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3 inflationary background in this section we shall

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3Inflationary backgroundIn this section we shall consider the inflationary solution for the dynamical equations of thetheory with the actionStotal=M2Pintegraldisplayd4xgR+¯Γ,(10)whereM2P= 1/16πGis the square of the Planck mass, and the quantum correction¯Γ istaken in the form (8). In what follows we setSc[gμν] = 0, including to it, when it is notindicated explicitly, also the classical vacuum term.7
Since we are going to look for the isotropic and homogeneous solution, the starting pointis to choose the metric in the formgμν=a2(η) ¯gμν, whereηis conformal time. It provesuseful to denote, as before,σ= lna. Now, one has to derive the equations for the three fields:ϕ,ψ,andσ. In the rest of this section we shall consider the conformally flat backgroundand thus set ¯gμν=ημν.The equations forϕandψhave especially simple formgbracketleftBiggΔ4ϕ+b8π(E23R)w8πbC2bracketrightBigg= 0,gbracketleftBiggΔ4ψw8πbC2bracketrightBigg= 0.One has to remind the transformation law for the quantities which enter the last expression:gC2=¯g¯C2,gΔ4=¯g¯Δ4,(11)g(E23R) =¯g(¯E23¯¯R+ 4¯Δ4σ).(12)Taking into account our choice for the fiducial metric ¯gμν=ημν, one arrives at the followingequations in flat space-time2ϕ+b2π2σ= 0,2ψ= 0.(13)The solutions of (13) can be presented in the formϕ=b2πσ+ϕ0,ψ=ψ0.(14)whereϕ00are general solutions of the homogeneous equations2ϕ0= 0,2ψ0= 0.Thus one meets an arbitrariness related to the choice of the initial conditions for the auxiliaryfieldsϕ,ψ. But, the inflationary solution does not depend onϕ00. Substituting (14) backinto the action and taking variation with respect toσwe arrive at the same equation forσthat follows directly from (6). We shall write this equation in terms of the physical timet,defined, as usual, througha(η)=dt. The useful variable isH(t) = ˙a(t)/a(t) = ˙σ(t), sincethe equation (completely equivalent to the one of [7]) is of the third order in this variable:...H+ 7..HH+ 4parenleftBigg1 +3bcparenrightBigg.HH2+ 4.H2+4bcH42M2PcparenleftBigH2+.HparenrightBig= 0.(15)It is easy to identify the special solution corresponding toH=const:H=±MPb,a(t) =a0·expHt.(16)8
Positive sign corresponds to inflation. The solution (16) had been first discovered in [4, 5]and studied in [4, 6]7.The detailed analysis shows [4, 7] that the special solution (16) is stable with respectto the variations (not necessary small) of the initial data fora(t), if the parameters of theunderlying quantum theory satisfy the conditionbc>0, that leads, according to (2), to therelationN1<13N1/2+118N0.(17)This constraint is not satisfied for the Minimal Standard Model (MSM) withN1= 12,N1/2= 24 andN0= 4. However, one can consider some aspects of the neutrino oscillationsas an indication that the MSM should be extended, and in this case the inequality (17) canbe readily satisfied. Below we consider two versions, each of which leads to stable inflation.

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