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.

我们研究了帕拉蒂尼形式主义中标量张量理论的宇宙学扰动。首先，我们介绍一个动作，其中里奇标量与标量场的函数及其动力学项保形地耦合，并且还存在一个由标量及其动力学项组成的k本质项。此动作具有彼此等效的三个框架：原始的Jordan框架，重新定义度量的Einstein框架和重新定义连接的Riemann框架。在文献中，我们首次计算了三个不同帧中的标量和张量摄动的二次作用以及声速，并明确表明它们是一致的。此外，我们表明，对于这种作用，引力波的声速是统一的。因此，即使存在非最小耦合中标量场动力学项的依赖关系，该模型也可以用作暗能量和充气子，这与度量形式主义的情况不同。然后，我们继续在Palatini形式主义中构建称为伽利略术语的L3动作并计算其扰动。我们发现，对于度量形式主义中的给定Galileon术语，Palatini形式主义中实质上有10个不同（不等价）的定义。我们还可以看到，一般而言，L3术语由于Ostrogradsky的不稳定性而产生了幻影，而重力波的声速可能会偏离统一性，这与公制形式主义的情况形成了鲜明的对比。有趣的是，一旦我们消除了这种鬼影，引力波的声速也会变得统一。因此，