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 $f(R)=R+R^2/(6M^2)$. If the same procedure is applied to general modified $f(R)$ theories in which the Ricci scalar $R$ is replaced by $R+A_{\mu }{A}^{\mu }+\beta {\nabla }_{\mu }{A}^{\mu }$ with constant $\beta$, we obtain the Brans-Dicke theory with a scalar potential and the Brans-Dicke parameter ${\omega }_{\mathrm{BD}}={\beta }^2/4$. 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 $\beta$ is constrained to be $\beta <25$ (68% confidence level) from the bounds of the scalar spectral index and the tensor-to-scalar ratio.

Figure 1

Evolution of the scalar field $\varphi$ and the Hubble parameter $\tilde{H}$ in the Einstein frame for $\beta =10$. We choose the initial conditions $\varphi =10M_{\text{pl}}$ and $\dot{\varphi }=0$ at $\tilde{t}=0$. In this case the inflationary period lasts with the number of $e$-foldings about 55, which is followed by the reheating stage with the oscillating inflaton field.

Figure 2

The 68% C.L. (inside) and 95% C.L. (outside) observational contours in the $(n_s,r)$ plane derived from the joint data analysis of Planck $\mathrm{TT}+\mathrm{lowP}+\mathrm{BKP}+\mathrm{BAO}$ (thick solid) and Planck TT, TE, $\mathrm{EE}+\mathrm{lowP}$ (thick dashed). The pivot scale is chosen to be $k_{*}=0.002$ ${\mathrm{Mpc}}^{-1}$. We also show the theoretical curves of the auxiliary vector modified Starobinsky model (3.2) as functions of $\beta$ for $N=50$ and 60 (thin solid).

Figure 3

The theoretical curves of the model (5.15) with $N=60$ in the $(n_s,r)$ plane for $\beta =0,10,{10}^4$. These curves are plotted as functions of the power $n(>1)$. The theoretical prediction for the $n=2$ case is shown as a thin dotted curve. For smaller $n$, the tensor-to-scalar ratio gets larger. The 68% C.L. and 95% C.L. observational contours are the same as those plotted in Fig. 2.