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5
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84927220700
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For a brief summary, see A. H. Guth, in Eleventh Texas Symposium on Relativistic Astrophysics, edited by D. S. Evans
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(1984)
Ann. N.Y. Acad. Sci.
, vol.422
, pp. 1
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17
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84927220682
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In the early models, the scalar field which drives the inflation was taken to be the same Higgs field which spontaneously breaks the grand unified gauge symmetry. However, it has been found (Ref. 5) that in order for the mass-density fluctuations to be small enough, the scalar field which drives inflation must have an extraordinarily flat potential-energy function. For this reason, most of the currently acceptable models (Ref. 45) introduce a new weakly coupled gauge-singlet scalar field for the purpose of driving the inflation. Ovrut and Steinhardt (Ref. 46) have proposed a model in which this field also serves the purpose of breaking supersymmetry, and Pi (Ref. 47) has suggested that inflation could be driven by the complex gauge-singlet field which gives rise to the axion.
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18
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84927220680
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A preliminary version of this material appeared as A. H. Guth and S. -Y. Pi, in Proceedings of the Inner Space/Outer Space Conference, Fermilab (Ref. 4); also S.-Y. Pi, in Proceedings of the Inner Space/Outer Space Conference, Fermilab (Ref. 4).
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19
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84927220678
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The false vacuum value of phi is equal to zero in most models, but any other value would do just as well.
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20
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84927220677
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The properties of a de Sitter space are well described in S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, England, 1973).
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30
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84927220675
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See, for example, G. Börner and E. Seiler, Max-Planck-Institut report, 1983 (unpublished);
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33
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84927220674
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See also the articles by Brandenberger in Particles and Gravity: Proceedings of the 8th Johns Hopkins Workshop on Current Problems in, edited by, G. Domokos, S. Kovesi-Domokos, World Scientific, Singapore
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(1984)
Particle Theory
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34
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84927220673
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in Proceedings of the Inner Space/Outer Space Conference, Fermilab (Ref. 4).
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43
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84927220658
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A rather thorough discussion of the upside-down harmonic oscillator is given by G. Barton, Ann. Phys. (N.Y.) (to be published). He does not, however, discuss the classical nature of the solution at large times.
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44
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84927220657
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When we say that a function f(x,t) is O ( g(t) ), we mean that | f( x,t)/ g(t) | is bounded for each x as t -> inf.
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49
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84927220656
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For the benefit of newcomers to de Sitter space, we point out that the expansion of this space provides an event horizon. If two observers that are comoving in the coordinate system described by (3.1) are separated at a given time by a physical distance χ-1 (i.e., a coordinate distance χ-1e-χ t ), then a light pulse emitted by one observer at the given time will never reach the other observer.
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53
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84927220655
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The much weaker coupling for our case will mean that thermalization does not have time to occur.
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55
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84927220654
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We thank Jim Peebles for calling our attention to this fact.
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56
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84927220653
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Guth and Pi (Ref. 5).
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57
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84927220651
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It is demonstrated in a footnote in Ref. 37 that this limit exists for arbitrary V( φ ). The key point is that for large times δ φ ( x vec ,t) and φ dot0(t) obey the same second-order differential equation. One of the two linearly independent solutions dominates at large times, so both δ phi and φ dot0 share the same large-time asymptotic behavior.
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58
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84927220649
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Note that Eq. (3.6) of Ref. 33 is misprinted, and should read ψk( η ) = α-1( π /4)1/2η3/2Hν(2)( k η ).
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59
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84927220640
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This matching condition, used in Ref. 37, is roughly equivalent to the frequently used rule of thumb that δ phi is of order chi when the wavelength equals the Hubble length.
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60
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84927220639
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Note that our expression differs from that of Ref. 33 in two compensating ways: the Hankel functions Hp(1)(z) and Hp(2)(z) are reversed, and so is the sign of z. We have chosen our conventions to avoid using the Hankel functions on the negative real axis, where they are usually defined to have a branch cut.
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61
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Note that the wave function (B7) can also be obtained quite easily by the operator methods of Sec. III. Using Eq. (3.30) for the annihilation operator aalpha and (3.22) for πα== -i hbar partial / partial σalpha, one can find PSIα( σα,t) by solving the equaton aα| PSIα> =0.
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79
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84927220637
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Ovrut and Steinhardt (Ref. 45).
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80
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84927220636
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Pi (Ref. 45).
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