Modified Starobinsky inflation by the $R\ln\left( \square\right) R$ term

In the context of effective theories of gravity, a minimalist bottom-up approach which takes into account $1$-loop quantum corrections leads to modifications in the Einstein-Hilbert action through the inclusion of four extra terms: $R^{2}$, $C_{\kappa\rho\alpha\beta}C^{\kappa\rho\alpha\beta}$, $R\ln\left( \square\right) R$ and $C_{\kappa\rho\alpha\beta}\ln\left( \square\right) C^{\kappa\rho\alpha\beta}$. The first two terms are necessary to guarantee the renormalizability of the gravitational theory, and the last two terms (nonlocal terms) arise from the integration of massless/light matter fields. This work aims to analyze how one of the nonlocal terms, namely $R\ln\left( \square\right) R$, affects the Starobinsky inflation. We consider the nonlocal term as a small correction to the $R^{2}$ term, and we demonstrate that the model behaves like a local model in this context. In addition, we show that the approximate model in the Einstein frame is described by a canonical scalar field minimally coupled to general relativity. Finally, we study the inflationary regime of this model and constrain its free parameters through observations of CMB anisotropies.