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Σ 0 y t 0 if we neglect those terms in the

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σ0.y[t] = 0.If we neglect those terms in the coefficients of the perturbations which tend to zero att→ ∞, thek- dependence disappears.Thus, we have learned two lessons from the analysis of the massless case.It is sufficient toconsider the case Λ = 0 andk= 0, for this does not modify the stability of the solution. Thesestatements can be used in the more realistic theory of inflation based on the effective action forthe massive fields, but with a proper caution. In fact, the negligible role of Λ depends on that it isa constant. As we shall see in the next section, this is not true for the effective action of massivefields.Until now, we have considered only the stable inflationary solution withc >0. In other words,we were interested in the high energy region where the supersymmetry is unbroken.Obviously,another end of the energy scale also represents a great interest, for it enables one to perform asimple and efficient test of the model. Let us consider, again, a nowaday universe withH=H010-42GeV. According to the recent data [16], this magnitude of the Hubble constant is mainly dueto the contribution of the cosmological constant. As we have already mentioned in the last section,the value ofcin the present-day universe is negative due to the contribution of the single masslessparticle - photon. Therefore, we have to check whether the higher derivative terms do not destroythe stability of the first solution in Eq. (24). Using (48), we arrive at the following characteristicequation for the perturbation ofHH+const·exp(λt).λ3+ 72+4 3-bcH2-M2P8πcλ-16bcH3+M2P2πcH= 0.(50)Since the explicit solution of this equation is tedious, let us start from the Λ = 0 case. ThenH= 0and the roots of the equation (50) areλ(0)1= 0,λ(0)2/3=±MPp8π|c|i .(51)At this level, we do not have definite answer to our question about stability. Now, let us look forthe solution of Eq. (50) making perturbations in (51). Due to the huge difference betweenHandMP, this approximation is perfect. The result isλ1=-4H ,λ2/3=-32H±MPp8π|c|i ,whereH=rΛ3>0.(52)It is very nice to see that the cosmological constant Λ>0 really stabilizes the solution in thelow-energy region, exactly as we should optimistically expect. The stability can be verified also for15
the earlier post-inflationary epoch, when the energy density of vacuum played smaller role than thedensity of radiation [25, 23]. For the later epoch, as we have already noticed, when we substitutethe FRW solution into Eq. (17), the “quantum” terms decrease as 1/a2compared to the Einstein-Hilbert and matter terms. Therefore, for the later epoch of the expanding post-inflationary universethe FRW is a very good approximation.4The stability in the interpolation regimeAt the very beginning of the inflationary period, the evolution goes exactly as in the masslesscase and the condition of stability must be the same (25).Later on, whenσ(t) becomes largeenough, the deviation from the exponential inflation becomes greater. Finally, the inflation slowsdown such that the Hubble parameter achieves the scaleM*, and the massive sparticles decouple.

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Term
Fall
Professor
NoProfessor
Tags
Quantum Field Theory, General Relativity, Big Bang, Physical cosmology

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