Inflation due to non-minimal coupling of $f(R)$ gravity to a scalar field

Abstract: In this work we investigate a inflationary scenario generated by a large scalar field $\phi$ that non-minimally couples to a $f(R)$ modified gravity model. For a Starobinsky's like model, it is found that along a particular flat direction, the scalar potential takes a simple form $V = (M_p^4/4) [V(\phi)/\alpha(\phi)^2]$ where $\alpha(\phi)$ is a non-minimal coupling to Ricci scalar $R$ in the model. The inflation, therefore, is effectively represented as a single field inflaton scenario. For a specific example, such as a scalar potential $V(\phi) = \mu_1\phi^2 + \mu_2\phi^4$, we found that the predictions match nicely in the $1\sigma$ confidence level of \textit{Plank} TT, TE, EE+lowP combination data of Planck 2015 CMB data for $ 0<\mu_3 \leq 100$, where $\mu_3 := |\mu_1|/(\mu_2 M_p^2)$. For example, taking $\mu_3 = 0.01$ the scalar-to-tensor ratio $r=0.0004$ and and spectral index $n_s = 0.96985$ for $N_\ast = 50$ while taking $\mu_3 = 100.0$ produces $r= 0.03$ and $n_s = 0.96359$ for $N_\ast = 60$.