Analytic extensions of Starobinsky model of inflation

We study several extensions of the Starobinsky model of inflation, which obey all observational constraints on the inflationary parameters, by demanding that both the inflaton scalar potential in the Einstein frame and the $F(R)$ gravity function in the Jordan frame have the explicit dependence upon fields {\it and} parameters in terms of elementary functions. Our models are continuously connected to the original Starobinsky model via changing the parameters. We modify the Starobinsky $(R+R^2)$ model by adding an $R^3$-term, an $R^4$-term, and an $R^{3/2}$-term, respectively, and calculate the scalar potentials, the inflationary observables and the allowed limits on the deformation parameters by using the latest observational bounds. We find that the tensor-to-scalar ratio in the Starobinsky model modified by the $R^{3/2}$-term significantly increases with raising the parameter in front of that term. On the other side, we deform the scalar potential of the Starobinsky model in the Einstein frame in powers of $y=\exp\left(-\sqrt{\frac{2}{3}}\phi/M_{Pl}\right)$, where $\phi$ is the canonical inflaton (scalaron) field, calculate the corresponding $F(R)$ gravity functions in the two new cases, and find the restrictions on the deformation parameters in the lowest orders with respect to the variable $y$ that is physically small during slow-roll inflation.