Planck constraints on inflation in auxiliary vector modified $f(R)$ theories

We show that the universal $\alpha$-attractor models of inflation can be realized by including an auxiliary vector field $A_{\mu }$ for the Starobinsky model with the Lagrangian $f(R)=R+R^2\mathrm{/}(6M^2)$. If the same procedure is applied to general modified $f(R)$ theories in which the Ricci scalar $R$ is replaced by $R+A_{\mu }{A}^{\mu }+\beta {\mathrm{\nabla }}_{\mu }{A}^{\mu }$ with constant $\beta$, we obtain the Brans-Dicke theory with a scalar potential and the Brans-Dicke parameter ${\omega }_{\mathrm{B}\mathrm{D}}={\beta }^2\mathrm{/}4$. We also place observational constraints on inflationary models based on auxiliary vector modified $f(R)$ theories from the latest Planck measurements of the Cosmic Microwave Background (CMB) anisotropies in both temperature and polarization. In the modified Starobinsky model, we find that the parameter $\beta$ is constrained to be $\beta <25$ (68,%,confidence level) from the bounds of the scalar spectral index and the tensor-to-scalar ratio.