We argue that an analogous conclusion holds whenever a patch of de Sitter is embedded in a larger theory with an infinite-dimensional Hilbert space, including semiclassical quantum gravity with false vacua or complementarity in theories with at least one Minkowski vacuum. This reasoning provides an escape from the Boltzmann brain problem in such theories. It also implies that vacuum states do not uptunnel to higher-energy vacua and that perturbations do not decohere while slow-roll inflation occurs, suggesting that eternal inflation is much less common than often supposed. On the other hand, if a de Sitter patch is a closed system with a finite-dimensional Hilbert space, there will be Poincaré recurrences and dynamical Boltzmann fluctuations into lower-entropy states. Our analysis does not alter the conventional understanding of the origin of density fluctuations from primordial inflation, since reheating naturally generates a high-entropy environment and leads to decoherence, nor does it affect the existence of non-dynamical vacuum fluctuations such as those that give rise to the Casimir effect.

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Notes

  1. See [23] for a discussion on the difficulties of measurements in a finite-dimensional asymptotic de Sitter space if a measuring device were indeed present.

  2. For the purposes of this paper, we are concerned with only the von Neumann entropy from entanglements. There is also the thermodynamic entropy associated with a mixed thermal density matrix, which sets an upper bound on the von Neumann entropy. As the quantum system thermalizes, the von Neumann entropy approaches the thermodynamic entropy [39].

  3. Even in stationary states, one can define an effective evolution with respect to correlations with a clock subsystem [40]. The effective time parameter \(\tau \) has nothing to do with the ordinary coordinate time t; all such time evolutions are present at every moment of (ordinary) time. From this perspective, a large number of Boltzmann brains and similar fluctuations actually exist at every moment in an apparently stationary spacetime. Such a conclusion would apply to Minkowski spacetime as well as to de Sitter, in conflict with the conventional understanding that dynamical fluctuations in de Sitter depend on the Gibbons–Hawking temperature (but see [41, 42]). This kind of effective evolution is fundamentally different from the ordinary evolution studied in this paper.

  4. We thank Alan Guth, Charles Bennett, and Jess Riedel for discussions on this point.

  5. We do not consider the massless case, since there is no (vacuum) state that is invariant under the full de Sitter group [47], which is problematic for the cosmic no-hair theorem in Sect. 3.2. However, if one assumes the shift invariance of the massless scalar field is just a global gauge transformation, then a fully de Sitter invariant vacuum can in fact be defined [48].

  6. Without the Hadamard condition [49], there are a continuum of de Sitter-invariant states, known as the \(\alpha \) vacua, which are related to one another via Bogoliubov transformations [47].

  7. As previously mentioned, the massless case is problematic, since there is no de Sitter-invariant vacuum in the noninteracting limit [47]. With nonvanishing interactions, correlation functions of the field at large timelike separations grow no faster than a polynomial function of \(H\tau \) at the perturbative level [57]. There is, however, evidence at one and two loops that the 2-point correlation function decays as a polynomial of H [57–59].

  8. For subtleties involving the use of the static Hamiltonian in quantum gravity, see [66].

  9. We thank Stefan Leichenauer and Paul Steinhardt for discussions of these issues.

  10. For related work that questions the validity of Boltzmann brains for decoherence-based reasons, see [42, 77, 78]. For the need for Hilbert space to be infinite-dimensional, see [79].

  11. One might imagine that decoherence occurs because modes become super-Hubble-sized, and we should trace over degrees of freedom outside the horizon. This reasoning is not quite right, as such modes could (and often do) later re-enter the observable universe; they become larger than the Hubble radius during inflation but never leave the true horizon.

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Acknowledgments

We have benefited from helpful discussions with Scott Aaronson, Charles Bennett, Alan Guth, James Hartle, Stefan Leichenauer, Spyridon Michalakis, Don Page, John Preskill, Jess Riedel, Charles Sebens, Paul Steinhardt, and several participants at the Foundational Questions Institute conference on The Physics of Information (though they might not agree with our conclusions). This research is funded in part by DOE Grant DE-FG02-92ER40701 and by the Gordon and Betty Moore Foundation through Grant 776 to the Caltech Moore Center for Theoretical Cosmology and Physics.

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  1. Physics Department, California Institute of Technology, Pasadena, CA, 91125, USA

    Kimberly K. Boddy, Sean M. Carroll & Jason Pollack

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Boddy, K.K., Carroll, S.M. & Pollack, J. De Sitter Space Without Dynamical Quantum Fluctuations. Found Phys 46, 702–735 (2016). https://doi.org/10.1007/s10701-016-9996-8

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