4. Inflationary bubble nucleation amplitude

The next case study is somewhat more subtle and we discuss it to illustrate the applicability of our effective method. To the extent that this system may arise as a fluctuation of the gravitational vacuum, we interpret it as a possible constituent of the Planckian spacetime foam even though it does not correspond to any class of wormholes. What we have in mind is the nucleation of a false vacuum deSitter bubble in a Minkowski background which represents the key mechanism proposed in Ref. [11], and expanded in Ref. [10], to generate quantum mechanically an inflationary domain. From our vantage point, the difficulty of that proposal is the presence of a virtual black hole, decaying through Hawking radiation, as an intermediate state between the initial Minkowski state and the final Minkowski$+$deSitter state. This intermediate state involves the still obscure issue of the final stage of black hole evaporation. Interestingly enough, it is possible to bypass this difficulty by choosing a negative surface tension and vanishing total mass energy for the original quantum fluctuation that triggers the process in the first place. Physically, this assumption of negative surface tension may be justified by a simple analogy with the multiphase vacuum of QCD. If the unified field theory undergoing primordial phase transitions is of the Ginzburg-Landau type, then, for some choice of the coupling constants, bags can form around test charges with positive volume and negative surface energy [16]. The sign of the surface tension follows from the negative condensation energy. The vacuum state for such a model behaves as a type II superconductor with maximal boundary surface between the normal (non-confining) phase and the ordered (confining) phase [17].

With the above choice of surface tension and vanishing total mass-energy, the initial and final states are degenerate in energy and a spontaneous transition between them is allowed without an intermediate blackhole state.

Some preliminary remarks on the classical dynamics of a deSitter bubble will be helpful in order to clarify our final result. The classical equation of motion for the bubble trajectory is

  

$$\dot{R} ^{2} = - 1 + \frac{R ^{2}}{R _{B} ^{2}} \quad ,$$ (21)

$$R _{B} = \frac{8 \pi \vert \rho \vert \, G _{N}} {16 \pi ^{2} \rho ^{2} G _{N} ^{2} + 1 / H ^{2}} = \frac{R _{0}}{1 + R _{0} ^{2} / 4 H ^{2}} \quad ,$$ (22)

where $H$ is the radius of the deSitter cosmological event horizon, and $R _{0} = 3 \vert \rho \vert / \epsilon _{in}$ is the nucleation radius in the absence of gravity. Equation (21) admits only a bounce solution which represents an infinite domain of deSitter vacuum collapsing down to a nonvanishing minimum radius $R _{B} < R _{0}$, and then re-expanding indefinitely. Note also that $R _{B} \le H$, and that the equality holds only if the surface tension and the internal vacuum energy density are tuned so that $\epsilon _{in} = 6 \pi \rho ^{2} G _{N}$.

Having assumed a negative surface tension, we retrace our steps as in the previous case study: the matching condition now is

  

$$\sigma _{in} R \, \sqrt{1 + \dot{R} ^{2} - R ^{2} / H ^{2}} - \sigma _{out} R \, \sqrt{1 + \dot{R} ^2} = 4 \pi \rho \, G _{N} R ^{2}$$ (23)

$$H ^{2} \equiv 3 / \Lambda _{in} \quad ,$$ (24)

and gives us

  

$$\sigma _{out} = + 1 \quad ,$$ (25)

$$\sigma _{in} = - \mathrm{sign} \left( 16 \pi ^{2} \rho ^{2} G _{N} ^{2} - \frac{1}{H ^{2}} \right) = - \mathrm{sign} (R _{0}-2 H) \quad .$$ (26)

Equations (25) and (26) show that, while the sign of $\sigma _{out}$ is fixed to $+1$ along the bubble trajectory, the sign of $\sigma _{in}$ depends on the relative size of $R _{0}$ and $2H$. The two nucleation modes actually correspond to two different Penrose diagrams, but we shall omit their discussion in this brief communication.

Presently, for reasons of clarity and conciseness, we choose to discuss the case $${R _{0} < 2 H \rightarrow \sigma _{in} = + 1}$$. The corresponding semi-classical solution, which describes the nucleation of an expanding deSitter bubble, is obtained by matching the expanding half of the classical bounce to a quantum tunneling solution. Then, the classical turning point acquires the meaning of nucleation radius. The corresponding lagrangian and hamiltonian are,

   

$$L ^{\mathrm{eff}} = \frac{R}{G _{N}} \left( \sqrt{\beta _{in} + \dot{R} ^{2}} - \sqrt{1 + \dot{R} ^2} - 4 \pi \rho \, G _{N} R \right) +$$

$$- \frac{R \dot{R}}{G _{N}} \left[ \sinh^{-1} \left( \frac{\dot{R}}{\sqrt{\beta _{in}}} \right) - \sinh^{-1} \dot{R} ^{2} \right] \quad ,$$ (27)

$$P _{R} = - \frac{R}{G _{N}} \left[ \sinh ^{-1} \left( \frac{\dot{R}}{\sqrt{\beta _{in}}} \right) - \sinh ^{-1} \dot{R} \right]$$ (28)

$${\mathcal{H}} = 4\pi \rho R ^{2} + \frac{R}{G _{N}} \left[ \beta _{in} + 1 + 2 \s... ...beta _{in}} \cosh \left( \frac{G _{N} P _{R}}{R} \right) \right] ^{1/2} \quad .$$ (29)

Note that we have fixed the phase of $\sqrt{\beta _{in}}$ and $\sqrt{\beta _{out}}$ by the condition that the above hamiltonian coincides with the hamiltonian (18) in the limit of vanishing internal energy density. The classical equation ${\mathcal{H}} = 0$, gives

$$P _{R} = \frac{R}{G _{N}} \cosh ^{-1} \left[ \frac{(4 \p... ...} - 1 - \beta _{in}} {2 \sqrt{\beta _{in}}} \right] \quad .$$ (30)

This enables us to evaluate the tunneling amplitude

 

$$B = 2 \int _{0} ^{R _{B}} dR \left\vert P _{E} (R) \right\vert$$

$$= - \frac{1}{G _{N}} \int _{0} ^{R _{B}} dR R ^{2} \frac{d}{dR} \ar... ...(4 \pi \rho \, G _{N} R) ^{2} - 1 - \beta _{in}} {2 \sqrt{\beta _{in}}} \right]$$ (31)

and an explicit calculation yields

 

$$B = 4 \pi \vert \rho \vert \int _{0} ^{R _{B}} dR R ^{2} \left( 1 - \... ...left( \frac{R ^{2}}{4 \pi \vert \rho \vert \, G _{N} R _{B} H ^{2}} - 1 \right)$$

$$= \frac{\pi ^{2} \vert \rho \vert}{2} R _{B} ^{2} R _{0} \quad .$$ (32)

Our last comment concern the physical interpretation of this result. The model discussed above describes the quantum birth of an inflationary bubble in a Minkowski background. In connection with this process, we find that there is a lingering ambiguity in the published literature. It is indeed interesting, and perhaps somewhat puzzling, that the initial radius and the nucleation rate in this case are the same as for the false vacuum decay, that is, the nucleation of a Minkowski bubble in a deSitter background, originally discussed by Coleman and DeLuccia [18]. With hindsight, this coincidence is hardly surprising since the two cases appear to be completely symmetrical due to the fact that the euclidean trajectory interpolating between the two vacuum states is the same in both cases. However, there is a difference, even at the classical level, and it lies in the global structure of the spacetime manifold in the two cases. The point is that, for an inflationary bubble in a Minkowski background, at a given instant, say the nucleation Minkowski time, all the points in the interval $0 \le r \le R _{B}$ suddenly undergo a phase transition from the Minkowski to the deSitter geometry. Then, the new vacuum domain, driven by the negative pressure of the false vacuum, expands exponentially, eventually filling up the whole spacetime. In contrast, a true vacuum bubble, no matter how large, will never fill up the whole deSitter manifold. As a matter of fact, in this difference lies the problem of the ``graceful exit'' from the inflationary stage [19].