7.1 Curvature perturbations

Since δ ϕ = 0 and δF = 0 in Eq. (6.12View Equation) we obtain
ℛ˙ α = ------------. (7.34 ) H + ˙F∕(2F )
Plugging Eq. (7.34View Equation) into Eq. (6.11View Equation), we have
[ ] 1 Δ 3H F˙ − κ2ω ˙ϕ2 A = − -----˙------- a2ℛ + ----------˙-------ℛ˙ . (7.35 ) H + F ∕(2F ) 2F {H + F ∕(2F )}
Equation (6.14View Equation) gives
( ) [ ] ˙ -F˙ 3F˙ 3F¨-+-6H-F˙+-κ2-ωϕ˙2- Δ-- A + 2H + 2F A + 2F α˙+ 2F + a2 α = 0, (7.36 )
where we have used the background equation (6.7View Equation). Plugging Eqs. (7.34View Equation) and (7.35View Equation) into Eq. (7.36View Equation), we find that the curvature perturbation satisfies the following simple equation in Fourier space
3 ˙ 2 ℛ¨+ (a-Qs)-ℛ˙+ k-ℛ = 0, (7.37 ) a3Qs a2
where k is a comoving wavenumber and
˙2 ˙2 2 Qs ≡ ω-ϕ--+-3F--∕(2κ-F-). (7.38 ) [H + ˙F∕ (2F )]2
Introducing the variables √--- zs = a Qs and u = zsℛ, Eq. (7.37View Equation) reduces to
( ′′) u′′ + k2 − zs- u = 0, (7.39 ) zs
where a prime represents a derivative with respect to the conformal time ∫ −1 η = a dt.

In General Relativity with a canonical scalar field ϕ one has ω = 1 and F = 1, which corresponds to Qs = ˙ϕ2∕H2. Then the perturbation u corresponds to u = a[− δϕ + (ϕ˙∕H )ψ]. In the spatially flat gauge (ψ = 0) this reduces to u = − aδϕ, which implies that the perturbation u corresponds to a canonical scalar field δχ = aδϕ. In modified gravity theories it is not clear at this stage that the perturbation --- u = a√ Qsℛ corresponds a canonical field that should be quantized, because Eq. (7.37View Equation) is unchanged by multiplying a constant term to the quantity Q s defined in Eq. (7.38View Equation). As we will see in Section 7.4, this problem is overcome by considering a second-order perturbed action for the theory (6.2View Equation) from the beginning.

In order to derive the spectrum of curvature perturbations generated during inflation, we introduce the following variables [315Jump To The Next Citation Point]

H˙- -ϕ¨- --F˙- -E˙-- 𝜖1 ≡ − H2 , 𝜖2 ≡ H ϕ˙, 𝜖3 ≡ 2HF , 𝜖4 ≡ 2HE , (7.40 )
where 2 2 2 E ≡ F[ω + 3F˙ ∕(2κ ϕ˙F )]. Then the quantity Qs can be expressed as
2 E Qs = ˙ϕ F-H2-(1 +-𝜖)2. (7.41 ) 3
If the parameter 𝜖1 is constant, it follows that η = − 1∕[(1 − 𝜖1)aH ] [573Jump To The Next Citation Point]. If ˙𝜖i = 0 (i = 1,2,3,4) one has
z′s′ ν2ℛ − 1∕4 2 1 (1 + 𝜖1 + 𝜖2 − 𝜖3 + 𝜖4)(2 + 𝜖2 − 𝜖3 + 𝜖4) ---= ----2----, with νℛ = --+ ----------------------2---------------. (7.42 ) zs η 4 (1 − 𝜖1)
Then the solution to Eq. (7.39View Equation) can be expressed as a linear combination of Hankel functions,
∘ ---- π |η | [ ] u = ------ei(1+2 νℛ )π∕4 c1H (ν1ℛ)(k|η|) + c2H (ν2ℛ)(k|η |) , (7.43 ) 2
where c1 and c2 are integration constants.

During inflation one has |𝜖i| ≪ 1, so that z′′s∕zs ≈ (aH )2. For the modes deep inside the Hubble radius (k ≫ aH, i.e., |kη | ≫ 1) the perturbation u satisfies the standard equation of a canonical field in the Minkowski spacetime: ′′ 2 u + k u ≃ 0. After the Hubble radius crossing (k = aH) during inflation, the effect of the gravitational term ′′ zs∕zs becomes important. In the super-Hubble limit (k ≪ aH, i.e., |kη| ≪ 1) the last term on the l.h.s. of Eq. (7.37View Equation) can be neglected, giving the following solution

∫ --dt- ℛ = c1 + c2 a3Q , (7.44 ) s
where c1 and c2 are integration constants. The second term can be identified as a decaying mode, which rapidly decays during inflation (unless the field potential has abrupt features). Hence the curvature perturbation approaches a constant value c1 after the Hubble radius crossing (k < aH).

In the asymptotic past (k η → − ∞) the solution to Eq. (7.39View Equation) is determined by a vacuum state in quantum field theory [88], as u → e− ikη∕√2k--. This fixes the coefficients to be c = 1 1 and c = 0 2, giving the following solution

∘ ---- u = --π|η|ei(1+2νℛ)π∕4H (1)(k|η|). (7.45 ) 2 νℛ

We define the power spectrum of curvature perturbations,

4πk3 𝒫 ℛ ≡ -----|ℛ |2. (7.46 ) (2π)3
Using the solution (7.45View Equation), we obtain the power spectrum [317]
1 ( Γ (ν ) H )2 ( |k η|)3−2νℛ 𝒫 ℛ = --- (1 − 𝜖1)----ℛ----- ---- , (7.47 ) Qs Γ (3 ∕2)2π 2
where we have used the relations H (1)(k|η|) → − (i∕π)Γ (ν)(k|η|∕2)−ν ν for kη → 0 and Γ (3∕2 ) = √ π-∕2. Since the curvature perturbation is frozen after the Hubble radius crossing, the spectrum (7.47View Equation) should be evaluated at k = aH. The spectral index of ℛ, which is defined by n ℛ − 1 = d ln 𝒫 ℛ∕d ln k|k=aH, is
nℛ − 1 = 3 − 2 νℛ, (7.48 )
where νℛ is given in Eq. (7.42View Equation). As long as |𝜖i| (i = 1,2,3,4) are much smaller than 1 during inflation, the spectral index reduces to
nℛ − 1 ≃ − 4𝜖1 − 2𝜖2 + 2𝜖3 − 2 𝜖4, (7.49 )
where we have ignored those terms higher than the order of 𝜖 i’s. Provided that |𝜖| ≪ 1 i the spectrum is close to scale-invariant (nℛ ≃ 1). From Eq. (7.47View Equation) the power spectrum of curvature perturbations can be estimated as
1 ( H )2 𝒫 ℛ ≃ --- --- . (7.50 ) Qs 2π
A minimally coupled scalar field ϕ in Einstein gravity corresponds to 𝜖3 = 0, 𝜖4 = 0 and Qs = ˙ϕ2∕H2, in which case we obtain the standard results n − 1 ≃ − 4𝜖 − 2𝜖 ℛ 1 2 and 𝒫 ≃ H4 ∕(4π2ϕ˙2) ℛ in slow-roll inflation [573390].
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