This paper extends the theory of cosmological perturbations to a more general Lagrangian that includes arbitrary functions of the scalar field and its first derivatives. This extension unifies the descriptions of known cases such as the usual scalar field and hydrodynamical perfect fluid, and applies to the recently proposed $k$-inflation, which is driven by non-minimal kinetic terms. The authors calculate the spectrum of quantum fluctuations for slow-roll and power-law $k$-inflation, finding that the "consistency relation" between the tensor spectral index and the relative amplitude of scalar and tensor perturbations is modified. This modification suggests that $k$-inflation can be phenomenologically distinguished from standard inflation. The paper also discusses the possibility of adjusting the speed of sound in power-law inflation to any given value, which could lead to scenarios where tensor modes dominate over scalar modes, a prediction that could be tested in future observations. The authors conclude with speculative remarks on the potential implications of these findings, including the possibility of communication with distant civilizations using quintessence waves.This paper extends the theory of cosmological perturbations to a more general Lagrangian that includes arbitrary functions of the scalar field and its first derivatives. This extension unifies the descriptions of known cases such as the usual scalar field and hydrodynamical perfect fluid, and applies to the recently proposed $k$-inflation, which is driven by non-minimal kinetic terms. The authors calculate the spectrum of quantum fluctuations for slow-roll and power-law $k$-inflation, finding that the "consistency relation" between the tensor spectral index and the relative amplitude of scalar and tensor perturbations is modified. This modification suggests that $k$-inflation can be phenomenologically distinguished from standard inflation. The paper also discusses the possibility of adjusting the speed of sound in power-law inflation to any given value, which could lead to scenarios where tensor modes dominate over scalar modes, a prediction that could be tested in future observations. The authors conclude with speculative remarks on the potential implications of these findings, including the possibility of communication with distant civilizations using quintessence waves.