Master's Thesis in Theoretical Physics - Universiteit Utrecht

value of ξ. With the explicit expression for the Ricci scalar in Eq. (4.94) the field equationbecomes,∑[k=1,2 ξ ik φ k¨φ i + 3H ˙φi +M 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ −2∑](δ lm − 6ξ lm )( ˙φ∗lφ˙m ) =kl(δ lm − 6ξ lm )φ m l,m=1,2∑[k=1,2 ξ ik φ kM 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ 6 ∑ξ lm [φ ∗ ∂Vlkl(δ lm − 6ξ lm )φ m l,m=1,2∂φ ∗ + ∂V]φ m ] − 4V − ∂Vm ∂φ l ∂φ ∗ (4.95) ,iwhere V = V (φ 1 ,φ 2 ). We have written the potential terms on the righthand side of theequation of motion. In the slow-roll approximation these potential terms dominate over thekinetic terms. As in the single field case, we assume the slow-roll conditions (4.12). Thisbasically means we neglect the terms proportional to ¨φ and ˙φ2 but keep the term 3H ˙φi andthe potential terms. Thus in the slow-roll approximation the field equation becomes3H ˙φi ≈∑k=1,2 ξ ik φ kM 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ kl(δ lm − 6ξ lm )φ m[6 ∑l,m=1,2ξ lm [(φ ∗ ∂Vl∂φ ∗ m+ ∂V ]φ m )] − 4V − ∂V∂φ l ∂φ ∗ .iThen we substitute this expression into the energy constraint equation (4.91) and also usethe slow-roll approximation to findH 2≈1{3(M 2 p − 2 ∑ 2i, j=1,2 ξ i j φ ∗ i φ V +j) M 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ (4.96)kl(δ lm − 6ξ lm )φ m×[(M 2 p − 2 ∑ )(φ ∗ k ξ klφ l −∑ξ i j [φ ∗ ∂V ∂V)ik,l=1,2i, j=1,2∂φ ∗ + φ j ] − 8V ∑φ ∗ i∂φ ξ i jξ jk φ k]}.j i i, j,k=1,2If we now substitute the potential (4.72) and take the large negative nonminimal couplinglimit ξ ii ≪ 0, we find a complicated expression for H 2 and the equations of motion. However,we can solve these equations numerically, which we will do in the next section.4.4.2 Numerical solutions of the field equationsIn Fig. 4.7 we have numerically calculated solutions for the fields φ 1 and φ 2 from the fieldequation (4.95). As a matter of simplicity we have chosen the fields and nonminimal couplingsto be real. All the nonminimal couplings have typical values ξ ii ∼ −10 3 and thequartic couplings λ i ∼ 10 −3 . One interesting observation is that inflation does not happenimmediately, i.e. the fields do not roll down from their initial value to their minimumstraight away. Instead, they interact (through the interaction terms with couplings ξ 12 , ξ 21and λ 3 ) and start to oscillate rapidly. This is shown in Fig. 4.8. After a short time the oscillationsare damped out and the fields acquire a "fixed" value. Then the fields do roll down totheir minimum and inflation works. The numerical solutions for φ 1 and φ 2 in Fig. 4.7 showthat during inflation the fields are proportional to each other. This suggest that we can takeone linear combination of the two fields that is the inflaton. The other linear combination isan oscillating mode that quickly decays. This could then be a source for particle productionand perhaps baryogenesis.In Fig. 4.9a we show the growth of the number of e-folds in time. For the typical nonminimalmodel as mentioned above, the maximum number of e-folds that is reached at the endof inflation is N ≃ 1200. The condition necessary to solve the flatness and horizon puzzlesis N ≥ 70, so this condition is easily satisfied. We could therefore relax the initial values ofthe fields or the values of the nonminimal couplings. Thus, our nonminimal couplings couldbe much smaller, or the initial values of the fields could be lower. The Hubble parameter