New mechanism for bubble nucleation: Classical transitions

Given a scalar field with metastable minima, bubbles nucleate quantum mechanically. When bubbles collide, energy stored in the bubble walls is converted into kinetic energy of the field. This kinetic energy can facilitate the classical nucleation of new bubbles in minima that lie below those of the “parent” bubbles. This process is efficient and classical, and changes the dynamics and statistics of bubble formation in models with multiple vacua, relative to that derived from quantum tunneling.

Figure 1

Schematic diagram of a 3-minima potential. The false vacuum at ${\varphi }_1$ quantum mechanically nucleates bubbles into regions of ${\varphi }_2$. Sufficiently energetic collisions of two ${\varphi }_2$ bubbles can classically nucleate a bubble of ${\varphi }_3$. The two barrier heights are defined with respect to $V({\varphi }_1)$. Superscript “$b$” means barrier.

Figure 2

Conformal diagram showing the values of $\varphi$ on the $X-\tau$ plane (left) and the $Y-\tau$ plane (right). Our conformal coordinates, $(\tau ,X,Y,Z)$, are chosen such that $X$ is the axis through the centers of the bubbles and $\tau$ is defined in the caption of Fig. 3. The collision occurs at $X=0$ (analogous to $x=0$ in the text). Regions of ${\varphi }_1$ appear off-white, regions of ${\varphi }_2$ appear grey and regions of ${\varphi }_3$ appear black. In this simulation, the centers of the two bubbles are $\sim 0.45H_0^{-1}$ apart at nucleation. When these bubbles collide (at $\tau \approx 17.5$) the walls have a Lorentz factor $\gamma \approx 1.5$. The slight ripples in the middle of each bubble are numerical artifacts.

Figure 3

Time evolution of two bubbles whose centers are $\sim .36H_0^{-1}$ apart at nucleation. When these bubbles collide (at $\tau \approx 9.0$) the walls have a Lorentz factor $\gamma \approx 1.3$. We use a conformal time $a(t)d\tau =\sqrt{\lambda }{\varphi }_0^{2}dt$, where $a(0)=1$ at the beginning of the simulation and $dt$ is the usual proper time of an Friedmann-Robertson-Walker spacetime. Notice the Lorentz contraction of the wall thickness as the bubble expands.

Figure 4

Time evolution of two bubbles whose centers are $\sim .45H_0^{-1}$ apart at nucleation. When these bubbles collide (at $\tau \approx 17.5$) the walls have a Lorentz factor $\gamma =1.8$. We use a conformal time $a(t)d\tau =\sqrt{\lambda }{\varphi }_0^{2}dt$, where $a(0)=1$ at the beginning of the simulation and $dt$ is the usual proper time of an Friedmann-Robertson-Walker spacetime.