Gravitational Kaluza-Klein modes in the string-cigar braneworld

In this work we analyze the properties of the gravitational Kaluza-Klein (KK) modes in two stringlike braneworlds, the thin Gherghetta-Shaposhnikov (GS) model and the thick string-cigar model. The string-cigar model is a smooth generalization of the GS model that undergoes a Ricci geometrical flow. We find a new massless mode in both models satisfying the respective Schrödinger equations. By means of a numerical analysis, we obtain the complete graviton spectrum and its respective eigenfunctions. The KK spectrum exhibits the usual linear regime for large discrete index $n$ and we find a new decreasing regime for small $n$. Moreover, there is an asymmetric mass gap between the massless mode and the massive KK tower. The mass gap in the GS model is bigger than in the string-cigar model. In addition, the mass gap remains invariant upon the geometrical flow. It turns out that in the string-cigar model the brane structure smoothes and amplifies the KK modes near the brane core. The presence of a potential well in the string-cigar scenario allows the existence of resonant massive gravitons for small masses.

Figure 1

Energy density of the string-cigar braneworld.

Figure 2

GS model mass spectrum obtained by the matrix method. The subgraph is the linear regime scale magnification.

Figure 3

Mass spectrum of the string-cigar model obtained by the matrix method. It is very similar to the GS model except for the absence of a transient state.

Figure 4

GS model numerical eigenfunction for $m=5.172\times 10^{-7}$.

Figure 5

GS model numerical eigenfunction for (a) $m=37.156$ and (b) $m=22.729$.

Figure 6

String-cigar model numerical eigenfunction for (a) $m=37.022$ and (b) $m=22.721$.

Figure 7

Numerical integral solution of the transformation $z(\rho )$.

Figure 8

The analogue quantum potential $U(z)$.

Figure 9

Plot of the relative probability $P(m)$. The peaks reveal the resonant massive states. For $m^2\gg U_{\max }$, a plateau is formed at $P=0.1$ corresponding to the plane wave regime.

Figure 10

Solutions of the Schrödinger-like equation for the masses corresponding to resonant peaks. The first solution has the highest probability to interact with the brane.