The Shape of Gravity in a Warped Deformed Conifold

Abstract

We study the spectrum of the gravitational modes in Minkowski spacetime due to a 6-dimensional warped deformed conifold, i.e., a warped throat, in superstring theory. After identifying the zero mode as the usual 4D graviton, we present the KK spectrum as well as other excitation modes. Gluing the throat to the bulk (a realistic scenario), we see that the graviton has a rather uniform probability distribution everywhere while a KK mode is peaked in the throat, as expected. Due to the suppressed measure of the throat in the wave function normalization, we find that a KK mode's probability in the bulk can be comparable to that of the graviton mode. We also present the tunneling probabilities of a KK mode from the inflationary throat to the bulk and to another throat. Due to resonance effect, the latter may not be suppressed as natively expected. Implication of this property to reheating after brane inflation is discussed.

Figures (6)

• Figure 1: A schematic picture of the Calabi-Yau manifold is presented here. The large circle given by dashed line represent the 3-cycle where NS-NS three form $H_3$ is turned on. The smaller circle in the throat stands for the 3-cycle where the R-R three form $F_3$ is turned on. Also shown are D7-branes wraping 4-cycles. There may exists a number of throats like the one shown here. There is a mirror image of the entire picture due to the ${\bf{IIB/Z_2}}$ orientifold operation.
• Figure 2: Here is a schematic picture of the conifold (dashed line) and the deformed conifold (solid line). The apex is at $r=0$. The conifold is deformed at the tip such that $r=\epsilon$ is now an $S^3$, where $S^{2}$ has shrinked to zero. The dashed circle at constant $r$ represents the base of the conifold which is a $T^{1,1}$. For large $r$, the base of the deformed conifold asymptotically approaches $T^{1,1}$.
• Figure 3: In the left figure $V_{eff}(\tau)$ is plotted for $\mu= j-1=l=0$, i.e. excited graviton with no angular momentum. In the right figure, two terms of $V_{eff}$ in Eq.(\ref{['VeffS']}) for the first excited state are plotted. It is clear that the last term in $V_{eff}$ quickly reaches the asymptotical value $4/9$, while the first term quickly falls off. This justifies our approximations used in Eq.(\ref{['approx1']}).
• Figure 4: In this figure the qualitative shape of $V_{eff}$ given in Eq.(\ref{['Vtotal']}) is plotted. $V_{eff}$ in the throat region quickly reaches the vale $4/9$. The throat is glued to the bulk at $\tau_c$. The bulk ends at $T$, the position of the O-3 plane. There is a mirror image of this potential for $T < \tau < 2\,T$.
• Figure 5: In Figure a, $\beta_{21}$ (measures the KK mass spectrum) versus $\beta_{13}$ is plotted by solving Eq.(\ref{['energy']}) numerically. The curve formed by box, big circle and small circle represent the first, second and third excited states respectively. Here $\beta_1=1/50$ but the shape of the plots do not change if other values for $\beta_1$ are chosen as input parameter. In the asymptotic region when $\beta_{21} \sim$ constant, the throat dominates and the spectrum is given by Eq.(\ref{['limit1,2']}) (the horizontal lines in b) . The other asymptotic region is when the bulk dominates and the spectrum is given by Eq.(\ref{['limit2,2']}) (the curved lines in b). In between the spectrum is given by the superposition of these 2 sets of curves. The transition between these two regions happens earlier for the excited modes. Also the excited modes halt temporarily when $\beta_{21}$ approaches the following lower roots of $J_2$ and afterwards drops as given by Eq.(\ref{['limit2,2']}). In the right figure the solutions of Eq.(\ref{['limit1,2']}) and Eq.(\ref{['limit2,2']}) versus $\beta_{13}$ are plotted. The agreement between our numerically obtained curve and the superposition of the theoretical curves is very good. Note that the energies of different KK modes never cross or become degenerate.