Inflation from $(R+\gamma R^n-2\mathrm{\Lambda })$ Gravity in Higher Dimensions

We propose a derivation of the inflaton scalar potential from the higher $(D)$ dimensional $(R+\gamma R^n-2\mathrm{\Lambda })$ gravity, with the new coupling constant $\gamma$ and the cosmological constant $\mathrm{\Lambda }$. We assume that a compactification of extra dimensions happened before inflation, so that the inflaton scalar potential in four spacetime dimensions appears to be dependent upon the parameters $(\gamma ,\mathrm{\Lambda },D,n)$. We find that consistency requires $n=D\mathrm{/}2$, while the dimension $D$ has to be a multiple of four. We calculate the potential for any $D$, and determine the values of $\gamma$ and $\mathrm{\Lambda }$ from observations. The cases of $D=8$ and $D=12$ are considered in more detail. Our approach results in the viable models of chaotic large-field inflation, and leads to the sharp predictions of an observable value of the tensor-to-scalar ratio $r$ of the Cosmic Microwave Background radiation.