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 phenomenology and consistency require $n=D/2$, 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.

Figure 1

The profile of the four-dimensional scalar potential obtained from $D=8$ dimensions. The green line describes the case of $\lambda =0$, the blue line is of $\lambda =1$, and the red line is of $\lambda =2$, respectively.

Figure 2

The profile of the scalar potential obtained from $D=12$ dimensions. The green line describes the case of $\lambda =0$, the blue line is of $\lambda =1$, and the red line is of $\lambda =2$, respectively.