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

We show that the universal α-attractor models of inflation can be realized by including an auxiliary vector field Aμ for the Starobinsky model with the Lagrangian f(R)=R+R2/(6M2). If the same procedure is applied to general modified f(R) theories in which the Ricci scalar R is replaced by R+AμAμ+β μAμ with constant β, we obtain the Brans-Dicke theory with a scalar potential and the Brans-Dicke parameter ωBD=β2/4. 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 anisotropies in both temperature and polarization. In the modified Starobinsky model, we find that the parameter β is constrained to be β<25 (68% confidence level) from the bounds of the scalar spectral index and the tensor-to-scalar ratio.