Inflation in Extra-Dimensions with one or two branes

TL;DR

This work investigates inflation in higher-dimensional settings by applying two inflationary potentials, monomial and α-attractor, to Dark Dimension and RS1/RS2 braneworlds. The authors derive the corresponding Friedmann equations and slow-roll parameters, then fit the data from Planck, BICEP, and ACT for the observables $n_s-1$, $α$, $Δ_s^2$, and $r$, finding monomial potentials severely disfavored while α-attractor models remain compatible, with extra dimensions offering additional flexibility. A key result is that RS2 can exhibit genuinely nonstandard cosmology at high energies, affecting the inflationary dynamics and perturbations, whereas DD and RS1 tend to reproduce 4D behavior at low energies. Overall, extra-dimensional realizations provide a consistent and sometimes advantageous framework for inflation, particularly for α-attractor models, with observational data preferentially supporting a modest number of e-folds and robust scalar tilts. The findings highlight how warping, brane tensions, and the 5D Planck scale modulate perturbation amplitudes and the tensor-to-scalar ratio, impacting the viability and parameter ranges of inflationary scenarios in higher dimensions.

Abstract

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.

Figures (4)

• Figure 1: Bounds on the coupling $\alpha$ and distance $\lambda$ at which deviations from the $1/r^2$ Newton's law should be observed, once new physics is cast in terms of a Yukawa potential $V = - G_{\rm N} \, m/r \left [ 1 + \alpha \, \exp \left ( - r/\lambda \right )\right ]$, from Ref. Lee:2020zjt.
• Figure 2: Left panel: The allowed parameter space in the RS1 model. A ratio $m_1/m_r = 10$ was assumed. The blue region is the current exclusion bound at the LHC Run II. The green-shaded region in the upper left corner is disfavored because the radion lifetime is too long. The gray-shaded region in the lower right corner is excluded, as the effective theory is not reliable for particles whose mass is larger than the effective scale of the model. Right panel: The allowed parameter space in the RS2 model. The two gray regions reflect our theoretical prejudice that the hierarchy $k \leq M_5 \leq M_{\rm P}$ should be satisfied. In the plot, the red solid line shows the values of $(k, M_5)$, which lead to the observed 4D $M_\text{P}$. We also show three lines for the $\sigma_0/M_5^4$ ratio as a function of $k$. The dot-dashed, dashed and dotted lines co correspond to $\sigma_0/M_5^4 = 0.1, 1$ and $10$, respectively. These values will be useful in Section \ref{['sect:inflapot']}, as in the region to the left of the line $\sigma_0/M_5^4 = 0.1$ the effect from extra-dimensional physics is dominant, according to Eq. \ref{['Hubble final']}. On the other hand, to the right of the line $\sigma_0/M_5^4 = 10$ the extra-dimensional corrections to the 4D scenario are negligible.
• Figure 3: Results for the simplest $\alpha$-attractor inflationary model for: $a)$ 4D (top left); $b)$ DD (top right); $c)$ RS1 (bottom left); $d)$ RS2 with $V \ll \sigma_0$, called RS2-A (bottom right). In all cases, the vertical axis represents $N_\star$ and the horizontal axis the inflaton mass $M$. The green contours have been obtained using Planck and BICEP data only, whereas the blue contours include the newest ACT dataset. The numerical results are given in Table \ref{['tab:alpharesults']}.
• Figure 4: Results for the simplest $\alpha$-attractor inflationary model for RS2 with $V \gg \sigma_0$, called RS2-B. The vertical axis represents $N_\star$ and the horizontal axis the ratio of $M_5$ and the inflaton mass $M$. The green contours have been obtained using Planck and BICEP data, only, whereas the blue contours include the newest ACT dataset. The numerical results are given in Table \ref{['tab:alpharesults']}.