Dynamics and instability of false vacuum bubbles

This paper examines the classical dynamics of false-vacuum regions embedded in surrounding regions of true vacuum, in the thin-wall limit. The dynamics of all generally relativistically allowed solutions—most but not all of which have been previously studied—are derived, enumerated, and interpreted. We comment on the relation of these solutions to possible mechanisms whereby inflating regions may be spawned from noninflating ones. We then calculate the dynamics of first-order deviations from spherical symmetry, finding that many solutions are unstable to such aspherical perturbations. The parameter space in which the perturbations on bound solutions inevitably become nonlinear is mapped. This instability has consequences for the Farhi-Guth-Guven mechanism for baby universe production via quantum tunneling.

Figure 1

Conformal diagram of the Schwarzschild–de Sitter geometry for $3M<{\Lambda }_{+}$.

Figure 2

Conformal diagram for the Schwarzschild–de Sitter geometry when $3M={\Lambda }_{+}$.

Figure 3

Conformal diagram for the de Sitter geometry.

Figure 4

The potential for various $Y$.

Figure 5

Potential for $A=9$ and $B=15$. The two dashed lines labeled $Q_{\mathrm{SdS}}$ and $Q_s$ represent the exterior and the interior horizon crossings, respectively. The vertical dotted lines denote the regions in which ${\beta }_{\mathrm{SdS}}$ and ${\beta }_{\mathrm{dS}}$ are positive and negative. Various trajectories are noted.

Figure 6

Potential for $A=1$ and $B=6$. For this choice of parameters, the sign change in ${\beta }_{\mathrm{SdS}}$ occurs to the left of the maximum in the potential. Various trajectories are noted.

Figure 7

Potential for $A=1$ and $B=2$. For this choice of parameters, the sign change in ${\beta }_{\mathrm{SdS}}$ occurs to the left of the maximum in the potential and there is no ${\beta }_{\mathrm{dS}}$ sign change. Various trajectories are noted.

Figure 8

Potential for $A=2.9$ and $B=3$. For this choice of parameters, the sign change in ${\beta }_{\mathrm{SdS}}$ occurs to the right of the maximum in the potential and there is no ${\beta }_{\mathrm{dS}}$ sign change. Various trajectories are noted.

Figure 9

Conformal diagrams for the allowed one-bubble spacetimes. The dS and SdS diagrams are matched across the bubble wall (line with arrow), and the physical regions shaded. Solutions 3 and 4 share the same SdS diagram. Regions which do not contain antitrapped surfaces are shaded green (light), regions which do are shaded blue (dark).

Figure 10

Conformal diagrams for the allowed one-bubble spacetimes. The dS and SdS diagrams are matched across the bubble wall (line with arrow), and the physical regions shaded. For solutions with qualitatively similar SdS diagrams, the various options for the dS interior are listed.

Figure 11

Tunneling spacetime.

Figure 12

Conformal diagrams for the thermal decay of de Sitter vacua. The upper diagram corresponds to $B<3(A-1)$, the lower diagram corresponds to $B>3(A-1)$.

Figure 13

Contour plot of ${log}_{10}[f(l=1,z_0,Q,T_{\max })]$ (left) and ${log}_{10}[g(l=1,z_0,Q,T_{\max })]$ (right) for $M_I={10}^{14}\mathrm{GeV}$ (top) and $M_I=100\mathrm{GeV}$ (bottom).

Figure 14

$f(l,z_0=0.5,Q=-{10}^{-4},T)$ for various $l$. The inset shows the oscillatory behavior of $f$ for large $l$.