7.3 The spectra of perturbations in inflation based on f (R) gravity

Let us study the spectra of scalar and tensor perturbations generated during inflation in metric f (R) gravity. Introducing the quantity E = 3FĖ™2āˆ•(2κ2), we have šœ–4 = ¨F āˆ•(H Ė™F) and
2 Q = ---6F-šœ–3---= ------E------. (7.60 ) s κ2(1 + šœ–3)2 F H2 (1 + šœ–3)2
Since the field kinetic term Ė™2 Ļ• is absent, one has šœ–2 = 0 in Eqs. (7.42View Equation) and (7.49View Equation). Under the conditions |šœ–i| ā‰Ŗ 1 (i = 1,3,4), the spectral index of curvature perturbations is given by n ā„› − 1 ā‰ƒ − 4šœ–1 + 2 šœ–3 − 2šœ–4.

In the absence of the matter fluid, Eq. (2.16View Equation) translates into

šœ–1 = − šœ–3(1 − šœ–4), (7.61 )
which gives šœ–1 ā‰ƒ − šœ–3 for |šœ–4| ā‰Ŗ 1. Hence we obtain [315]
nā„› − 1 ā‰ƒ − 6šœ–1 − 2šœ–4. (7.62 )
From Eqs. (7.50View Equation) and (7.60View Equation), the amplitude of ā„› is estimated as
( )2 š’«ā„› ā‰ƒ --1-- -H-- 1-. (7.63 ) 3πF mpl šœ–23

Using the relation šœ–1 ā‰ƒ − šœ–3, the spectral index (7.57View Equation) of tensor perturbations is given by

nT ā‰ƒ 0, (7.64 )
which vanishes at first-order of slow-roll approximations. From Eqs. (7.58View Equation) and (7.63View Equation) we obtain the tensor-to-scalar ratio
r ā‰ƒ 48šœ–2 ā‰ƒ 48 šœ–2. (7.65 ) 3 1

7.3.1 The model n f (R) = αR (n > 0)

Let us consider the inflation model: f(R ) = αRn (n > 0). From the discussion given in Section 3.1 the slow-roll parameters šœ–i (i = 1,3,4) are constants:

2 − n n − 2 šœ–1 = ---------------, šœ–3 = − (n − 1)šœ–1, šœ–4 = -----. (7.66 ) (n − 1)(2n − 1) n − 1
In this case one can use the exact results (7.48View Equation) and (7.56View Equation) with νā„› and νt given in Eqs. (7.42View Equation) and (7.54View Equation) (with šœ– = 0 2). Then the spectral indices are
2(n − 2)2 nā„› − 1 = nT = − --2----------. (7.67 ) 2n − 2n − 1

If n = 2 we obtain the scale-invariant spectra with nā„› = 1 and nT = 0. Even the slight deviation from n = 2 leads to a rather large deviation from the scale-invariance. If n = 1.7, for example, one has n ā„› − 1 = nT = − 0.13, which does not match with the WMAP 5-year constraint: n ā„› = 0.960 ± 0.013 [367Jump To The Next Citation Point].

7.3.2 The model 2 2 f (R) = R + R āˆ•(6M )

For the model f(R ) = R + R2 āˆ•(6M 2), the spectrum of the curvature perturbation ā„› shows some deviation from the scale-invariance. Since inflation occurs in the regime R ā‰« M 2 and |H Ė™| ā‰Ŗ H2, one can approximate F ā‰ƒ R āˆ•(3M 2) ā‰ƒ 4H2 āˆ•M 2. Then the power spectra (7.63View Equation) and (7.58View Equation) yield

( ) ( ) 1 M 2 1 4 M 2 š’«ā„› ā‰ƒ ---- ---- -2, š’«T ā‰ƒ -- ---- , (7.68 ) 12π mpl šœ–1 π mpl
where we have employed the relation šœ–3 ā‰ƒ − šœ–1.

Recall that the evolution of the Hubble parameter during inflation is given by Eq. (3.9View Equation). As long as the time tk at the Hubble radius crossing (k = aH) satisfies the condition 2 (M āˆ•6)(tk − ti) ā‰Ŗ Hi, one can approximate H (tk) ā‰ƒ Hi. Using Eq. (3.9View Equation), the number of e-foldings from t = tk to the end of inflation can be estimated as

1 Nk ā‰ƒ -------. (7.69 ) 2 šœ–1(tk)
Then the amplitude of the curvature perturbation is given by
2( )2 š’« ā‰ƒ N-k -M-- . (7.70 ) ā„› 3π mpl
The WMAP 5-year normalization corresponds to −9 š’« ā„› = (2.445 ± 0.096) × 10 at the scale −1 k = 0.002 Mpc [367Jump To The Next Citation Point]. Taking the typical value Nk = 55, the mass M is constrained to be
−6 M ā‰ƒ 3 × 10 mpl. (7.71 )
Using the relation F ā‰ƒ 4H2 āˆ•M 2, it follows that šœ–4 ā‰ƒ − šœ–1. Hence the spectral index (7.62View Equation) reduces to
( ) −1 -2- − 2 Nk- n ā„› − 1 ā‰ƒ − 4šœ–1 ā‰ƒ − Nk = − 3.6 × 10 55 . (7.72 )
For Nk = 55 we have nā„› ā‰ƒ 0.964, which is in the allowed region of the WMAP 5-year constraint (nā„› = 0.960 ± 0.013 at the 68% confidence level [367Jump To The Next Citation Point]). The tensor-to-scalar ratio (7.65View Equation) can be estimated as
( ) −2 12- −3 Nk- r ā‰ƒ N 2 ā‰ƒ 4.0 × 10 55 , (7.73 ) k
which satisfies the current observational bound r < 0.22 [367]. We note that a minimally coupled field with the potential V(Ļ• ) = m2 Ļ•2 āˆ•2 in Einstein gravity (chaotic inflation model [393]) gives rise to a larger tensor-to-scalar ratio of the order of 0.1. Since future observations such as the Planck satellite are expected to reach the level of r = š’Ŗ(10 −2), they will be able to discriminate between the chaotic inflation model and the Starobinsky’s f (R) model.

7.3.3 The power spectra in the Einstein frame

Let us consider the power spectra in the Einstein frame. Under the conformal transformation &tidle;gμν = F gμν, the perturbed metric (6.1View Equation) is transformed as

d&tidle;s2 = F ds2 &tidle;2 &tidle; &tidle; &tidle; &tidle; i = − (1 + 2α&tidle;)dt − 2&tidle;a(t)(∂iβ − Si)dtdx&tidle; + &tidle;a2(&tidle;t)(δij + 2ψ&tidle;δij + 2∂i∂j&tidle;γ + 2∂jF&tidle;i + &tidle;hij)d&tidle;xid&tidle;xj. (7.74 )
We decompose the conformal factor into the background and perturbed parts, as
( ) δF-(t,x-) F (t,x) = F¯(t) 1 + ¯F(t) . (7.75 )
In what follows we omit a bar from F. We recall that the background quantities are transformed as Eqs. (2.44View Equation) and (2.47View Equation). The transformation of scalar metric perturbations is given by
&tidle;α = α + δF-, β&tidle;= β, ψ&tidle; = ψ + δF-, &tidle;γ = γ. (7.76 ) 2F 2F
Meanwhile vector and tensor perturbations are invariant under the conformal transformation (&tidle;Si = Si, &tidle; Fi = Fi, &tidle; hij = hij).

Using the above transformation law, one can easily show that the curvature perturbation ā„› = ψ − H δFāˆ•FĖ™ in f (R) gravity is invariant under the conformal transformation:

ā„›&tidle; = ā„›. (7.77 )
Since the tensor perturbation is also invariant, the tensor-to-scalar ratio &tidle;r in the Einstein frame is identical to that in the Jordan frame. For example, let us consider the model f (R ) = R + R2 āˆ•(6M 2). Since the action in the Einstein frame is given by Eq. (2.32View Equation), the slow-roll parameters &tidle;šœ–3 and &tidle;šœ–4 vanish in this frame. Using Eqs. (7.49View Equation) and (3.27View Equation), the spectral index of curvature perturbations is given by
2-- n&tidle;ā„› − 1 ā‰ƒ − 4&tidle;šœ–1 − 2&tidle;šœ–2 ā‰ƒ − &tidle;N , (7.78 ) k
where we have ignored the term of the order of 2 1 āˆ• &tidle;Nk. Since N&tidle;k ā‰ƒ Nk in the slow-roll limit (|FĖ™āˆ•(2HF )| ā‰Ŗ 1), Eq. (7.78View Equation) agrees with the result (7.72View Equation) in the Jordan frame. Since Qs = (dĻ•āˆ•d &tidle;t)2āˆ• &tidle;H2 in the Einstein frame, Eq. (7.59View Equation) gives the tensor-to-scalar ratio
64 π( d Ļ•)2 1 12 &tidle;r = --2- --- --2 ā‰ƒ 16&tidle;šœ–1 ā‰ƒ --2, (7.79 ) m pl d&tidle;t H&tidle; N&tidle;k
where the background equations (3.21View Equation) and (3.22View Equation) are used with slow-roll approximations. Equation (7.79View Equation) is consistent with the result (7.73View Equation) in the Jordan frame.

The equivalence of the curvature perturbation between the Jordan and Einstein frames also holds for scalar-tensor theory with the Lagrangian ā„’ = F (Ļ•)R āˆ•(2κ2) − (1āˆ•2)ω(Ļ• )gμν∂μĻ•∂νĻ• − V(Ļ• ) [411240]. For the non-minimally coupled scalar field with F (Ļ•) = 1 − ξκ2Ļ•2 [269241] the spectral indices of scalar and tensor perturbations have been derived by using such equivalence [366590].


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