Master's Thesis in Theoretical Physics - Universiteit Utrecht

of the field and expand the action up to quadratic order in fluctuations, which allows us towrite the action for the quantum fluctuations∫S Q = d 4 x −g(− ∑g µν (∂ µ δφ i ) ∗ (∂ ν δφ i ) −∑ξ i j Rδφ ∗ i δφ j−∑i, j=1,2i=1,2i, j=1,2[δφ ∗ ∂Vi∂φ ∗ i ∂φ δφ j + 1j 2 δφ ∂Vi δφ j + 1 ]∂V )∂φ i ∂φ j 2 δφ∗ i∂φ ∗ δφ ∗i ∂φ∗ j,jwhere φ i (x) = φ 0 i (t)+δφ i(x, t), with φ 0 i (t) the classical inflaton field and δφ i(x, t) the quantumfluctuation. The potential terms might suggest that we have some nontrivial terms ∝(δφ ∗ ) 2 and ∝ δφ 2 . These terms are in fact real, because our potential is also real, i.e. itcontains only terms φ ∗ i φ i. We can also write these terms as ∝ δφ ∗ δφ by writing the complexfield φ = |φ|exp(iθ) and move the phase into the potential derivatives. The action we cantherefore write as∫S Q = d 4 x −g(− ∑g µν (∂ µ δφ i ) ∗ (∂ ν δφ i ) −∑ξ i j Rδφ ∗ i δφ j −∑)δφ ∗ i m2 i j δφ j , (5.109)i=1,2i, j=1,2i, j=1,2wherem 2 i j = 2 ∂V∂φ ∗ i ∂φ . (5.110)jNote that (m 2 i j )∗ = m 2 . The mass term (5.110) contains only the classical inflaton fieldsjiφ 0 because higher order fluctuation terms vanish. So we can say that the classical inflatonifields φ 0 i effectively generate a mass for the quantum fluctuations δφ i. From now on we willomit the δ’s in the quantum fields for notational convenience. Since we only work with thequantum fluctuations in this section, we do not experience any notational problems. Justremember that from this point on the fields φ i are actually quantum fluctuations δφ i .We can write the quantum action (5.109) in matrix notation as∫S Q =d 4 x ()−g −g µν ∂ µ Φ † ∂ ν Φ − RΦ † ΞΦ − Φ † M 2 Φ , (5.111)where( ) ( ) (φ1ξ11 ξ 12Φ = , Ξ =φ 2 ξ ∗ , M 2 =12ξ 22m 2 11m 2 12(m 2 12 )∗ m 2 22). (5.112)The field vector Φ contains the quantum fluctuations φ i . The matrices nonminimal couplingmatrix and the mass matrix are hermitean, i.e. Ξ † = Ξ, (M 2 ) † = M 2 .Let us now derive the field equations for the quantum fluctuations from the action (5.111).This gives−□Φ + (RΞ + M 2 )Φ = 0.The field equation for the complex conjugate fields φ ∗ is obtained by complex conjugation ofithis equation. Substituting the conformal FLRW metric g µν = a 2 (η)η µν , we get1a 2 (a 2 Φ ′) ′− ∇ 2 Φ + a 2 (RΞ + M 2 )Φ = 0.Again we can do a conformal rescaling of the fields to get(∂ 2 η − ∇2 + a 2[ R(Ξ − 1 6 ) + M2]) (aΦ) = 0. (5.113)