General Expressions for $n_{\mathrm{s}}$ and $r$ for Several Minimally Coupled Single Scalar Field Models

We derive a generalized expressions for the scalar power spectrum and the observational parameters of inflation (the scalar spectral index $n_{\mathrm{s}}$ and the tensor-to-scalar ratio $r$) for several single field scalar models minimally coupled with gravity. We work in the framework where the perturbed Einstein equations, at linear order, are of the same form to those obtained in the standard $\mathit{\text{k}}$-inflation, with generalized form of the pump field $z$. By generalizing the expressions for $z$ and the speed of sound squared $c_{\mathrm{s}}^2$, using the slow-roll approximation, we calculate $n_{\mathrm{s}}$ and $r$ up to the second order in the Hubble flow parameters.