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7
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2142719004
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For an early attempt to apply anthropic arguments to the cosmological constant, see also P. C. W. Davis and S. Unwin, Proc. R. Soc. London A377, 147 (1981).
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(1981)
Proc. R. Soc. London
, vol.377
, pp. 147
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Davis, P.C.W.1
Unwin, S.2
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13
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85037239272
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A similar argument for the coincidence of the present time (Formula presented) with the curvature domination time (Formula presented) in an open universe was given in
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A similar argument for the coincidence of the present time (Formula presented) with the curvature domination time (Formula presented) in an open universe was given in 11.
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17
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85037198402
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The expansion rate during inflation can be written as (Formula presented), where (Formula presented). The (Formula presented)-dependent correction to H will tend to distort the flat distribution, favoring larger values of (Formula presented). This effect becomes important on a time scale (Formula presented). On the other hand, quantum “diffusion” of (Formula presented) will tend to keep the distribution flat. Quantum fluctuations of (Formula presented) can be pictured as a random walk of step (Formula presented) per Hubble time (Formula presented), with steps taken independently in each horizon-size region (Formula presented). The typical variation of (Formula presented) due to this quantum random walk in a time interval (Formula presented) is (Formula presented), where (Formula presented) is the number of steps during the interval (Formula presented). The back reaction can be neglected if (Formula presented) is much greater than the relevant interval of (Formula presented). This gives (Formula presented)Consistency with Eq. (13) requires (Formula presented), which is satisfied with a wide margin for anthropically allowed values of (Formula presented) and all acceptable values of the inflationary expansion rate H. Hence, there is a wide range of slopes of the potential for which the back reaction effects are negligible. However, for sufficiently small (Formula presented), the above condition is violated, and the a priori probability (Formula presented) should grow faster with (Formula presented) than indicated by Eq. (19)
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The expansion rate during inflation can be written as (Formula presented), where (Formula presented). The (Formula presented)-dependent correction to H will tend to distort the flat distribution, favoring larger values of (Formula presented). This effect becomes important on a time scale (Formula presented). On the other hand, quantum “diffusion” of (Formula presented) will tend to keep the distribution flat. Quantum fluctuations of (Formula presented) can be pictured as a random walk of step (Formula presented) per Hubble time (Formula presented), with steps taken independently in each horizon-size region (Formula presented). The typical variation of (Formula presented) due to this quantum random walk in a time interval (Formula presented) is (Formula presented), where (Formula presented) is the number of steps during the interval (Formula presented). The back reaction can be neglected if (Formula presented) is much greater than the relevant interval of (Formula presented). This gives (Formula presented)Consistency with Eq. (13) requires (Formula presented), which is satisfied with a wide margin for anthropically allowed values of (Formula presented) and all acceptable values of the inflationary expansion rate H. Hence, there is a wide range of slopes of the potential for which the back reaction effects are negligible. However, for sufficiently small (Formula presented), the above condition is violated, and the a priori probability (Formula presented) should grow faster with (Formula presented) than indicated by Eq. (19).
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20
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0542371327
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For a discussion and references see A. Vilenkin, Phys. Rev. D 58, 067301 (1998).
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(1998)
Phys. Rev. D
, vol.58
, pp. 67301
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Vilenkin, A.1
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31
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85037186440
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The function (Formula presented) can be found as an eigenvalue of the Fokker-Planck operator, as discussed in Refs
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The function (Formula presented) can be found as an eigenvalue of the Fokker-Planck operator, as discussed in Refs. 2627. With a fixed-time cutoff, as in Eq. (43), this function depends on the choice of the time variable t 2728. This gauge dependence is usually regarded as a problem and some attempts have been made to define gauge-independent probabilities in the case of a single eternally inflating universe 29. It appears, however, that in the approach we adopted here for an ensemble of universes it would be unreasonable to require gauge-independence. The quantity (Formula presented) characterizes the rate of growth of the thermalized volume. Clearly, it depends on the time variable with respect to which the rate is calculated. The most natural choice appears to be the proper time which we have used throughout the paper. It is conceivable, however, that there may be other consistent and reasonable approaches which may give different answers for the probability distribution in the case of disconnected universes. Investigation of this issue is left for further research.
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32
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0000986201
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H. J. de Vega, N. Sanchez
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A. A. Starobinsky, in Current Topics in Field Theory, Quantum Gravity and Strings, edited by H. J. de Vega and N. Sanchez, Lecture Notes in Physics Vol. 246 (Springer, Heidelberg, 1986), pp. 107–126.
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(1986)
Current Topics in Field Theory, Quantum Gravity and Strings
, pp. 107-126
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Starobinsky, A.A.1
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42
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85037248485
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astro-ph/9908023
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L. Amendola, astro-ph/9908023.
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Amendola, L.1
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