the expectation value of the energy-momentum tensor being quantized in some state [@BD; @9509057; @9707062; @BF; @PT]. $$\tensor{R}{_\mu_\nu} -\frac{1}{2} \tensor{g}{_\mu_\nu} R = 8 \pi G \langle \tensor{\hat{T}}{_\mu_\nu} \rangle \, ,  \label{1loop}$$ where $\langle \tensor{\hat{T}}{_\mu_\nu} \rangle$ includes $\infty^{4}$, $\infty^{2}$, and $\ln \infty$ divergences. The first and the second divergences can be eliminated by the renormalization of the cosmological constant and the Newtonian coupling constant, respectively. However, the logarithmic divergence can be eliminated only if one introduces an additional Lagrangian density of the form $\sqrt{-g} \left(c_1 R^2 + c_2 \tensor{R}{_\mu_\nu} \tensor{R}{^\mu^\nu} \right)$ [@Nariai73].

Among these quantum corrections to general relativity, if one includes only the quadratic terms in the Riemann tensors, then these terms are expressed as [@Mijic86] $$\alpha R^2 + \beta \tensor{R}{_\mu_\nu} \tensor{R}{^\mu^\nu} + \gamma \tensor{R}{_\mu_\nu_\lambda_\sigma} \tensor{R}{^\mu^\nu^\lambda^\sigma} = \epsilon R^2 + \xi \tensor{C}{_\mu_\nu_\lambda_\sigma} \tensor{C}{^\mu^\nu^\lambda^\sigma} + \eta \chi_{\text{E}} \label{quadraticR} \, ,$$ where $C$ is the Weyl tensor and $\chi_{\text{E}}$ is the density of the Euler number for the manifold. If one considers the Robertson-Walker metric, then $C^2$-term vanishes because the metric is conformally flat. Thus, Starobinsky includes the curvature-squared correction to the Einstein-Hilbert action and this term is important for the early universe [@Star80]. At the early universe ([*i.e.*]{} when the curvatures 