Inflation in Extra-Dimensions with one or two branes

In this paper, we study two inflationary models, namely, monomial inflation and the simplest $α$-attractor inflation, within extra-dimensional frameworks. We consider three extra-dimensional setups: Dark Dimension, which embeds one flat extra-dimension to explain the observed smallness of the 4D cosmological constant $Λ_4$; and the two Randall-Sundrum scenarios with one warped extra-dimension, namely RS1 with two branes and RS2 with one brane. We derive the corresponding Friedmann equations, compute the slow-roll parameters in each case, and we fit the experimental data for ($n_s - 1$, $α$, $Δ_s^2$, $r$), using Planck, BICEP, and ACT data. We find that monomial inflation is strongly disfavored in all scenarios, while $α$-attractor inflation provides an excellent fit to current observations, with extra-dimensional setups offering additional flexibility compared to the standard 4D case.