7.4 The Lagrangian for cosmological perturbations
In Section 7.1 we used the fact that the field which should be quantized corresponds to .
This can be justified by writing down the action (6.1) expanded at second-order in the perturbations .
We recall again that we are considering an effective single-field theory such as f (R) gravity and
scalar-tensor theory with the coupling . Carrying out the expansion of the action (6.2) in second
order, we find that the action for the curvature perturbation (either or ) is given by 
where is given in Eq. (7.38). In fact, the variation of this action in terms of the field
gives rise to Eq. (7.37) 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 [237, 209, 208, 127] for this approach in the context of f (R) gravity and modified gravitational
Introducing the quantities and , the action (7.80) can be written as
where a prime represents a derivative with respect to the conformal time . The action (7.81) leads
to Eq. (7.39) in Fourier space. The transformation of the action (7.80) to (7.81) gives rise to the effective
We have seen in Eq. (7.42) that during inflation the quantity can be estimated as
in the slow-roll limit, so that . For the modes deep inside the Hubble
radius () the action (7.81) reduces to the one for a canonical scalar field in the flat
spacetime. Hence the quantization should be done for the field , as we have done in
From the action (7.81) we understand a number of physical properties in f (R) theories and
scalar-tensor theories with the coupling listed below.
- 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 for the high- modes
in Fourier space.
- The sign of corresponds to the sign of the kinetic energy of . The negative sign
corresponds to a ghost (phantom) scalar field. In f (R) gravity (with ) the ghost appears
for . In Brans–Dicke theory with and  (where
) the condition for the appearance of the ghost () translates
into . In these cases one would encounter serious problems related to vacuum
instability [145, 161].
- The field has the effective mass squared given in Eq. (7.82). In f (R) gravity it can be written as
where we used the background equation (2.16) to write in terms of and . In Fourier
space the perturbation obeys the equation of motion
For , the field propagates with speed of light. For small satisfying
, we require a positive to avoid the tachyonic instability of perturbations. Recall
that the viable dark energy models based on f (R) theories need to satisfy (i.e.,
) 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
which is consistent with the result (5.2) derived by the linear analysis about the Minkowski
background. Together with the ghost condition , this leads to . Recall that these
correspond to the conditions presented in Eq. (4.56).