  We investigate cosmological perturbations of scalar-tensor theories in
Palatini formalism. First we introduce an action where the Ricci scalar is
conformally coupled to a function of a scalar field and its kinetic term and
there is also a k-essence term consisting of the scalar and its kinetic term.
This action has three frames that are equivalent to one another: the original
Jordan frame, the Einstein frame where the metric is redefined, and the Riemann
frame where the connection is redefined. For the first time in the literature,
we calculate the quadratic action and the sound speed of scalar and tensor
perturbations in three different frames and show explicitly that they coincide.
Furthermore, we show that for such action the sound speed of gravitational
waves is unity. Thus, this model serves as dark energy as well as an inflaton
even though the presence of the dependence of the kinetic term of a scalar
field in the non-minimal coupling, different from the case in metric formalism.
We then proceed to construct the L3 action called Galileon terms in Palatini
formalism and compute its perturbations. We found that there are essentially 10
different(inequivalent) definitions in Palatini formalism for a given Galileon
term in metric formalism. We also see that,in general, the L3 terms have a
ghost due to Ostrogradsky instability and the sound speed of gravitational
waves could potentially deviate from unity, in sharp contrast with the case of
metric formalism. Interestingly, once we eliminate such a ghost, the sound
speed of gravitational waves also becomes unity. Thus, the ghost-free L3 terms
in Palatini formalism can still serve as dark energy as well as an inflaton,
like the case in metric formalism.
