3.1 Inflationary dynamics

We consider the models of the form
f(R ) = R + αRn, (α > 0,n > 0 ), (3.1 )
which include the Starobinsky’s model [564Jump To The Next Citation Point] as a specific case (n = 2). In the absence of the matter fluid (ρM = 0), Eq. (2.15View Equation) gives
n− 1 2 1- n n−2 ˙ 3(1 + nαR )H = 2(n − 1)αR − 3n(n − 1)αHR R. (3.2 )
The cosmic acceleration can be realized in the regime n− 1 F = 1 + nαR ≫ 1. Under the approximation F ≃ nαRn −1, we divide Eq. (3.2View Equation) by 3n αRn −1 to give
( ) 2 n − 1 R˙ H ≃ ------ R − 6nH -- . (3.3 ) 6n R

During inflation the Hubble parameter H evolves slowly so that one can use the approximation 2 |H˙∕H | ≪ 1 and |H ¨∕(H H˙)| ≪ 1. Then Eq. (3.3View Equation) reduces to

H˙ 2 − n --- ≃ − 𝜖1, 𝜖1 = ----------------. (3.4 ) H2 (n − 1)(2n − 1 )
Integrating this equation for 𝜖 > 0 1, we obtain the solution
1 H ≃ ---, a ∝ t1∕𝜖1. (3.5 ) 𝜖1t
The cosmic acceleration occurs for 𝜖1 < 1, i.e., √ -- n > (1 + 3)∕2. When n = 2 one has 𝜖1 = 0, so that H is constant in the regime F ≫ 1. The models with n > 2 lead to super inflation characterized by ˙ H > 0 and −1∕|𝜖1| a ∝ |t0 − t| (t0 is a constant). Hence the standard inflation with decreasing H occurs for √ -- (1 + 3)∕2 < n < 2.

In the following let us focus on the Starobinsky’s model given by

2 2 f (R) = R + R ∕(6M ), (3.6 )
where the constant M has a dimension of mass. The presence of the linear term in R eventually causes inflation to end. Without neglecting this linear term, the combination of Eqs. (2.15View Equation) and (2.16View Equation) gives
˙2 ¨H − H---+ 1-M 2H = − 3H H˙, (3.7 ) 2H 2 ¨R + 3H R˙ + M 2R = 0. (3.8 )
During inflation the first two terms in Eq. (3.7View Equation) can be neglected relative to others, which gives H˙ ≃ − M 2∕6. We then obtain the solution
H ≃ Hi − (M 2∕6)(t − ti), (3.9 ) [ 2 2] a ≃ ai exp Hi (t − ti) − (M ∕12)(t − ti) , (3.10 ) R ≃ 12H2 − M 2, (3.11 )
where Hi and ai are the Hubble parameter and the scale factor at the onset of inflation (t = ti), respectively. This inflationary solution is a transient attractor of the dynamical system [407Jump To The Next Citation Point]. The accelerated expansion continues as long as the slow-roll parameter
H˙ M 2 𝜖1 = − H2-≃ 6H2-, (3.12 )
is smaller than the order of unity, i.e., H2 ≳ M 2. One can also check that the approximate relation 3H R˙ + M 2R ≃ 0 holds in Eq. (3.8View Equation) by using R ≃ 12H2. The end of inflation (at time t = tf) is characterized by the condition 𝜖f ≃ 1, i.e., √ -- Hf ≃ M ∕ 6. From Eq. (3.11View Equation) this corresponds to the epoch at which the Ricci scalar decreases to R ≃ M 2. As we will see later, the WMAP normalization of the CMB temperature anisotropies constrains the mass scale to be M ≃ 1013 GeV. Note that the phase space analysis for the model (3.6View Equation) was carried out in [40724131].

We define the number of e-foldings from t = ti to t = tf:

∫ tf M 2 N ≡ Hdt ≃ Hi (tf − ti) −----(tf − ti)2. (3.13 ) ti 12
Since inflation ends at t ≃ t + 6H ∕M 2 f i i, it follows that
3H2i 1 N ≃ ---2-≃ ------, (3.14 ) M 2𝜖1(ti)
where we used Eq. (3.12View Equation) in the last approximate equality. In order to solve horizon and flatness problems of the big bang cosmology we require that N ≳ 70 [391Jump To The Next Citation Point], i.e., 𝜖1(ti) ≲ 7 × 10− 3. The CMB temperature anisotropies correspond to the perturbations whose wavelengths crossed the Hubble radius around N = 55 –60 before the end of inflation.
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