7.4 The Lagrangian for cosmological perturbations

In Section 7.1 we used the fact that the field which should be quantized corresponds to √ -- u = a Qs ℛ. This can be justified by writing down the action (6.1View Equation) expanded at second-order in the perturbations [437]. We recall again that we are considering an effective single-field theory such as f (R) gravity and scalar-tensor theory with the coupling F (ϕ)R. Carrying out the expansion of the action (6.2View Equation) in second order, we find that the action for the curvature perturbation ℛ (either ℛ δF or ℛ δϕ) is given by [311]
∫ [ ] (2) 3 3 1- 2 1-1-- 2 δS = dtd xa Qs 2ℛ˙ − 2 a2(∇ ℛ ) , (7.80 )
where Qs is given in Eq. (7.38View Equation). In fact, the variation of this action in terms of the field ℛ gives rise to Eq. (7.37View Equation) in Fourier space. We note that there is another approach called the Hamiltonian formalism which is also useful for the quantization of cosmological perturbations. See [237209208127] for this approach in the context of f (R) gravity and modified gravitational theories.

Introducing the quantities u = zsℛ and √-- zs = a Qs, the action (7.80View Equation) can be written as

∫ [ ] (2) 3 1 ′2 1 2 1 z′′s 2 δS = d ηd x 2u − 2(∇u ) + 2-z-u , (7.81 ) s
where a prime represents a derivative with respect to the conformal time ∫ η = a−1dt. The action (7.81View Equation) leads to Eq. (7.39View Equation) in Fourier space. The transformation of the action (7.80View Equation) to (7.81View Equation) gives rise to the effective mass6
′′ ˙2 ¨ ˙ M 2s ≡ − 1-zs-= Q-s-− -Qs-− 3H-Qs-. (7.82 ) a2zs 4Q2s 2Qs 2Qs

We have seen in Eq. (7.42View Equation) that during inflation the quantity ′′ zs∕zs can be estimated as z′s′∕zs ≃ 2 (aH )2 in the slow-roll limit, so that M s2≃ − 2H2. For the modes deep inside the Hubble radius (k ≫ aH) the action (7.81View Equation) reduces to the one for a canonical scalar field u in the flat spacetime. Hence the quantization should be done for the field u = a√Qs-ℛ, as we have done in Section 7.1.

From the action (7.81View Equation) we understand a number of physical properties in f (R) theories and scalar-tensor theories with the coupling F(ϕ )R listed below.

1.
Having a standard d’Alambertian operator, the mode has speed of propagation equal to the speed of light. This leads to a standard dispersion relation ω = k∕a for the high-k modes in Fourier space.
2.
The sign of Qs corresponds to the sign of the kinetic energy of ℛ. The negative sign corresponds to a ghost (phantom) scalar field. In f (R) gravity (with ˙ϕ = 0) the ghost appears for F < 0. In Brans–Dicke theory with F (ϕ ) = κ2ϕ and ω (ϕ) = ωBD ∕ϕ [100Jump To The Next Citation Point] (where ϕ > 0) the condition for the appearance of the ghost (ω ˙ϕ2 + 3 ˙F2∕(2κ2F ) < 0) translates into ωBD < − 3∕2. In these cases one would encounter serious problems related to vacuum instability [145Jump To The Next Citation Point161Jump To The Next Citation Point].
3.
The field u has the effective mass squared given in Eq. (7.82View Equation). In f (R) gravity it can be written as
( ) 72F 2H4 1 288H3 − 12HR 1 f2 R˙2 7 M s2= − -----------------+ -F ---------------+ ----- + -,RR-2--− 24H2 + -R, (7.83) (2FH + f,RRR˙)2 3 2F H + f,RRR˙ f,RR 4F 6
where we used the background equation (2.16View Equation) to write ˙ H in terms of R and 2 H. In Fourier space the perturbation u obeys the equation of motion
u′′ + (k2 + M 2a2)u = 0. (7.84) s
For k2∕a2 ≫ M 2s, the field u propagates with speed of light. For small k satisfying k2∕a2 ≪ M 2 s, we require a positive M 2 s to avoid the tachyonic instability of perturbations. Recall that the viable dark energy models based on f (R) theories need to satisfy Rf,RR ≪ F (i.e., m = Rf,RR ∕f,R ≪ 1) at early times, in order to have successful cosmological evolution from radiation domination till matter domination. At these epochs the mass squared is approximately given by
2 F M s ≃ -----, (7.85) 3f,RR
which is consistent with the result (5.2View Equation) derived by the linear analysis about the Minkowski background. Together with the ghost condition F > 0, this leads to f,RR > 0. Recall that these correspond to the conditions presented in Eq. (4.56View Equation).


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