Super-renormalizable or finite completion of the Starobinsky theory

The recent Planck data of Cosmic Microwave Background (CMB) temperature anisotropies support the Starobinsky theory in which the quadratic Ricci scalar drives cosmic inflation. We build up a multi-dimensional quantum consisted ultraviolet completion of the model in a phenomenological ``bottom-up approach". We present the maximal class of theories compatible with unitarity and (super-)renormalizability or finiteness which reduces to the Starobinsky theory in the low-energy limit. The outcome is a maximal extension of the Krasnikov-Tomboulis-Modesto theory including an extra scalar degree of freedom besides the graviton field. The original theory was afterwards independently discovered by Biswas-Gerwick-Koivisto-Mazumdar starting from first principles. We explicitly show power counting super-renormalizability or finiteness (in odd dimensions) and unitarity (no ghosts) of the theory. Any further extension of the theory is non-unitary confirming the existence of at most one single extra degree of freedom, the scalaron. A mechanism to achieve the Starobinsky theory in string (field) theory is also investigated at the end of the paper.