Vacuum Decay in Time-Dependent Backgrounds

Patrick<EMAIL_ADDRESS>Manthos
<EMAIL_ADDRESS>and Hao<EMAIL_ADDRESS>

Illinois Center for Advanced Studies of the Universe & Department of Physics,

University of Illinois, 1110 West Green St., Urbana IL 61801, U.S.A.

We develop semiclassical methods for studying bubble nucleation in models with
parameters that vary slowly in time. Introducing a more general rotation of
the time contour allows access to a larger set of final states, and typically
a non-Euclidean rotation is necessary in order to find the most relevant
tunneling solution. We work primarily with effective quantum mechanical models
parametrizing tunneling along restricted trajectories in field theories, which
are sufficient, for example, to study thin wall bubble nucleation. We also
give one example of an exact instanton solution in a particular Kaluza-Klein
cosmology where the circumference of the internal circle is changing in time.

##  1 Introduction

Almost fifty years ago, Coleman and collaborators developed powerful Euclidean
path integral methods to compute vacuum decay rates in a semiclassical
expansion [1, 2, 3]. The phenomenon is of clear cosmological and existential
relevance, and continues to drive interesting theoretical work, most
importantly the development of new calculational tools [4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14].

Euclidean path integral techniques are most straightforwardly applied to
systems with time-independent couplings. However, if the tunneling field is
coupled to other dynamically evolving fields, the tunneling potential may
become effectively time dependent. In this paper we discuss generalizations of
the semiclassical approach that can be used to compute tunneling probabilities
in systems with time dependent parameters.

In Sec. 2 we describe a formulation of the semiclassical tunneling problem
convenient for analyzing scenarios with explicit time dependence. The time
dependence must be adiabatic, so that it makes sense to focus on the
instantaneous semiclassical false and true vacua. The main differences with
the usual Euclidean approach are $(a)$ the introduction of a generalized time
contour, and $(b)$ a reformulation of the variational problem, from fixed
initial position and fixed energy, to fixed wavepacket-like initial and final
states at fixed times. Actually, the wavepacket formulation is mainly useful
for interpreting the results of computations where the initial energy is still
provided: it maps the real and imaginary parts of the initial and final
momenta in the semiclassical solution to simple features of the initial and
final packet wavefunctions.

In Sec. 3 we examine the role of the rotation angle of the time contour in
time-independent systems. We find that adjusting this contour allows us to
identify different final states outside the potential barrier, all related to
each other by classical time evolution. Another way to view this is that the
time contour controls the time at which barrier penetration begins. In Sec. 4
we turn to time-dependent systems, beginning with a toy model that can be
solved analytically. In this case the generalized time contour turns out to be
essential: controlled solutions are only obtained if the time contour is not
too close to the Euclidean axis. Next we study thin-wall bubble nucleation in
3+1 dimensions with time dependent parameters. The leading correction to the
semiclassical action can be obtained analytically, and solutions for the
tunneling trajectories are easily obtained nnumerically. We describe how to
make an optimal choice of continuation parameter depending on the path
integral of interest.

In Sec. 5 we discuss gravitational effects. We outline a general treatment,
then consider some particular cases where approximations can make the problem
more tractable. We find, for example, that the leading ${\cal O}(H)$
contribution to the real part of the action vanishes for thin-wall nucleation
on a fixed FRW background. We also describe, in Sec. 6, an example of a
Kaluza-Klein cosmology where a full bubble solution for the fields can be
obtained. In this case the Lorentzian bubble metric is real, but the instanton
is quasi-Euclidean, corresponding to a complex metric.

Sec. 7 contains some comments on directions for further development.

##  2 Semiclassical Formulation

In the usual semiclassical treatment of vacuum decay, one searches for saddle
points of the Euclidean path integral that interpolate between the false
vacuum and a bubble of true vacuum, or in quantum mechanics, between the false
vacuum and a point outside the well.

To get a different perspective on the problem, let us consider real-time
quantum mechanical transition amplitudes of the form

$\displaystyle\langle\psi_{f}(R,$ $\displaystyle
T_{f})|\psi_{i}(R,T_{i})\rangle=N\int
dR_{i}dR_{f}\int_{R(T_{i})=R_{i}}^{R(T_{f})=R_{f}}DR\,e^{iS/\hbar}\,,$
$\displaystyle
S=S_{i}(R_{i})-S_{f}(R_{f})+\int_{T_{i}}^{T_{f}}dt\,L(\partial_{t}R,R,t).$ (1)

The initial and final states are given by wavefunctions parametrized as
$\psi(R)=e^{iS_{i,f}(R)/\hbar}$, where $S_{i,f}$ are complex functions. In
tunneling problems these wavefunctions should be localized inside and outside
of a false vacuum well. We have written explicit factors of $\hbar$ to
indicate that we are interested in states where both the real and imaginary
parts of $S_{i,f}$ contain contributions of order one in the $\hbar$
expansion, like coherent states, but subsequently we will drop the $\hbar$s. A
similar starting point was used to study complexified path integral
trajectories in [15].

Next, we perform a general rotation of the time contour, $t\rightarrow
e^{i\gamma}\tau$. The Euclidean continuation corresponds to $\gamma=-\pi/2$,
but it turns out to be useful to keep $\gamma$ arbitrary. This technique was
applied to the symmetric double well tunneling problem in [16].111Among the
interesting phenomena observed in [16] is that the ordinary Euclidean
instanton, with a hyperbolic tangent profile, becomes a complex space-filling
curve in the real-time limit $\gamma\rightarrow 0$. $\gamma$ will play a
somewhat different role in the decay problems considered here.

After rotation of the time contour, the path integral appearing in the
amplitude is

$\displaystyle\int
dR_{i}dR_{f}\int_{R(T_{i})=R_{i}}^{R(T_{f})=R_{f}}DR\,e^{-S_{c}}\,,$
$\displaystyle S_{c}=-iS_{i}($ $\displaystyle
R_{i})+iS_{f}(R_{f})-i\int_{T_{i}}^{T_{f}}d\tau\,L_{c}(\partial_{\tau}R,R,\tau)\,$
$\displaystyle L_{c}$
$\displaystyle(\partial_{\tau}R,R,\tau)=e^{i\gamma}L(e^{-i\gamma}\partial_{\tau}R,R,e^{i\gamma}\tau).$
(2)

In the leading semiclassical approximation we are interested in solutions to
the the bulk Euler-Lagrange equation, $\delta L_{c}/\delta
R-\partial_{\tau}(\delta L_{c}/\delta\partial_{\tau}R)=0$. The equation of
motion can be replaced by the conservation of energy condition if $L$ does not
depend explicitly on $t$.

Finally, we relax the Dirichlet boundary conditions used in ordinary
semiclassical treatments of the path integral. Since we have introduced
general initial and final states, we can consider unrestricted boundary
variations. Stationarity of the action then gives rise to the following
boundary relations:

$\displaystyle p_{i}\equiv\frac{\delta L_{c}}{\delta\partial_{\tau}R}$
$\displaystyle\bigg{|}_{\tau=T_{i}}=S_{i}^{\prime}(R_{i})$ $\displaystyle
p_{f}\equiv\frac{\delta L_{c}}{\delta\partial_{\tau}R}$
$\displaystyle\bigg{|}_{\tau=T_{f}}=S_{f}^{\prime}(R_{f}).$ (3)

Eq. (3) will be important for understanding the various semiclassical
tunneling solutions we obtain below. These conditions relate the initial and
final momenta of the semiclassical trajectories to the gradient of the
wavefunction at the initial and final positions. A totally generic initial and
final wavefunction, and a given solution $R(\tau)$, will not in general
satisfy Eq. (3), and so will not provide a single saddle point approximation
to the full path integral, including the initial and final states as described
above. Our approach will be to find solutions to the bulk equations of motion
first, and then use Eq. (3) to infer wavefunctions with consistent properties,
such that those solutions provide genuine saddle points of the full path
integral. These are generally weak constraints on the initial and final
wavefunctions, only constraining their gradients at points, but we must
determine them after the fact.

These self-consistency conditions have very natural physical interpretations.
For example, the real part of the momentum at the semiclassical endpoints has
to match the momentum of the initial and final state wavefunctions in the
eikonal approximation.

At leading order in the semiclassical expansion, then, the task is to solve
the equations of motion subject to the boundary conditions (3). The canonical
variables in the solution are generally complex-valued. In addition to the
$e^{i\gamma}$, $S_{i,f}$ are complex if, say, the states defined on real
$R_{i,f}$ are modeled as Gaussian wavepackets or some similar perturbative
ground state. To estimate a total decay probability, one must also extremize
over semiclassically inequivalent final states, and integrate over any zero
modes.

Although this is an acceptable formulation of a tunneling problem, generally
it is not what is done in standard, time-independent tunneling problems. One
instead solves a variational problem with fixed energy:

$\displaystyle E_{E}=E_{FV}$ (4)

where $E_{E}$ is the Euclidean energy and $E_{FV}=V(R_{FV})$ is the value of
the potential in the classical false vacuum $R_{FV}$. Typically there is a
solution to Eq. (4) that starts at $R_{FV}$ and reaches the classical turning
point $R=R_{\tiny TP}$ with zero momentum at some later time, which is
determined by the solution. For example, in an inverted quartic double well
model,

$\displaystyle L=\frac{1}{2}\dot{R}^{2}-R^{2}+\lambda R^{4},$ (5)

the relevant zero-energy Euclidean solution is

$\displaystyle R=\frac{1}{\sqrt{\lambda}}{\rm sech}(\sqrt{2}\tau).$ (6)

(In this case $R_{FV}$ and $R_{TP}$ are only reached asymptotically, but this
is not important.)

From such a solution, we may work backward and identify problems of the type
(1)-(3) to which it is also a solution. In a corresponding problem of the type
(1)-(3), one needs a low-energy initial state, like the perturbative ground
state wavefunction in the false vacuum, for which $S_{i}^{\prime}(R_{\tiny
FV})=0$, and similarly the final state can be some zero-momentum wavepacket
peaked at $R_{\tiny TP}$. These requirements are typical of time-independent
problems analytically continued to Euclidean time.

We will see that other final states arise naturally in two more general
situations: first, we consider time-independent problems with arbitrary time
continuation parameters $\gamma$, and second, cases with explicit time-
dependence in the action. We solve the semiclassical equations of motion with
initial energy as input, finding trajectories that end, typically, at some
$R_{f}\neq R_{\tiny TP}$ with $p_{f}\neq 0$. From these solutions we may read
off amplitudes of the form (1) for which they also provide saddle point
trajectories. In particular Eq. (3) provides relations between the complex
semiclassical momenta and features of suitable wavepackets.

Let us pause and make a couple of tangential comments. First, in relating a
complex-time solution to the original real-time amplitude (1), there is a
little more that needs to be said about the continuation of the parameters
$T_{i,f}$ back to real time. We will discuss this issue below. Second, we note
that it is important _not_ to consider a path integral with fixed initial and
final times and fixed initial and final values of $R$. Position eigenstates
have completely uncertain momentum, and there will always be solutions,
unrelated to tunneling, that have enough energy to summit the barrier
classically. The fixed-energy formulation avoids these complications, as does
the wavepacket formulation, as long as the gradients in the wavepacket are not
too large.

##  3 Time-Independent Tunneling with $t\rightarrow e^{i\gamma}\tau$

In this section we consider the general Wick rotation with parameter $\gamma$.
For illustration, we work with examples of the form

$\displaystyle L=-R_{0}R^{p-1}\sqrt{1-(\partial_{t}R)^{2}}+R^{p}.$ (7)

A large class of field theoretic vacuum decay processes are described by this
effective Lagrangian at the leading semiclassical order, including thin-wall
bubble nucleation in $d=p$ spatial dimensions, Schwinger pair production in a
constant background electric field for $p=1$, and bubble of nothing decays for
$p=d-1$. (Therefore $R$ is assumed to be $\geq 0$, since it plays the role of
a radial collective coordinate.) For example, consider a real scalar field in
3+1 dimensions with a double-well potential and a small symmetry breaking
parameter $\epsilon$,

$\displaystyle
V(\phi)=\frac{1}{2}\lambda(\phi^{2}-a^{2})^{2}-(\epsilon/2a)\phi.$ (8)

It is well-known that the effective theory (7) with $p=3$ is obtained from a
Born-Oppenheimer approximation to the Klein-Gordon field theory with potential
(8). Let us briefly review the construction which will be useful to have in
mind when we include slow time dependence.

For small $\epsilon$ the semiclassical states relevant for tunneling in the
theory (8) are large spherical domain walls. Neglecting the slow accelerations
driven by curvature and the energy gap $\epsilon$, the equation of motion for
the “heavy” degrees of freedom is
$\partial_{t}^{2}\phi-\partial_{r}^{2}\phi+2\lambda(\phi^{2}-a^{2})\phi=0$,
for which the relevant solutions are

$\displaystyle\phi\approx
a\tanh\left[\frac{\sqrt{\lambda}a(R+vt-r)}{\sqrt{1-v^{2}}}\right].$ (9)

Here $R$ is a large radius, $v$ is the wall velocity, the wall tension is
given by $\sigma\propto\sqrt{\lambda}a^{3}$, and the factor of
$\frac{1}{\sqrt{1-v^{2}}}$ accounts for length contraction of the wall
thickness. To obtain an effective action for slow variations of $v$ driven by
curvature and pressure, we replace $R+vt\rightarrow R(t)$,
$v\rightarrow\dot{R}(t)$, and plug these field configurations into the Klein-
Gordon action. Upon so doing each term in the Lagrangian is found to be
proportional to $f(r)\,{\rm
sech}^{4}\left[\frac{\sqrt{\lambda}a(R-r)}{\sqrt{1-\dot{R}^{2}}}\right]$. In
the thin-wall limit $\frac{\sqrt{\lambda}a}{\sqrt{1-\dot{R}^{2}}}R\gg 1$ the
${\rm sech}^{4}$ is sharply peaked around $r=R$, so we may replace
$f(r)\rightarrow f(R)$ and do the integral over space. One obtains an
effective Lagrangian for the collective coordinate $R(t)$ which is, apart from
a overall multiplicative constant, given by Eq. (7) with $p=3$ and
$R_{0}=3\sigma/\epsilon.$ (We will be lax about such multiplicative constants,
which do not affect the equations of motion and can be restored easily in the
on-shell action.)

Since (7) does not depend explicitly on time, there is a conserved energy.
After continuation $t\rightarrow e^{i\gamma}\tau$, the conserved energy is

$\displaystyle
E=e^{i\gamma}\left(\frac{R_{0}R^{p-1}}{\sqrt{1-e^{-2i\gamma}(\partial_{\tau}R)^{2}}}-R^{p}\right).$
(10)

There is a false vacuum at $R_{FV}=0$ with $E_{FV}=0$, and the classical
turning point is $R_{TP}=R_{0}$. It is also informative to write the energy in
terms of the momentum
$P=e^{-i\gamma}R_{0}R^{p-1}\left(\partial_{\tau}R/\sqrt{1-e^{-2i\gamma}(\partial_{\tau}R)^{2}}\right)$,

$\displaystyle
E=e^{i\gamma}\left(\sqrt{P^{2}+R_{0}^{2}R^{2(p-1)}}-R^{p}\right)\,.$ (11)

For $p>1$, $T_{f}=0$, and sufficiently early $T_{i}$, the solutions to Eq. (4)
relevant for tunneling are quite simple. In the Euclidean case
$\gamma=-\pi/2$, they are

$\displaystyle R=\Theta(\tau+R_{0})\sqrt{R_{0}^{2}-\tau^{2}}.$ (12)

The solution is real in the relevant range of $\tau$, and the nucleation point
is $R(0)=R_{0}$, where the momentum vanishes. For other $\gamma$, $R$ is
generally not real, so we seek a zero energy solution for which $R$ is real at
$\tau=0$. Eq. (12) generalizes to

$\displaystyle R=\Theta(\tau-R_{0}\csc\gamma)\sqrt{R_{0}^{2}+(e^{i\gamma}\tau-
R_{0}\cot\gamma)^{2}}.$ (13)

The particle sits at the $R=0$ false vacuum until $\tau=\tau_{0}\equiv
R_{0}\csc\gamma$, then moves in the complex $R$ plane to the nucleation point
at $\tau=0$,

$\displaystyle R(0)=R_{0}|\csc\gamma|.$ (14)

Curiously, the nucleation point only coincides with the classical turning
point in the Euclidean case. Examples are shown in Fig. 1.

Figure 1: Tunneling trajectories (13) with $R_{0}=1$ and a range of $\gamma$
between $-\frac{\pi}{2}$ (Euclidean) and $-\frac{\pi}{10}$. As
$\gamma\rightarrow 0^{-}$ the tunneling starts earlier and ends further
outside the well, with finite classical momentum such that the total energy is
conserved.

We should say a word about the sense in which Eqs. (12), (13) are saddle
points of the action. Since $\dot{R}$ is not defined at
$\tau=\tau_{0}=R_{0}\csc\gamma$, we must give it a definition so that the
trajectories are in the domain of the action functional. Any definition of
$\dot{R}$ as a discontinuous function is sufficient and results in the same
value of the action. The discontinuity in $\dot{R}$ gives rise to a delta
distribution in $\ddot{R}$, which could spoil the stationarity of the action
under variations. However, for $p>1$ the delta function is multiplied by a
positive power of $R$ in the action density, which vanishes at
$\tau=R_{0}\csc\gamma$.

We have assumed that the initial time is early enough,
$T_{i}<R_{0}\csc\gamma$, so that the instanton can “fit.” Apart from this
constraint, the action of the semiclassical solution is independent of
$T_{i}$. The particle sits as long as it needs to in the false vacuum, and
then at $\tau=R_{0}\csc\gamma$ it starts the jump to the nucleation point.

To better understand the solutions for $\gamma\neq-\pi/2$, we can ask what
wavepacket problems of the form (1)-(3) they also solve. Let us examine the
boundary conditions (3). The final momentum of the solutions (13) is

$\displaystyle p_{f}=-R_{0}^{p}|\csc\gamma|^{p-1}\cot\gamma.$ (15)

In particular, it is real, and so is related to the gradient of the phase of
the final wavefunction by Eq. (3). For wavepackets the gradient of the phase
is just the classical momentum $P$.

Furthermore is easily seen that the _real-time_ energy, Eq. (11) with
$\gamma=0$, vanishes for all final states labeled by the real positions
$R\rightarrow R(0)$ given in Eq. (14) and real momenta $P=p_{f}$ given in Eq.
(15). This is because Eq. (11) only depends explicitly on $\gamma$ in an
overall multiplicative factor, and Eq. (3) instructs us to map the
semiclassical trajectory momentum $p_{f}$ and the semiclassical final state
momentum $P$ to each other. The final states at $\tau=0$ correspond to
different points in phase space, related by real-time classical evolution.

We conclude that different values of $\gamma$ correspond to equivalent
tunneling processes in an interesting way. For $-\pi<\gamma<-\pi/2$, the
classical state is a particle somewhere to the right of the potential barrier,
moving up the hill toward the classical turning point. For $-\pi/2<\gamma<0$,
the particle is moving away from the turning point. Classical evolution,
either forward or backward in time, takes the particle to the state at rest at
the turning point, which is the nucleation point for $\gamma=-\pi/2$. We may
interpret this as meaning that different $\gamma$ (in the range
$-\pi/2<\gamma<0$) correspond to different times at which a bubble nucleates
at rest. The semiclassical path integral computes the amplitude to evolve from
“no bubble” at $T_{i}$ to “some bubble” at $T_{f}$, and different $\gamma$
appear to naturally account for the fact that we would also find a nonzero
semiclassical amplitude, to nucleate a bubble with a smaller $p_{f}$, if we
chose earlier $T_{i}$.

Consistent with this interpretation, all of the trajectories have the same on-
shell value of the real part of the action. For example, for $p=3$, one finds

$\displaystyle{\rm
Re}\left[-i\int_{T_{i}}^{T_{f}}d\tau\,L_{c}(\partial_{\tau}R,R,\tau)\right]=\frac{\pi
R_{0}^{4}}{16}$ (16)

independent of $\gamma$.

According to Eq. (2), the tunneling exponent ${\rm Re}(S_{c})$ also receives
contributions from the initial and final state via ${\rm
Im}(S_{i,f}(R_{i,f}))$. These factors can also be taken to be independent of
$\gamma$. For example, the final states can be chosen as wavepackets related
by translation in $R$. (As long as the packets are localized away from the
origin, the fact that $R$ is bounded from below is of no practical
consequence.)

Finally, we must consider the analytic continuation of the amplitude back to
real time. This is different from taking the limit $\gamma\rightarrow 0$ in
the semiclassical solutions, because we have identified solutions with
different values of $\gamma$ as saddle points describing tunneling between
different final states. Instead we want to continue the amplitude in the
complex $T_{i}$-plane back to the real time axis, for _fixed_ final states.
Here we will be content with a plausibility argument. We have already seen
that for fixed $\gamma$ the tunneling exponent is independent of $T_{i}$ (if
$|T_{i}|>R_{0}|\csc\gamma|$, so that the trajectory described by the instanton
has enough time to complete.) Furthermore on physical grounds the real time
decay rate is expected to be constant over timescales long compared to the
perturbative timescales in the false vacuum and short compared to the
lifetime. Therefore we make the plausible assumption that $T_{i}$ can be
continued back to the real axis, trivially, for any final state corresponding
to some $\gamma$. Furthermore we have seen that we obtain the same real part
of the exponent, which controls the lifetime, for any $\gamma$.

##  4 Time-Dependent Tunneling

The formulations of the tunneling problem described above admit a
straightforward generalization to time-dependent systems. We consider two
explicit examples, a toy model and thin-wall bubble nucleation in 3+1
dimensions.

###  4.1 Toy model

The first example is simple enough that we can solve it more or less
analytically:

$\displaystyle L=(\partial_{t}R)^{2}-\left((R_{0}+vt)R-R^{2}\right).$ (17)

As before the degree of freedom is constrained to $R\geq 0$, so that on the
domain of $R$, the classical potential has a false vacuum at $R=0$. We assume
$v>0$ so that the barrier is growing with time. We are free to choose the time
parametrization so that the nucleation time is $T_{f}=0$, and we assume that
$R_{0}$ is large enough so that $R_{0}+vT_{i}>0$. The model (17) has been
scaled so that time is dimensionless and $v$ has the dimensions of $R$.

We perform the same general continuation as previously, $t\rightarrow
e^{i\gamma}\tau$, and seek solutions to the equations of motion on which the
classical Hamiltonian vanishes at some $\tau=\tau_{0}$. The tunneling solution
is

$\displaystyle R(\tau)=\frac{1}{2}\left(R_{0}+e^{i\gamma}\tau
v\right)-\frac{1}{2}\left(R_{0}+e^{i\gamma}\tau_{0}v\right)\cosh\left(e^{i\gamma}\left(\tau-\tau_{0}\right)\right)-\frac{1}{2}v\sinh\left(e^{i\gamma}\left(\tau-\tau_{0}\right)\right).$
(18)

Some examples are shown in the complex-$R$ plane in Fig. 2.

Figure 2: Tunneling trajectories (18) with $R_{0}=1$, $v=1/20$, and a range of
$\gamma$ between $-\frac{\pi}{6}$ and $-\frac{2\pi}{3}$.

To understand the decay process described by the solutions (18), we must
determine the tunneling time, the relevant initial and final states, and the
real part of the on-shell action. We consider each in turn.

Demanding that $R$ is real again at $\tau=T_{f}=0$ determines the tunneling
time $\tau_{0}$ via the somewhat messy relation

$\displaystyle R_{0}\sin\left(\tau_{0}\sin\gamma\right)$
$\displaystyle\sinh\left(\tau_{0}\cos\gamma\right)=v\bigg{(}\sin\left(\tau_{0}\sin\gamma\right)\cosh\left(\tau_{0}\cos(\gamma)\right)$
$\displaystyle-\tau_{0}\left[\sin\gamma\left(\cos\left(\tau_{0}\sin\gamma\right)\right)\cosh\left(\tau_{0}\cos\gamma\right)+\cos\gamma\sin\left(\tau_{0}\sin\gamma\right)\sinh\left(\tau_{0}\cos\gamma\right)\right]\bigg{)}$
(19)

For nonzero $v$ and Euclidean time, $\gamma=-\pi/2$, (19) reduces to
$\tan(\tau_{0})=\tau_{0}$, which has no nontrivial solution. Conversely it is
satisfied by any $\tau_{0}$ if $v=0$ and $\gamma=-\pi/2$. We solve it
perturbatively in small $v$:

$\displaystyle\tau_{0}=\pi\csc\gamma\bigg{(}1-\frac{v}{R_{0}}\text{coth}(\pi\cot\gamma)+{\cal
O}(v^{2})\bigg{)}$ (20)

where the higher order coefficients are lengthy but easily calculable. Near
$\gamma=-\pi/2$, the solution (20) behaves as

$\displaystyle\tau_{0}\sim-\left(\pi+(1/6+\pi^{2})\frac{v^{2}}{\pi
R_{0}^{2}}\right)-\frac{v}{R_{0}}\left(\gamma+\frac{\pi}{2}\right)^{-1}+\frac{v^{2}}{\pi
R_{0}^{2}}\left(\gamma+\frac{\pi}{2}\right)^{-2}+\dots$ (21)

so the small-$v$ expansion is only controlled for

$\displaystyle|\gamma+\pi/2|\gg\frac{v}{\pi R_{0}}.$ (22)

It is also possible to obtain $\tau_{0}$ as an expansion in $\gamma+\pi/2$ for
fixed nonzero $v$, but one finds that $R_{f}$ is small, of order
$\gamma+\pi/2$, and the solutions typically do not describe tunneling out of
the well. In other words, we find the surprising property that in this problem
we must not work too close to the imaginary time axis in order to obtain
classical solutions relevant for tunneling.

Now we examine the initial and final states. To describe tunneling between two
fixed times, we set $R=0$ for $T_{i}\leq\tau\leq\tau_{0}$, and $R$ is given by
Eq. (18) for $\tau>\tau_{0}$. Despite its piecewise nature, it is easy to
check that this trajectory is still a genuine stationary point of the
continued action.222Actually, by constructing a model simple enough to solve
analytically, we have inadvertently complicated this step, because the
potential is not differentiable at the origin. Its derivative must be defined
to be discontinuous, set to zero for all $t$ at $R=0$, and $R_{0}+vt-2R$ for
$R>0$. This is necessary for the vacuum solution $R=0$ to exist. This
technical complication will not arise in the bubble nucleation problem
considered in the next section. As for the final state, we find an ${\cal
O}(v)$ correction to the nucleation point,

$\displaystyle
R_{f}(0)=R_{0}+\frac{1}{2}R_{0}(\cosh(\pi\cot\gamma)-1)-\frac{1}{2}v\sinh(\pi\cot\gamma)+\dots.$
(23)

The $v=0$ contribution exhibits the same behavior as the time-independent
model of the previous section: as $\gamma\rightarrow 0^{-}$ it diverges,
indicating nucleation far outside the barrier. On the other hand, near the
Euclidean time axis (keeping leading terms of order $v^{2}$),

$\displaystyle
R_{f}(0)=R_{0}+\frac{(5\pi^{2}-2)v^{2}}{24R_{0}}+\frac{1}{2}v\pi(\gamma+\pi/2)-\frac{v^{2}}{4R_{0}}(\gamma+\pi/2)^{-2}+\dots.$
(24)

The third term is negative if $\gamma<-\pi/2$ and the fourth term is negative
definite. The second term is positive but for $|\gamma+\pi/2|\gg v/(\pi
R_{0})$ the third term is larger in absolute value so the sum of two is also
negative. Within the small-$v$ approximation, all of these terms must be much
smaller in magnitude than $R_{0}$. So a novel feature of time dependent
systems, which do not conserve energy, is that it is possible to obtain a
nucleation point less than the classical turning point at the nucleation time.
However, in the present model this effect is atypical and marginal in the
small-$v$ expansion.

The final momenta of the trajectories are

$\displaystyle{\rm Re}(p_{f})$ $\displaystyle=-R_{0}\sinh(\pi\cot\gamma)+{\cal
O}(v)$ $\displaystyle{\rm Im}(p_{f})$ $\displaystyle=\pi v\,{\rm
csch}(\pi\cot\gamma)+{\cal O}(v^{2}).$ (25)

As before, the final state is characterized by a nonzero classical momentum
${\rm Re}(p_{f})$ for general $\gamma$. The new feature is a nonzero imaginary
component of $p_{f}$ for nonzero $v$. For typical $\gamma$ not too close to
the Euclidean limit, ${\rm Im}(p_{f})$ is exponentially small, of order
$ve^{-\pi/\gamma}$. ${\rm Im}(p_{f})$ is related to the gradient of the final-
state wavepacket at the nucleation point, in problems of the type (1), by Eq.
(3). The imaginary part of $S_{f}$ and its gradient, for a wavepacket peaked
at some $R_{\star}$, satisfy

$\displaystyle{\rm Im}(S_{f})$
$\displaystyle=(R_{f}-R_{\star})^{2}/2\sigma^{2},$ $\displaystyle{\rm
Im}(S_{f}^{\prime})$ $\displaystyle=(R_{f}-R_{\star})/\sigma^{2}.$ (26)

A typical size of $\sigma$ is $\lesssim{\rm Re}(p_{f}^{-1})$. Consequently the
typical distance from $R_{f}$ to the peak of the wavepacket, for
$|\gamma|\lesssim{\cal O}(1)$, is also exponentially small in $\pi/\gamma$. So
to a good approximation we can still think of the final state as being peaked
at $R_{f}$, if $\gamma$ is not too close to the Euclidean limit, and the state
possesses finite classical momentum.

In the opposite limit, close to Euclidean time (while maintaining the bound
(22)), the classical momentum of the final state ${\rm Re}(p_{f})$ goes to
zero like $\gamma+\pi/2$: the particle nucleates approximately at rest. The
gradient of the final state wavefunction at the nucleation point grows like
$(\gamma+\pi/2)^{-1}$: the stationary point of the wavepacket problem balances
a decrease in the tunneling distance against a decrease in the amplitude of
the final state wavefunction.

The decay amplitude is governed by the real part of the on-shell action,

$\displaystyle{\rm Re}\,S_{c}={\rm Im}\,S_{i}(0)-{\rm
Im}\,S_{f}(R_{f})+\frac{\pi
R_{0}^{2}}{4}+\frac{1}{2}\pi^{2}R_{0}v\cot(\gamma)+{\cal O}(v^{2}).$ (27)

The first two terms depend on the wavepackets at the starting and nucleation
points. Since typical semiclassical solutions permit $R=0$ and $R=R_{f}$ to
lie close to the packet peaks, these terms may be neglected. (There is a
contribution from the wavefunction normalizations, but it is of higher order
in the $\hbar$ expansion.) The rest of (27) corresponds to the real part of
the action ${\rm Re}\,(-i\int L_{c}d\tau)$ evaluated on the semiclassical
trajectory.

The leading term in the $v$ expansion, $\pi R_{0}^{2}/4$, corresponds to the
time-independent semiclassical action. The ${\cal O}(v)$ correction can
actually be computed just from the ${\cal O}(v)$ term in the action, evaluated
on the $v=0$ solution. This is not completely obvious because the perturbation
also changes the solution at the initial and final times, as well as the
initial time. The full perturbation to the on-shell action at order $v$ takes
the form:

$\displaystyle\Delta S_{c}=v\left[\left(ie^{2i\gamma}\int_{\tau_{0}}^{0}d\tau\
\tau
R\right)+\left(iL_{c}(R,\tau=\tau_{0})\partial_{v}\tau_{0}\right)+\left(ip\partial_{v}R\right)\bigg{|}_{\tau=\tau_{0}}^{\tau=0}\right]\bigg{|}_{v=0}$
(28)

The second term arises from the ${\cal O}(v)$ correction to the initial time
$\tau_{0}$. It vanishes because $L_{c}=0$ at the initial time for $v=0$. The
third term involves the momentum $p=2e^{-i\gamma}\partial_{\tau}R$. It arises
from the integration by parts in the kinetic term when the on-shell action is
varied with respect to $v$. Physically, it translates the initial and final
$R$ by $\delta R=v\partial_{v}R$. The $\tau=\tau_{0}$ contribution from this
term vanishes because $\partial_{\tau}R$ vanishes at the initial time. The
contribution from $\tau=0$ is nonzero, but purely imaginary because $p_{f}$ is
real for $v=0$. (Of course, these boundary variations must also cancel if Eq.
(3) is satisfied.) Thus the first, “bulk” term in (28) accounts for the full
${\cal O}(v)$ term in (27), as can be verified explicitly,

$\displaystyle\Delta{\rm Re}S_{c}|_{{\cal
O}(v)}=\frac{1}{2}\pi^{2}R_{0}v\cot(\gamma).$ (29)

Physically, the ${\cal O}(v)$ correction to the tunneling exponent is sourced
by several effects. First, the tunneling takes place at earlier times, when
the potential barrier was smaller by an amount of order $R_{0}v\csc\gamma$.
(Recall that we assume $v>0$, so that the barrier is growing in time; it is
straightforward to reinterpret for negative $v$.) This effect decreases the
action. Second, the tunneling duration $|\tau_{0}|$ (cf. Eq. (20)) has a
${\cal O}(v)$ correction, which also affects the action at ${\cal O}(v)$. The
duration is longer for $\gamma<-\pi/2$, increasing the action, and shorter for
$\gamma>-\pi/2$, further decreasing it. The net ${\cal O}(v)$ shift in the
action is positive for $\gamma<-\pi/2$ and negative for $\gamma>-\pi/2$.

The magnitude of the ${\cal O}(v)$ contribution appears to decrease (increase)
without bound as $\gamma\rightarrow 0^{-}$ ($\gamma\rightarrow-\pi^{+}$), in
both cases because $\tau_{0}$ is diverging to $-\infty$. This reflects two
related facts. First the $v$ expansion of the action breaks down at large
$\tau_{0}\sim R_{0}/v$, and so (27) is not reliable unless $\gamma\ll-\pi
v/R_{0}$ and $\gamma\gg\pi+\pi v/R_{0}$. Then the correction is under control
and the real part of the action is positive definite. Second, we should
incorporate the bounds $R_{0}/v>|T_{i}|>|\tau_{0}|$, so that the tunneling
processes have time to complete, and the barrier is already present at the
initial time $T_{i}$ when the system was known to be in its vacuum state. This
implies the constraint $\pi v|\csc\gamma|<R_{0}$ for small $v$, which limits
$\gamma$ in the same way.

Apart from these bounds, what $\gamma$ should we pick? The smallest action
arises when tunneling starts right away, so that $T_{i}=\tau_{0}$ and the
barrier is minimized. For small $v$ this means we have $\pi\csc\gamma\approx
T_{i}$. Then we find

$\displaystyle\Delta{\rm Re}S_{c}|_{{\cal O}(v),{\rm max}}=-\frac{1}{2}\pi
R_{0}v\sqrt{T_{i}^{2}-\pi^{2}}$ (30)

for any $-\pi>T_{i}>-R_{0}/v.$ The upper bound on $T_{i}$ requires that at
least one instanton “fits” before the nucleation time $T_{f}=0$, and the lower
bound on $T_{i}$ corresponds to the time when the barrier vanishes; note that
here the correction to the action is of the same order as the leading term and
of opposite sign.

The correction (30) is straightforward to understand. Part of the effect is
translating the coupling $R_{0}$ appearing in the leading time independent
action, $\frac{\pi}{4}R_{0}^{2}$, back to the starting time, $R_{0}\rightarrow
R_{0}-vT_{i}$, at first order in $v$. This effect is dominant if
$T_{i}\gg\pi$. The rest of the correction is the leading effect of time
dependence while the particle is moving under the barrier. It vanishes in the
Euclidean case, $T_{i}=\pi$, because here there is an extra $-i\tau$ in the
${\cal O}(v)$ Lagrangian, which makes the contribution purely imaginary.

As we found in the time-independent case, generic values of $\gamma$
correspond to bubbles in motion at $T_{f}=0$. As in the time-independent case
it is natural to conclude that the solution is telling us that nucleation has
effectively already occurred at an early time: we can evolve the
characteristic position and momentum of a suitable wavepacket backward in real
time to a state of smaller momentum closer to the classical turning point at
that time. Our path integral only “asks” about events at $T_{f}$, but it
“knows” about decays that occured earlier.

Holding $\gamma$ fixed, the on-shell action Eq. (27) is independent of $T_{i}$
as long as the instanton fits. Therefore we may trivially continue the
semiclassical amplitude $e^{-S_{c}}$ back to real time, $T_{i,f}\rightarrow
e^{-i\gamma}T_{i,f}$. This continuation only acts nontrivially on $T_{i}$,
since $T_{f}=0$.

We conclude that the solutions (18) provide a semiclassical approximation to
the probability amplitude (1), with some low-energy initial state peaked at
$R=0$ at a specified initial time, and a final wavepacket localized near
$R_{f}$ at $T_{f}=0$, with properties related to the real and imaginary parts
of the final momentum by Eq. (3). As before, we have avoided the more
difficult problem of solving the boundary value problem (3) with specified
$S_{i,f}$ directly, and instead solve the initial value problem with specified
$R$ and $\dot{R}$. The price is that we did not know exactly what family of
wavepacket transition amplitudes we solved until we computed the total
tunneling time, the nucleation point, and final momentum of the solution.

###  4.2 Thin-wall bubble nucleation

Our second example is the 3+1 dimensional thin-wall bubble nucleation model.
We consider the case of a time-dependent domain wall tension proportional to
$\sigma(t)$ and vacuum energy splitting $\epsilon(t)$ (as before, geometric
factors have been absorbed to reduce clutter):

$\displaystyle
L=-\sigma(t)R^{2}\sqrt{1-(\partial_{t}R)^{2}}+\epsilon(t)R^{3}.$ (31)

We assume that $\sigma(t),\,\epsilon(t)$ are positive for all relevant times.

Before analyzing the tunneling processes in this model, let us discuss how
(31) might arise as an effective action in a field theory coupled to a
dynamical sector. For example, we can weakly couple the model (8) to another
scalar $\chi$ rolling in a shallow potential,

$\displaystyle
V(\phi,\xi)=\frac{1}{2}\lambda(\phi^{2}-a^{2})^{2}-\frac{\epsilon_{0}}{2a}(1+g\chi/a)\phi+V(\chi).$
(32)

Suppose that on the timescales of interest we can linearize $V(\chi)\approx
c/a\chi$. The specifications of “weak coupling” and “shallow potential” mean
we take the hierarchy $\lambda a^{4}\gg(c\chi/a,\epsilon_{0})\gg
g\epsilon_{0}\chi/a$ over all relevant $\chi$. Then we can add the effects of
$\chi$ to the Born-Oppenheimer treatment described in Sec. 3. In the first
step, we solve the approximate “heavy” dynamics, obtaining the profile (9) at
zeroth order in $\epsilon_{0}$, $c$, and $1/R^{2}$. Subsequently we take the
thin-wall limit. Then we obtain the “light” dynamics, finding a homogeneous
rolling solution for $\chi$ at leading order in the hierarchy $c\gg
g\epsilon_{0}$,

$\displaystyle\chi=\chi_{0}+vt+\frac{1}{2}(c/a)t^{2},$ (33)

for some $\chi_{0}$ and $v$. Evaluating the Lagrangian on the thin-wall
profile and Eq. (33) and integrating over space, we obtain the quantum
mechanical model (31) with constant $\sigma=16\pi\sqrt{\lambda}a^{3}/3$ and
$\epsilon(t)=4\pi/3[\epsilon_{0}+(g/a)(\chi_{0}+vt+\frac{1}{2}ct^{2})].$

With other weak couplings and potentials we may obtain arbitrary functions
$\sigma(t)$ and $\epsilon(t)$, and possibly more general couplings. It is also
straightforward to include backreaction of the $\phi$ profile on $\chi$ in the
matching of the effective parameters, although this is higher order in $g$ and
may be subleading to corrections to the thin-wall limit.

Now let us study the tunneling solutions of (31), along the same lines as our
analysis of the toy model. Unfortunately, in the present model, typically an
analytic solution is no longer available. However, it is straightforward to
construct numerical solutions, and the correction to the tunneling exponent
may be obtained analytically when the parameters vary slowly.

To construct numerical solutions we require initial conditions. At early times
the relevant nontrivial solutions behave as

$\displaystyle R$ $\displaystyle=c\sqrt{\tau-\tau_{0}}\left(1+{\cal
O}(\tau-\tau_{0})\right)\,,$ $\displaystyle c$
$\displaystyle=ie^{i\gamma/2}\sqrt{\frac{6(\sigma_{0}+e^{i\gamma}\tau_{0}v_{\sigma})}{3i(\epsilon_{0}+e^{i\gamma}\tau_{0}v_{\epsilon})+v_{\sigma}}}$
(34)

where $v_{\sigma,\epsilon}$ are defined by the early time expansions

$\displaystyle\sigma(e^{i\gamma}\tau)$
$\displaystyle=\sigma_{0}+e^{i\gamma}v_{\sigma}(\tau-\tau_{0})+\dots,$
$\displaystyle\epsilon(e^{i\gamma}\tau)$
$\displaystyle=\epsilon_{0}+e^{i\gamma}v_{\epsilon}(\tau-\tau_{0})+\dots.$
(35)

Using (34) as initial data, we may integrate the equations of motion with
specified $\sigma(t)$, $\epsilon(t)$ until $R$ becomes real again, which
defines a nucleation point $R_{f}$. We may adjust $\tau_{0}$ so that the
nucleation time is $T_{f}=0$. Call such a solution $\hat{R}(\tau)$. As before,
we can also attach an arbitrary period of the solution $R=0$ to the beginning
of the solution $\hat{R}(\tau)$ to obtain a tunneling solution of longer total
duration,

$\displaystyle R(\tau)=\theta(\tau)\hat{R}(\tau)\,;\;\;\;\;T_{i}<\tau<0.$ (36)

In Fig. 3 we show example trajectories in a model with linear time dependence
in $\sigma$ and $\epsilon$, such that the barrier height grows linearly in
time.

Figure 3: Left: Tunneling trajectories with $R_{0}=1$,
$R_{0}v_{\sigma}/\sigma=R_{0}v_{\epsilon}/\epsilon=\frac{1}{20}$, and a range
of $\gamma$ between $-\pi/6$ and $-2\pi/3$. Right: the $v$-dependent
correction to the real part of the on-shell action (solid) compared to the
analytic result linearized in $v$ (dashed, Eq. (38)), for $\gamma=-\pi/4$ and
a range of $v$. (Here $\Delta{\rm Re}\ S_{c}$ is normalized to
$R_{0}^{3}\sigma_{0}$ which controls the leading $v$-independent term in the
on-shell action.)

In some cases, we can compute corrections to the tunneling exponent without
knowledge of the full solution. Suppose that the parameters vary slowly enough
that the linearizations (34) hold to a good approximation between
$\tau=\tau_{0}$ and $\tau=0$. The ${\cal O}(v_{\epsilon},v_{\sigma})$
corrections to the on-shell action are given by the ${\cal
O}(v_{\epsilon},v_{\sigma})$ terms in the action evaluated on the
$v_{\epsilon}=v_{\sigma}=0$ solution, Eq. (13).333The reason is essentially
the same as in the analysis of the toy model, Eq. (28). In the bulk time
window the corrections to the solution itself only affect the action at ${\cal
O}(v^{2})$. The various boundary contributions either vanish because $R$
vanishes at the initial time, or are purely imgainary because the final
momentum for $v_{\epsilon,\sigma}=0$ is real, cf. Eq. (15). Thus the entire
${\cal O}(v)$ correction to the real part of the action comes from the bulk
${\cal O}(v)$ term evaluated on the $v=0$ solution (13). We obtain:

$\displaystyle\Delta{\rm Re}\leavevmode\nobreak\ S_{c}|_{{\cal
O}(v)}=\frac{\pi}{4}\cot(\gamma)R_{0}^{4}\left(v_{\sigma}-\frac{3}{4}R_{0}v_{\epsilon}\right).$
(37)

In Fig. 3 we compare the linear approximation to the full numerical result in
some examples, finding agreement consistent with ${\cal O}(v^{2})$
corrections.

As in the toy model studied in the previous section, we can map these
solutions to wavepacket transition amplitudes as in (1). The solution begins
at $R=0$ at an initial $\tau=T_{i}$. The particle remains at rest at the
origin until $\tau=\tau_{0}$, then moves to the nucleation point $R_{f}$ at
$T_{f}=0$, where it possesses some final momentum $p_{f}$. The action is
independent of $T_{i}$ as long as $T_{i}<\tau_{0}$, and so we trivially
continue the semiclassical amplitude $e^{-S_{c}}$ back to real time,
$T_{i}\rightarrow e^{-i\gamma}T_{i}$.

For fixed $T_{i}$, and assuming the barrier is growing monotonically
sufficiently rapidly, the largest tunneling amplitude corresponds to $\gamma$
such that $\tau_{0}=T_{i}$. For slow time dependence, this implies the
relation $R_{0}\csc\gamma\approx T_{i}$, and the correction to the tunneling
exponent

$\displaystyle\Delta{\rm Re}\leavevmode\nobreak\ S_{c}|_{{\cal O}(v),\rm
max}=-\frac{\pi}{4}\sqrt{T_{i}^{2}-R_{0}^{2}}R_{0}^{3}\left(v_{\sigma}-\frac{3}{4}R_{0}v_{\epsilon}\right).$
(38)

Again the correction (38) is easy to interpret. Part of the effect is
translating the couplings $\sigma_{0}$ and $\epsilon_{0}$ appearing in the
leading time independent action,
$\frac{\pi}{16}\sigma_{0}^{4}/\epsilon_{0}^{3}$, back to the starting time
$T_{i}$ in the linearized approximation. This effect is dominant if $T_{i}\gg
R_{0}$. The rest of the correction is the leading effect of time dependence
while the particle is moving under the barrier. The former correction is
captured by the replacements $\sigma_{0}\rightarrow\sigma(t)$,
$\epsilon_{0}\rightarrow\epsilon(t)$, in the leading on-shell action at the
linearized level. However, we see that this is an incomplete accounting of the
${\cal O}(v)$ terms, and corrections that arise from time dependence during
the main part of barrier penetration are obtained in the more systematic
analysis. The correction vanishes for the Euclidean case $T_{i}=R_{0}$ because
the shift in the action is purely imaginary; this is a consequence of
linearizing the time dependence, and nonzero corrections should be expected
even in the Euclidean case if the time dependence is nonlinear.

More generally, the leading correction to the tunneling exponent should be
obtained from the leading time-dependent perturbation to the time-independent
action, integrated over the time-independent trajectories, and minimizing the
real part over $\gamma$.

##  5 Coupling to gravity

In principle, the thin-wall bubble nucleation formalism used above may be
extended to include gravitational effects. In static cases, like the thin-wall
limit of (8), one can construct a suitable membrane effective action. We will
discuss this case below and generalize slightly an effective action
constructed by Visser [17]. The same action was used in [18] to compute decay
rates in special cases using euclidean methods. In the presence of time-
dependent driving fields, however, the construction of a complete membrane
effective action is more cumbersome, and it is convenient to adopt some
approximations. In one class of problems the gravitational field is the
primary source of time dependence, and in favorable circumstances we can
neglect the backreaction of the bubble on the gravitational field. In another
class of problems the backreaction of the bubble on the field is accounted
for, while the effects of time-dependent sources on the gravitational field
can be neglected. We consider each case in turn.

As an example of the first type, suppose that we place the model (8) on a
rigid FRW background, $ds^{2}=-dt^{2}+a(t)^{2}dx_{i}^{2}$. Slow time
dependence corresponds to tunneling processes that proceed quickly compared to
the Hubble time $a/\dot{a}$. Suppose that we are interested in tunneling
between times $t_{i}$ and $t_{f}$, and we normalize the scale factor to 1 at
$t_{f}$. Then we may set $a\approx 1+H(t-t_{f})$ where $H$ is the Hubble
parameter at $t_{f}$. To match to our framework in which tunneling ends at
time zero, we shift the time coordinate, $t\rightarrow t+t_{f}$. Then

$\displaystyle a=1+Ht$ (39)

and the Klein-Gordon action expanded to ${\cal O}(H)$ is

$\displaystyle S\approx\int d^{4}x\left[{\cal L}_{flat}+Ht(3{\cal
L}_{flat}+(\partial_{i}\phi)^{2})\right].$ (40)

The corresponding effective Lagrangian is

$\displaystyle L=-(1+3Ht)\sigma
R^{2}\sqrt{1-(\partial_{t}R)^{2}}+(1+3Ht)\epsilon R^{3}+\frac{\sigma
HtR^{2}}{\sqrt{1-(\partial_{t}R)^{2}}}$ (41)

with $\sigma$ and $\epsilon$ given by their flat-space values,
$\sigma=16\pi\sqrt{\lambda}a^{3}/3$ and $\epsilon=4\pi\epsilon_{0}/3$. We see
that in addition to time-dependent $\epsilon$ and $\sigma$, as in the model
(31), there is also a new term.

The real part of the ${\cal O}(H)$ correction to the on-shell action, obtained
from the ${\cal O}(H)$ term in Eq. (41), is found to vanish. In typical FRW
spacetimes the acceleration is of order $H^{2}$, so if there are ${\cal
O}(H^{2})$ corrections they must be found retaining both the acceleration and
terms of order $H^{2}$ in the action and terms of order $H$ in the solution.
This problem can be studied numerically, but we will not pursue it further
here.

Instead we consider another class of problems, where the gravitational field
is static, but some gravitational backreaction from the bubble is
incorporated. We start with the fully static problem. The Einstein-Hilbert
action coupled to a brane is 444Other boundary terms besides ${\cal T}$ will
be neglected since they do not affect effective action.

$S^{EHGH}_{1,2}=\frac{1}{16\pi G_{N}}\int_{{\cal
M}_{1,2}}\sqrt{|g|}d^{4}x\left({\cal
R}_{1,2}-2\Lambda_{1,2}\right)+\frac{1}{8\pi G_{N}}\int_{{\cal
T}}\sqrt{|h|}d^{3}{y}K_{1,2}-\mu\int_{{\cal T}}\sqrt{|h|}d^{3}y\,,$ (42)

where $K$ is the trace of the extrinsic curvature tensor and ${\cal T}$
denotes the brane worldvolume. The effective action for the location of the
brane $R$ is obtained by integrating out the bulk fields. We assume that the
relevant true and false vacuum spacetimes are approximately de Sitter in the
static slicing. The appropriate action is:

$\begin{split}S^{\text{eff}}_{\text{bubble}}=\frac{4\pi}{8\pi G_{N}}\int
d\lambda&\bigg{[}2R\dot{R}\,\text{sinh}^{-1}\left(\frac{\dot{R}}{\sigma_{\text{dS}_{F}}\sqrt{N^{2}f_{\text{dS}_{F}}}}\right)-2R\dot{R}\,\text{sinh}^{-1}\left(\frac{\dot{R}}{\sigma_{\text{dS}_{T}}\sqrt{N^{2}f_{\text{dS}_{T}}}}\right)\\\
&+2\sigma_{\text{dS}_{T}}R\sqrt{f_{\text{dS}_{T}}N^{2}+\dot{R}^{2}}-2\sigma_{\text{dS}_{F}}R\sqrt{f_{\text{dS}_{F}}N^{2}+\dot{R}^{2}}-8\pi
G_{N}\mu NR^{2}\bigg{]}\,,\end{split}$ (43)

where $\dot{R}=\frac{dr}{d\lambda}$ and the subscripts $T$ and $F$ refer to
the true and false vacua with $f_{dS_{T,F}}=1-R^{2}/L_{T,F}^{2}$. In general
the value of each $\sigma_{dS_{T,F}}$ could be $\pm 1$. This action slightly
generalizes that of Visser [17] (to which we refer for a more detailed
derivation) by the inclusion of a metric degree of freedom on the brane, $N$
which allows manifest reparametrization invariance. Varying this degree of
freedom, $\frac{\delta S_{bubble}}{\delta N}=0$, we find 555We have chosen
$\sigma_{dS_{T,F}}=+1$ which is the physically relevant case when
$0<R<L_{F}<L_{T}$, with same time orienation inside and outside the bubble,
and for small enough positive $\mu$. For an extensive analysis of different
cases see [19] and references therein.

$\sqrt{f_{dS_{T}}+N^{-2}\dot{R}^{2}}-\sqrt{f_{dS_{F}}+N^{-2}\dot{R}^{2}}-4\pi\mu
G_{N}R=0\,.$ (44)

The above equation is the only independent differential equation, since it
turns out that the Euler-Lagrange equation obtained by varying $r$ is simply
the $\lambda$ derivative of (44). Keeping the lapse in the action is important
to obtain the first-order equation (44) from first principles. Having derived
it as an Euler-Lagrange equation, we may now fix the gauge on the brane by
setting $N=1$, which identifies $\lambda$ with the proper time $s$ along
worldlines of fixed angular coordinates on the brane. Then Eq. (44) matches
the junction condition one would obtain by variation of the fields at
${\cal{T}}$ in (42).

We now set $N=1$ and continue $s\to e^{i\gamma^{\prime}}s$. (We use
$\gamma^{\prime}$ instead of $\gamma$ because the proper time is not exactly
the same as the coordinate time which was continued in earlier parts of this
paper; we will derive the relationship below.) The energy is

$E=e^{i\gamma^{\prime}}\frac{\pi}{G_{N}}\left[R\sqrt{f_{dS_{F}}+e^{-2i\gamma^{\prime}}\dot{R}^{2}}-R\sqrt{f_{dS_{T}}+e^{-2i\gamma^{\prime}}\dot{R}^{2}}+4\pi
G_{N}\mu R^{2}\right]\,.$ (45)

Setting $N=1$ and $s\to e^{i\gamma^{\prime}}s$ into Eq. (44) we note that the
energy vanishes on-shell, indicative of time reparametrization invariance. To
find the classical solutions we first note that $R(s)=0$ is a trivial solution
with vanishing energy. To find the other solution branch we rearrange (44)
into the form

$e^{-2i\gamma^{\prime}}\left(\frac{dR}{ds}\right)^{2}+V(R)=0\,,$ (46)

where $V(R)=1-\alpha^{2}R^{2}$, and
$\alpha^{2}=L_{F}^{-2}+\frac{\left(L_{F}^{-2}-L_{T}^{-2}-16\pi^{2}G_{N}^{2}\mu^{2}\right)^{2}}{64\pi^{2}G_{N}^{2}\mu^{2}}$.
The solution with vanishing energy at $s_{0}$ is

$R(s)=\alpha^{-1}\text{cosh}\left[\alpha\,e^{i\gamma^{\prime}}(s-s_{0})+i(\pi/2)\right]\,.$
(47)

The initial proper time $s_{0}=\frac{\pi}{2}\alpha^{-1}\csc(\gamma^{\prime})$
is the largest negative time for which $\text{Im}\,R(0)=0$. The final position
at $s=0$ is $R(0)=\alpha^{-1}\text{cosh}((\pi/2)\text{cot}\gamma^{\prime})$
which falls withing the classical allowed region $R>\alpha^{-1}$ for any
$\gamma^{\prime}$. Similar to the models of Sec. 3 the real time energy
vanishes for the final position and momenta, and final states with different
$\gamma^{\prime}$ are connected to each other at the classical level, by real
time evolution. The on-shell action is given by the following integral
$-i\int_{T_{i}}^{T_{f}}d\tau\,L_{c}=-ie^{i\gamma}\int_{0}^{R(0)}P(R)dR$, where
$P(R)$ is the on-shell momenta

$P(R)=\frac{1}{G_{N}}R\ln\left[\frac{\sqrt{1-L_{T}^{-2}R^{2}}}{\sqrt{1-L_{F}^{-2}R^{2}}}\leavevmode\nobreak\
\left(\frac{\sqrt{\alpha^{2}R^{2}-1}+R\sqrt{\alpha^{2}-L_{F}^{-2}}}{\sqrt{\alpha^{2}R^{2}-1}+R\sqrt{\alpha^{2}-L_{T}^{-2}}}\right)\right]\,.$
(48)

As one might expect the bubble action in Eq. (43) reduces to Eq. (7) with
$p=3$ to leading order in $R^{2}/G_{N}\to 0$ limit [18]. If we set $N=1$ and
expand (43) to leading order in $G_{N}$ we find the following action

$S^{\text{eff}}_{\text{bubble}}=\int ds\left(\frac{4}{3}\pi
R^{3}(\epsilon_{F}-\epsilon_{T})\sqrt{1+\dot{R}^{2}}-4\pi\mu R^{2}\right)\,,$
(49)

where $\Lambda_{T,F}=3L_{T,F}^{-2}=8\pi G_{N}\epsilon_{T,F}$ and $\dot{R}$
means derivation with respect to proper time $s$. To match with (43) we also
need to express the action in terms of the Lorentzian time $t$. To leading
order in $G_{N}$ the relation between $s$and $t$ is usual time dilation
relation $dt\approx ds\sqrt{1+\left(\frac{dR}{ds}\right)^{2}}$. In terms of
Lorentzian time $t$ the Lagrangian of (49) is proportional to (43) with $p=3$
and $R_{0}=\frac{3\mu}{\epsilon_{F}-\epsilon_{T}}\approx\alpha^{-1}$.

Similarly we can recover the solution (13) from (47) in the $R^{2}/G_{N}\to 0$
limit. If we continue the time dilation relation we find

$e^{i\gamma}d\tau\approx
e^{i\gamma^{\prime}}ds\sqrt{1+e^{-2i\gamma^{\prime}}\left(\frac{dR}{ds}\right)^{2}}\,.$
(50)

If we now integrate the time dilation relation, use $\alpha^{-1}\approx R_{0}$
to leading order and the initial conditions $s_{0}$,$\tau_{0}$ we find

$e^{i\gamma}\tau-R_{0}\text{cot}\gamma=\alpha^{-1}\text{sinh}\left(\alpha
e^{i\gamma^{\prime}}s-\frac{\pi}{2}\text{cot }\gamma^{\prime}\right)\,.$ (51)

Making use of (51) it is easy to check that the solution (47) reduces to (13).
The relation between the coordinate time continuation parameter $\gamma$ and
the proper time parameter $\gamma^{\prime}$ can also be found, if we set
$\tau=s=0$ into (51). We find
$\text{cot}\gamma=\text{sinh}\left(\frac{\pi}{2}\text{cot}\gamma^{\prime}\right)$,
which together with (51) implies $R(0)=R_{0}|\csc\gamma|$ as expected.

Now let us include some time dependence. A field theory model that can be
mapped more or less straightforwardly to a time-dependent generalization of
(43) is given by

$\displaystyle
V=\frac{1}{2}[\lambda_{0}+(g/a)\chi](\phi^{2}-a^{2})^{2}-(\epsilon/2a)\phi+V_{0}+V(\chi).$
(52)

As in the nongravitational example, and for similar reasons, we consider the
hierarchy $\lambda_{0}a^{4}\gg V_{0}\gg(V(\chi),\epsilon)>0$, so that the
bubble profile is dominated by the $\phi^{4}$ potential, and the state of the
gravitational fields in the true and false vacua can be approximated by de
Sitter space. We assume $\chi$ and $\phi$ are weakly coupled, which means both
that $g$ is small and that the metric in the bubble background does not
substantially alter the background solution for $\chi$, which is consistent
with the small $\epsilon$ approximation.

The relevant rolling solution for $\chi$ at ${\cal O}(g^{0})$ is homogeneous
in the flat slicing of dS. The static slicing is more convenient for the
present problem, so $\chi$ is a function of the form
$\chi(t+L\log\sqrt{1-r^{2}/L^{2}})$, with $L^{2}\approx 3M_{p}^{2}/V_{0}$ and
$r$ and $t$ the static radial and time coordinates, respectively.

This model is obviously not the most general case, since the background
spacetimes are approximated as time-independent and the underlying field
theory model is certainly fine-tuned. Its main virtue is that the effective
action is almost entirely recycled from the time-independent case. In the
thin-wall limit the bubble dynamics of the model (52) maps to a membrane
effective action of the form (43) with $\mu\rightarrow\mu(t_{dS_{F}},R)$. The
spacetime dependence of the tension appears at ${\cal O}(g)$ and is inherited
from the background solution for $\chi$ in static coordinates.

If we set lapse variation of (43) to zero with $\mu$ time dependent, rearrange
and apply a $\lambda$ derivative on both sides we find

$\frac{d}{d\lambda}\left[\frac{R}{\left(\frac{\partial q}{\partial
N}\right)}\left(2\sqrt{f_{dS_{T}}+N^{-2}\dot{R}^{2}}-2\sqrt{f_{dS_{F}}+N^{-2}\dot{R}^{2}}-\tilde{\mu}R\right)\right]=-NR^{2}\frac{\partial\tilde{\mu}}{\partial
t_{dS_{F}}},$ (53)

where we defined $q:=\sqrt{f^{-1}_{dS_{F}}N^{2}+f^{-2}_{dS_{F}}\dot{R}^{2}}$
and $\tilde{\mu}:=8\pi G_{N}\mu$. Eq. (44) is a first order equation analogous
to a conservation of energy law. Eq. (53) is a second order equation
describing the non-conservation of energy by the varying $\mu$.

From this point one may proceed to rotate the time contour and seek suitable
tunneling solutions connecting the vacuum to nucleated bubble states.
Similarly to the nongravitational cases, at leading order (in the derivative
of $\mu$ with respect to the flat time slicing) corrections to the tunneling
probability may be found from the unperturbed solutions (47) and extremization
over $\gamma$.

##  6 An instanton in a Kaluza-Klein cosmology

We conclude with an example where an exact solution of the field equations in
a time-dependent background may be obtained. The setting is purely
gravitational and describes tunneling in a certain Kaluza-Klein cosmology. We
begin with the metric

$\displaystyle ds^{2}=-dy^{2}+y^{2}d\phi^{2}+dt^{2}+dx^{2}+x^{2}d\psi^{2},$
(54)

with timelike coordinate $y$ and non-standard Kaluza-Klein identification

$\displaystyle(t,\phi)\sim(t+2\pi n\frac{\mu}{\sqrt{\mu-a^{2}}},\phi-2\pi
n\frac{a}{\sqrt{\mu-a^{2}}})$ (55)

and $0\leq a^{2}\leq\mu$. For nonzero $a$ the KK circle is twisted with
spatial Milne-type coordinate $\phi$. Indeed, the metric (54) can be extended
in a similar way to the Milne extension to Minkowski. First we define

$\displaystyle\tilde{\phi}=\phi+(a/\mu)t.$ (56)

At fixed $\tilde{\phi}$ the periodicity is simply $t\sim t+2\pi
n\frac{\mu}{\sqrt{\mu-a^{2}}}$. We then transform $y^{2}=z^{2}-w^{2}$,
$\tanh\tilde{\phi}=z/w$ to obtain

$\displaystyle
ds^{2}=-dz^{2}+\left(1+\frac{a^{2}}{\mu^{2}}(z^{2}-w^{2})\right)dt^{2}+\frac{2a}{\mu}dt(wdz-
zdw)+dw^{2}+dx^{2}+x^{2}d\psi^{2}.$ (57)

This will play the role of our vacuum spacetime. Locally, it is just flat
space, but globally there is a KK circle with interesting properties. For
$a=0$, (57) is ordinary KK spacetime which can decay by nucleating a bubble of
nothing [20]. The instanton is the Euclidean continuation of 5D Schwarzschild-
Tangherlini. For nonzero $a$, the vacuum spacetime is more exotic: it is
cylindrically symmetric, time dependent, and exhibits a Killing horizon for
$\partial_{t}$ at $w^{2}=w_{s}^{2}\equiv z^{2}+\mu^{2}/a^{2}$, beyond which
the KK circles are closed timelike curves (CTCs). The presences of CTCs is
certainly an upsetting feature, and we will assume that (57) is only valid out
to some cutoff $\bar{w}^{2}$, with

$\displaystyle\bar{w}^{2}<w_{s}^{2},$ (58)

beyond which the spacetime is “completed” in a more physical way. (This is
clearly a strong assumption. We will see at least a physical argument below
that for $a^{2}\ll\mu$ there is a parametric separation between the spacetime
region supporting bubble nucleation and the region exhibiting CTCs, lending
some support to the idea that the regions can be studied independently.) The
most interesting feature is that for $w^{2}<w_{s}^{2}$ the proper
circumference of the KK circle is time $(z)$ dependent, shrinking to a minimum
size $2\pi\sqrt{\frac{\mu^{2}-a^{2}w^{2}}{\mu-a^{2}}}$ at $z=0$. Then an
explicit instanton solution describing the decay of this approximate spacetime
may be obtained by analytic continuation of a Myers-Perry black hole
[21].666Ref. [22] also studied continuations of Myers-Perry solutions. We will
differ in the choice of continuation, and therefore also in the
interpretation.

The 5D Myers-Perry solution with one nonzero angular momentum parameter is
given in Boyer-Lindquist-type coordinates by the Lorentzian metric

$\displaystyle\begin{split}ds^{2}=&-dt^{2}+\sin^{2}\theta(r^{2}+a^{2})d\phi^{2}+\frac{\mu}{\rho^{2}}(dt+a\sin^{2}\theta
d\phi)^{2}\\\
&+\frac{\rho^{2}}{r^{2}+a^{2}-\mu}dr^{2}+\rho^{2}d\theta^{2}+r^{2}\cos^{2}\theta
d\psi^{2}\,,\end{split}$ (59)

where $\rho^{2}=r^{2}+a^{2}\cos^{2}\theta$, $a^{2}<\mu$, and the horizon is at
$r_{H}=\sqrt{\mu-a^{2}}$. Let us review the limiting behaviors.
Asymptotically, the metric takes the form

$\displaystyle ds^{2}=-dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}+\cos^{2}\theta d\psi^{2}).$ (60)

The angular part describes an $S^{3}$ in “Hopf” coordinates. $\phi,\psi$ range
from 0 to $2\pi$ and $\theta$ ranges from 0 to $\pi/2$. Next we inspect the
near-horizon limit, following the notation of [22]. Define the Killing vector
$\ell=\partial_{t}-(a/\mu)\partial_{\phi}$. Then the horizon is a fixed
surface of $\ell$, $\ell^{2}|_{r_{H}}=0$. Introducing the coordinate
$\tilde{\phi}=\phi+(a/\mu)t$ as above, which is constant on the orbits of
$\ell$, the near-horizon metric at fixed $\theta,\tilde{\phi},\psi$ takes the
form

$\displaystyle
ds^{2}=2\left(\mu-a^{2}\sin^{2}\theta\right)\left(-\frac{r_{H}}{\mu^{2}}(r-r_{H})dt^{2}+\frac{1}{4r_{H}}\frac{dr^{2}}{r-r_{H}}\right),$
(61)

and in terms of a proper radial coordinate $v$ it is proportional to

$\displaystyle
ds^{2}=dv^{2}-v^{2}\left(\frac{\mu-a^{2}}{\mu^{2}}\right)dt^{2}.$ (62)

Now we perform the analytic continuation

$\displaystyle t\rightarrow it,\leavevmode\nobreak\ \leavevmode\nobreak\
\phi\rightarrow i\phi,\leavevmode\nobreak\ \leavevmode\nobreak\
\theta\rightarrow i\theta.$ (63)

Under this map $\theta$ and $\phi$ become noncompact, and $\theta$ becomes
timelike. The continued metric is

$\displaystyle ds^{2}=\frac{r^{2}+a^{2}\cosh^{2}\theta}{r^{2}+a^{2}-\mu}$
$\displaystyle
dr^{2}+dt^{2}-\frac{\mu}{r^{2}+a^{2}\cosh^{2}\theta}(dt-a\sinh^{2}\theta
d\phi)^{2}$ $\displaystyle+r^{2}$
$\displaystyle\left(-\left[1+\frac{a^{2}\cosh^{2}\theta}{r^{2}}\right]d\theta^{2}+\sinh^{2}\theta\left[1+\frac{a^{2}}{r^{2}}\right]d\phi^{2}+\cosh^{2}\theta
d\psi^{2}\right).$ (64)

Asymptotically in $r$ it behaves as

$\displaystyle ds^{2}=dr^{2}+dt^{2}+r^{2}(-d\theta^{2}+\sinh^{2}\theta
d\phi^{2}+\cosh^{2}\theta d\psi^{2}).$ (65)

Transforming $\tanh^{2}\theta=(z^{2}-w^{2})/x^{2},\leavevmode\nobreak\
r^{2}=x^{2}-z^{2}+w^{2},\leavevmode\nobreak\ \tanh\tilde{\phi}=z/w$, we
recover the vacuum metric (57). The periodicities also match, as can be seen
from the continuation of the near-horizon metric (62). Absence of a conical
singularity at $r=r_{H}$ implies the periodicity $t\sim
t+2\pi\frac{\mu}{\sqrt{\mu-a^{2}}}$ at fixed $\tilde{\phi}$. Then in terms of
the original coordinates we recover precisely the periodicity (55). We also
conclude that the radial coordinate is bounded, $r\geq r_{H}$, capping off
smoothly at $r=r_{H}$.

Thus the spacetime (64) describes something bubble-like embedded in the
cosmology (57). The induced metric on the bubble wall worldvolume
$r\rightarrow r_{H}$ is

$\displaystyle
ds^{2}=-(\mu+a^{2}\sinh^{2}\theta)d\theta^{2}+(\mu-a^{2})\cosh^{2}\theta
d\psi^{2}+\frac{\mu^{2}\sinh^{2}\theta}{\mu+a^{2}\sinh^{2}\theta}d\tilde{\phi}^{2}$
(66)

which is somewhat opaque. On the wall we make the coordinate transformation

$\displaystyle\sinh\tau$ $\displaystyle=\sinh\theta\cosh\tilde{\phi}$
$\displaystyle\cosh\tau\cos\chi$ $\displaystyle=\sinh\theta\sinh\tilde{\phi}.$
(67)

$\tau$ is a timelike coordinate on the wall, and the induced metric at fixed
$\tau$ becomes, to ${\cal O}(a^{2})$,

$\displaystyle ds^{2}=\cosh^{2}\tau\left[\left(\mu+\frac{a^{2}}{4}\left((\cos
2\chi-1)\cosh 2\tau-(3\cos
2\chi+1)\right)\right)d\chi^{2}+\left(\mu-a^{2}\right)\sin^{2}(\chi)d\psi^{2}\right].$
(68)

At $\tau=0$ it has the form of the induced metric on a prolate spheroid
embedded in $\mathbb{R}^{3}$, in spherical coordinates with polar angle $\chi$
and azimuthal angle $\psi$, and ratio of semi-axis lengths
$c/b=1/\sqrt{1-a^{2}/\mu}$. It expands exponentially and deforms as time
evolves.

To summarize, after the continuation (63), the 5D black hole becomes a
Lorentzian geometry corresponding to an expanding bubble, embedded in a KK
space with nonstandard identification of the KK circle. It is therefore a
candidate decay product for the classical solution (57). More precisely, since
we assume the solution (57) is only valid for
$w^{2}<\bar{w}^{2}\ll(\mu/a)^{2}$, we have a candidate decay product so long
as the bubble fits, $r_{H}^{2}\ll\bar{w}^{2}$. For small $a$ the cutoff must
fall between $\mu\ll\bar{w}^{2}<(\mu/a)^{2}$.

Is there a saddle point of the gravitational action with action $S$, such that
the leading order decay amplitude is $e^{-Re(S)}$?

Because the black hole is stationary instead of static, there is no real
Euclidean section that can be obtained by analytic continuation of the
coordinates alone. We may obtain a suitable candidate from (59) by making the
$\tilde{\phi}=\phi+(a/\mu)t$ substitution and continuing only $t\rightarrow
it$. The result is a “quasi-Euclidean” metric with a bubble wall at $r=r_{H}$
and the Euclidean vacuum, Eq. (57) with $z\rightarrow-iz$, at large radius
$\sqrt{x^{2}+w^{2}}$. The induced metric on the $z=0$ nucleation hypersurface
agrees between the quasi-Euclidean instanton and the Lorentzian nucleated
bubble 64.

The action of the quasi-Euclidean solution is

$\displaystyle S=\frac{\pi^{2}\mu^{2}}{G_{5}\sqrt{\mu-a^{2}}}\,.$ (69)

It recovers the action of Witten’s bubble in the limit $a\rightarrow 0$ and
diverges in the extremal limit $a\rightarrow\sqrt{\mu}$, where the KK circle
size diverges.

The quasi-Euclidean solution describes the amplitude to evolve from a state
with no bubble at some early time $z_{i}$ to a state with a bubble at some
late time $z_{f}$. The action controls the amplitude for the bubble to
nucleate, and the analytic continuation describes the subsequent bubble
evolution. Since the background is time dependent there is no time translation
symmetry and the instanton computes a probability rather than a rate. Note
that the transition occurs when the KK circle is smallest near the origin of
the spatial coordinates. This corresponds to the smallest potential barrier.

We conclude this section with a few comments.

  * •

The bubble of nothing example studied in this section is similar to previous
examples we have seen, in that it involves a saddle point off in the
complexified phase space. Quasi-euclidean metrics based on the continuation of
Kerr-like solutions are known to play a variety of useful other roles [23,
24].

  * •

In a few other ways, the present example is different. First, the presence of
a time reversal symmetry means there is a time when the quasi-Euclidean
solution returns to the real axis with zero momentum. (Indeed, in the notation
of previous sections we have only considered the Euclidean continuation
$\gamma=-\pi/2$.) Also, we have only found an explicit solution when the
initial time is early, so that the KK circle is shrinking, and the final time
is late, when it is expanding again. In previous examples we were able to find
families of instantons for more general initial and final times. Finally, the
bubble is neither thin wall nor spherically symmetric. In all other cases, the
background was rotationally invariant and the bubble was thin-wall and
spherical by assumption.

  * •

At first sight, a more natural starting point would be the background (54)
with standard periodicity $\phi\sim\phi+2\pi$. However, this spacetime is
singular at $y=0$, where the circle shrinks to zero size. The instanton in
this case is also singular and has zero action – there is no barrier to
tunneling at this moment. The purpose of the twisted periodicity is to
regulate the circle so that it is never smaller than $\mu/\sqrt{\mu-a^{2}}$ at
the origin.

##  7 Discussion

We have surveyed the effects of slow explicit time dependence on barrier
penetration and bubble nucleation in some simple models. The problem is
interesting not because the effects are typically dramatic, but because the
usual semiclassical techniques require some development in order to calculate
the effects. It appears to be useful generalize the Wick rotation to other
contours in the complex time plane, which provides semiclassical access to
different tunneling times during a fixed time window. “Nucleation,” in the
sense of the bubble in the final state, need not occur at rest, and
semiclassical trajectories with nonzero final momentum can be interpreted as
providing a saddle point solution for a path integral with wavepacket initial
and final states, with the real and imaginary parts of the final momentum
mapping to phases and gradients of the wavepacket.

Reduction to effective quantum mechanical models simplified the problem
enormously, essentially from a nonlinear PDE back to a nonlinear ODE, as one
has in the fully field-theoretic treatment of $O(4)$-symmetric problems. It is
unclear if there is a practical generalization of our methods to full field
theories, although in some favorable cases exact solutions can be guessed.

We conclude with a few directions for future work. First, our analysis was
entirely at the leading semiclassical order. The fluctuation determinant has
interesting technical features, requiring the solution of flow equations to
identify suitable middle-dimensional integration cycles [25]. Although we have
seen a variety of physical features that suggest the complex trajectories
studied here indeed compute meaningful tunneling probabilities, explicit
computation of the fluctuation determinant in a model would provide further
support for the interpretation.

Second, in the effective model of membrane nucleation coupled to gravity, the
construction of the effective action itself is somewhat nontrivial, and we
only illustrated the construction in some relatively simple classes of models.
Further analysis and generalization could be of interest for cosmological
applications.

Finally, our study of exact solutions was limited to a single example. Some
continuations of other black hole and black ring solutions in higher
dimensions presumably describe bubble of nothing decays in more exotic Kaluza-
Klein cosmologies, and it would be interesting to catalog them.

## Acknowledgements

We thank Szilard Farkas, Adam Nahum, LianTao Wang, and Yikun Wang for useful
discussions. This work was supported in part by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics under award number DE-
SC0015655.

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