3.2 Dynamics in the Einstein frame

Let us consider inflationary dynamics in the Einstein frame for the model (3.6View Equation) in the absence of matter fluids (ℒM = 0). The action in the Einstein frame corresponds to (2.32View Equation) with a field ϕ defined by
∘ -- ∘ -- ( ) 31 31 R ϕ = 2κ-lnF = 2κ-ln 1 + 3M--2 . (3.15 )
Using this relation, the field potential (2.33View Equation) reads [408Jump To The Next Citation Point6163]
2 ( √--- ) V (ϕ) = 3M--- 1 − e− 2∕3κϕ 2 . (3.16 ) 4κ2
View Image

Figure 1: The field potential (3.16View Equation) in the Einstein frame corresponding to the model (3.6View Equation). Inflation is realized in the regime κ ϕ ≫ 1.

In Figure 1View Image we illustrate the potential (3.16View Equation) as a function of ϕ. In the regime κϕ ≫ 1 the potential is nearly constant (V(ϕ ) ≃ 3M 2∕(4κ2)), which leads to slow-roll inflation. The potential in the regime κϕ ≪ 1 is given by V (ϕ) ≃ (1∕2)M 2ϕ2, so that the field oscillates around ϕ = 0 with a Hubble damping. The second derivative of V with respect to ϕ is

2 −√2-∕3κϕ ( −√2-∕3κϕ) V,ϕϕ = − M e 1 − 2e , (3.17 )
which changes from negative to positive at ∘ ---- ϕ = ϕ1 ≡ 3∕2(ln2)∕κ ≃ 0.169mpl.

Since F ≃ 4H2 ∕M 2 during inflation, the transformation (2.44View Equation) gives a relation between the cosmic time &tidle;t in the Einstein frame and that in the Jordan frame:

∫ [ ] t√ -- 2 M 2 2 &tidle;t = F dt ≃ --- Hi (t − ti) −---(t − ti) , (3.18 ) ti M 12
where t = ti corresponds to &tidle;t = 0. The end of inflation (tf ≃ ti + 6Hi ∕M 2) corresponds to &tidle;tf = (2∕M )N in the Einstein frame, where N is given in Eq. (3.13View Equation). On using Eqs. (3.10View Equation) and (3.18View Equation), the scale factor √ -- &tidle;a = F a in the Einstein frame evolves as
( M 2 ) &tidle;a (t&tidle;) ≃ 1 − ----2M &tidle;t &tidle;aieM&tidle;t∕2, (3.19 ) 12H i
where &tidle;ai = 2Hiai∕M. Similarly the evolution of the Hubble parameter √-- &tidle;H = (H∕ F )[1 + ˙F∕(2HF )] is given by
[ 2 ( 2 ) −2] H&tidle;(&tidle;t) ≃ M-- 1 − -M--- 1 − -M---M &tidle;t , (3.20 ) 2 6H2i 12H2i
which decreases with time. Equations (3.19View Equation) and (3.20View Equation) show that the universe expands quasi-exponentially in the Einstein frame as well.

The field equations for the action (2.32View Equation) are given by

[ ( )2 ] 3H&tidle;2 = κ2 1- dϕ- + V (ϕ) , (3.21 ) 2 d&tidle;t 2 d-ϕ-+ 3 &tidle;H dϕ-+ V,ϕ = 0. (3.22 ) d &tidle;t2 d &tidle;t
Using the slow-roll approximations (dϕ ∕d&tidle;t)2 ≪ V (ϕ) and |d2ϕ ∕d&tidle;t2| ≪ | &tidle;Hd ϕ∕d &tidle;t| during inflation, one has &tidle;2 2 3H ≃ κ V (ϕ) and &tidle; 3H (d ϕ∕d&tidle;t) + V,ϕ ≃ 0. We define the slow-roll parameters
( )2 2 2 &tidle;𝜖 ≡ − dH&tidle;∕d-&tidle;t≃ -1-- V,ϕ , &tidle;𝜖 ≡ -d-ϕ-∕d&tidle;t-- ≃ &tidle;𝜖 − V,ϕϕ. (3.23 ) 1 &tidle;H2 2κ2 V 2 H&tidle;(d ϕ∕d&tidle;t) 1 3H&tidle;2
For the potential (3.16View Equation) it follows that
√ --- 2 √ --- √ --- &tidle;𝜖1 ≃ 4(e 2∕3κϕ − 1 )− 2, &tidle;𝜖2 ≃ &tidle;𝜖1 + M---e− 2∕3κϕ(1 − 2e − 2∕3κϕ), (3.24 ) 3 3 &tidle;H2
which are much smaller than 1 during inflation (κϕ ≫ 1). The end of inflation is characterized by the condition {&tidle;𝜖1,|&tidle;𝜖2|} = 𝒪 (1). Solving &tidle;𝜖1 = 1, we obtain the field value ϕf ≃ 0.19mpl.

We define the number of e-foldings in the Einstein frame,

∫ &tidle;tf ∫ ϕi V N&tidle; = H&tidle;d &tidle;t ≃ κ2 --- dϕ, (3.25 ) &tidle;ti ϕf V,ϕ
where ϕ i is the field value at the onset of inflation. Since H&tidle;d &tidle;t = Hdt [1 + F˙∕(2HF )], it follows that N&tidle; is identical to N in the slow-roll limit: ˙ ˙ 2 |F ∕(2HF )| ≃ |H ∕H | ≪ 1. Under the condition κϕi ≫ 1 we have
3 √ --- N&tidle; ≃ -e 2∕3κϕi. (3.26 ) 4
This shows that ϕi ≃ 1.11mpl for N&tidle; = 70. From Eqs. (3.24View Equation) and (3.26View Equation) together with the approximate relation H&tidle; ≃ M ∕2, we obtain
-3--- -1- &tidle;𝜖1 ≃ 4 &tidle;N 2, &tidle;𝜖2 ≃ N&tidle; , (3.27 )
where, in the expression of &tidle;𝜖2, we have dropped the terms of the order of &tidle;2 1∕N. The results (3.27View Equation) will be used to estimate the spectra of density perturbations in Section 7.
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