Three-form inflation in type II Randall-Sundrum

It has been shown in the last few years that three-form fields present viable cosmological solutions for inflation and dark energy with particular observable signatures distinct from those of canonical single scalar field inflation. The aim of this work is to explore the dynamics of a single three-form in a five-dimensional type II Randall-Sundrum braneworld scenario, in which a three-form is confined to the brane and only gravity propagates in the bulk. We compare the solutions with the standard four-dimensional case already studied in the literature. In particular, we evaluate how the spectral index and the ratio of tensor-to-scalar perturbations are influenced by the presence of the bulk and put constraints on the parameters of the models in light of the recent Planck 2015 data.

Figure 1

Phase space $(\tanh (x),w)$ for $V=e^{{\chi }^2}-1$ at $\mathrm{\Theta }=0.3$.

Figure 2

Phase space $(\tanh (x),w)$ for $V=e^{{\chi }^2}-1$ at $\mathrm{\Theta }=0.9$.

Figure 3

Potential $V(\chi )$ (solid line) and effective potential $V_{\mathrm{eff}}$ (dashed lines) for the potential $V=e^{{\chi }^2}-1$ for different values of $\mathrm{\Theta }$.

Figure 4

Potential $V(\chi )$ (solid line) and effective potential $V_{\mathrm{eff}}$ (dashed lines) for the potential $V=({\chi }^2-{0.5}^2{)}^2$ for different values of $\mathrm{\Theta }$.

Figure 5

Solutions for the system (19)–(20) for the four-dimensional case (dashed line) i.e. for $\mathrm{\Theta }=1$ already studied in [16] and for the RSII model (solid line) when $\mathrm{\Theta }$ is given by Eq. (14) for $V=V_0(e^{{\chi }^2}-1)$, $V_0=10^{-14}$, $\lambda =10^{-12}$ and for the initial conditions $(x_0,w_0)=(2,0.9055)$. The smaller panel shows the change in $\mathrm{\Theta }$, for the RSII model, as the system evolves.

Figure 6

Change in the slow roll parameter $\epsilon$ for the solutions for the system (19)–(20) for the RSII model for $V=V_0(e^{{\chi }^2}-1)$, $V_0=10^{-14}$, $\lambda =10^{-12}$ and for the initial conditions $(x_0,w_0)=(2,0.9055)$. The dashed line marks $\epsilon =1$ just for reference.

Figure 7

Comparison of the spectral index and the tensor-to-scalar ratio against the recent Planck 2015 data [26] for 50 (small dot) and 60 (large dot) $e$-folds for different values of the brane tension $\lambda$. We considered the potential in Eq. (32) with $b=-0.245$. The bars represent, from bottom to top, the solutions with $\lambda =10^{-5}$, $\lambda =3\times {10}^{-9}$ and $\lambda =10^{-10}$ in units ${\kappa }^2=1$).

Figure 8

$\log \lambda$ vs $r$, for the potential (32), with $b=-0.245$, for 60 $e$-folds, for different values of the brane tension $\lambda$.

Figure 9

$\log \lambda$ vs $n_s$, for the potential (32), with $b=-0.245$, for 60 $e$-folds, for different values of the brane tension $\lambda$.

Figure 10

$\log \lambda$ vs $V_0^{*}=V_0\times {10}^{12}$, for the potential (32), with $b=-0.245$, for 60 $e$-folds, for different values of the brane tension $\lambda$.