### 12.2 Ghosts

What is a ghost for these theories? A ghost mode is a propagating degree of freedom with a
kinetic term in the action with opposite sign. In order to see if a ghost is propagating on a given
background, one needs to expand the action at second order around the background in terms of the
perturbation fields. After integrating out all auxiliary fields, one is left with a minimal number of
gauge-invariant fields . These are not unique, as we can always perform a field redefinition (e.g., a
field rotation). However, no matter which fields are used, we typically need – for non-singular
Lagrangians – to define the kinetic operator, the operator which in the Lagrangian appears as
[186, 185]. Then the sign of the eigenvalues of the matrix defines whether a mode is
a ghost or not. A negative eigenvalue would correspond to a ghost particle. On a FLRW background the
matrix will be in general time-dependent and so does the sign of the eigenvalues. Therefore
one should make sure that the extra scalar modes introduced for these theories do not possess
wrong signs in the kinetic term at any time during the evolution of the Universe, at least up to
today.
An overall sign in the Lagrangian does not affect the classical equations of motion. However, at the
quantum level, if we want to preserve causality by keeping the optical theorem to be valid, then the ghost
can be interpreted as a particle which propagates with negative energy, as already stated above. In other
words, in special relativity, the ghost would have a four-momentum with . However it
would still be a timelike particle as , whether is negative or not. The problem
arises when this particle is coupled to some other normal particle, because in this case the
process with can be allowed. This means in general that
for such a theory one would expect the pair creation of ghost and normal particles out of the
vacuum. Notice that the energy is still conserved, but the energy is pumped out of the ghost
particle.

Since all the particles are coupled at least to gravity, one would think that out of the vacuum particles
could be created via the decay of a couple of gravitons emitted in the vacuum into ghosts and non-ghosts
particles. This process does lead to an infinite contribution unless one introduces a cutoff for the
theory [145, 161], for which one can set observational constraints.

We have already seen that, for metric f (R) gravity, the kinetic operator in the FLRW background
reduces to given in Eq. (7.60) with the perturbed action (7.80). Since the sign of is
determined by , one needs to impose in order to avoid the propagation of a ghost
mode.