♮♮institutetext: Department of Physics, Kobe University,
Kobe 657-8501, Japan††institutetext: International Center for Quantum-field
Measurement Systems for Studies of the Universe and Particles (QUP),
KEK, Tsukuba 305-0801, Japan ♭♭institutetext: Particle Theory and Cosmology
Group, Center for Theoretical Physics of the Universe,
Institute for Basic Science (IBS), Daejeon, 34126, Korea

# Stochastic tunneling in de Sitter spacetime

Taiga Miyachi ♮,† Jiro Soda ♭ and Junsei Tokuda<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Tunneling processes in de Sitter spacetime are studied by using the stochastic
approach. We exploit the Martin–Siggia–Rose–Janssen–de Dominicis (MSRJD)
functional integral to obtain the tunneling rate. The applicability conditions
of this method are clarified using the Schwinger-Keldysh formalism. In the
case of a shallow potential barrier, we reproduce the Hawking-Moss (HM)
tunneling rate. Remarkably, in contrast to HM picture, the configuration
derived from the MSRJD functional integral satisfies physically natural
boundary conditions. We also discuss the case of a steep potential barrier and
find an interesting Coleman-de Luccia (CDL) bubble-like configuration. Our
results demonstrate how the bubble nucleation process could be described in
the stochastic approach. Our method turns out to be useful for investigating
various tunneling processes during inflation.

KOBE-COSMO-23-09 CTPU-PTC-23-33

##  1 Introduction

Tunneling is an important non-perturbative phenomenon in cosmology. Hence, it
is crucial to understand tunneling processes in curved spacetime. In
particular, tunneling in de Sitter spacetime is worth studying in detail. The
reason is that de Sitter spacetime is the simplest non-trivial curved
spacetime. Moreover, the results obtained there can be applicable to tunneling
phenomena during inflation. Since both Minkowski and de Sitter spacetime have
the maximal symmetry, there exist a similarity. Indeed, the Euclidean
instanton method for evaluating the tunneling rate in Minkowski spacetime
Coleman:1977py ; Callan:1977pt  is applicable to tunneling processes in de
Sitter spacetime Coleman:1980aw . However, the extension of Euclidean method
to de Sitter spacetime has a difficulty in interpretation. In the case of de
Sitter spacetime, there are two saddle solutions: Coleman-De Luccia (CDL)
Coleman:1980aw  and Hawking-Moss (HM) instantons Hawking:1981fz . The latter
instanton is a homogeneous solution. Therefore, it is not straightforward to
interpret the tunneling rate as that for the tunneling process from a false
vacuum to a true vacuum Rubakov:1999ir ; Brown:2007sd .

To circumvent the difficulty, a real time formalism might be useful. In the
case of de Sitter spacetime in the flat chart, the stochastic formalism has
been used for investigating time evolution of fluctuations during inflation
Starobinsky:1986fx ; Starobinsky:1994bd . In this formalism, Langevin
equations describe the dynamics of long wavelength modes. Stochastic noises
stem from short wavelength quantum modes. The stochastic formalism has been
also used to analyze the tunneling processes in de Sitter spacetime
Goncharov:1987ir ; Linde:1991sk ; Tolley:2008qv ; Noorbala:2018zlv ;
Hashiba:2020rsi ; Camargo-Molina:2022ord ; Camargo-Molina:2022paw . However,
most of the previous works treated only the HM transition by utilizing the
Fokker-Plank (FP) equation for a homogeneous scalar field, which can be
derived from the Langevin equation. Obviously, the HM transition cannot
describe the bubble nucleation in contrast to CDL tunneling. To describe the
bubble, we have to include the spatial gradient term and the FP equation
becomes the functional differential equation instead of the partial
differential equation. Hence, at first glance, it seems difficult to treat the
CDL bubble by the stochastic approach. However, we make an observation that
there exists a path integral representation to solutions of the FP equation.
In condensed matter physics, it is called Martin–Siggia–Rose–Janssen–de
Dominicis (MSRJD) functional integral Martin:1973zz ; DeDominicis:1975gjb ;
Janssen:1976aa ; DeDominicis:1977fw .

Figure 1: The schematic picture of the potential $V(\phi)$ which has a false
vacuum at $\phi=\phi_{\text{false}}$, true vacuum at $\phi=\phi_{\text{true}}$
and the top of the potential at $\phi=\phi_{\text{top}}$.

In this paper, we use the MSRJD path integral formula to study the tunneling
processes on the fixed de Sitter background. The applicability conditions of
this method are clarified based on the Schwinger-Keldysh formalism. We
consider a potential $V(\phi)$ which has a local minimum at
$\phi=\phi_{\text{false}}$, a true minimum at $\phi_{\text{true}}$, and a
local maximum at $\phi_{\text{top}}$ as shown in FIG. 1. We use the saddle
point method and evaluate the tunneling rate by computing the action for the
tunneling configurations PhysRevE.70.041106 . We investigate both HM and CDL
tunneling processes. In the case of a shallow potential barrier, the
conventional HM tunneling rate is reproduced. Remarkably, the tunneling
configuration in the MSRJD functional integral method has physically natural
boundary conditions in contrast to the Euclidean method. We then clarify that
the HM tunneling rate is the tunneling rate of a (roughly Hubble-sized)
coarse-grained patch transitioning from $\phi_{\text{false}}$ to
$\phi_{\text{top}}$. In the case of a steep potential barrier, we find an
interesting CDL bubble-like configuration which is obtained by solving the
solution of a scalar field equation in Euclidean anti-de Sitter space even
though we do not work in the imaginary time. Our results show how the bubble
nucleation process could be described in the stochastic approach.

This paper is organized as follows. In sec. 2, we present the setup and review
the stochastic approach. In sec. 3, the MSRJD functional integral is
constructed from the Langevin equations. We also outline the derivation of
MSRJD functional integral starting from the Schwinger-Keldysh formalism. In
sec. 4, we apply the MSRJD functional integral to the case of the shallow
potential barrier and study the HM tunneling process. In sec. 5, we consider
the case of the steep potential barrier and find the CDL bubble-like
configuration. We derive the tunneling rate using the configuration and
compare it with the one obtained in sec. 4. We also compare our results with
those in the Euclidean method. Sec. 6 is devoted to the conclusion. Some
technical details are presented in appendices.

##  2 Stochastic approach

In this section, we review the stochastic approach Starobinsky:1986fx ;
Starobinsky:1994bd .

###  2.1 Setup

We consider a real scalar field $\phi$ in (3+1) dimensional de Sitter
background. The background geometry is described by the metric

$\displaystyle\mathrm{d}s^{2}=-\mathrm{d}t^{2}+a(t)^{2}\mathrm{d}\bm{x}^{2}=a(\eta^{2})(-\mathrm{d}\eta^{2}+\mathrm{d}\bm{x}^{2}),$
(1)

with the scale factor

$a(t)=e^{Ht}=-\frac{1}{H\eta},$ (2)

where $H$ is the Hubble constant. The action is given by

$S=\int\mathrm{d}t\mathrm{d}^{3}x\,a^{3}\bigg{[}\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}a^{-2}(\nabla\phi)^{2}-V(\phi)\bigg{]},$
(3)

where a dot denotes a derivative respect to $t$ and $V(\phi)$ is a potential
function. From this action, the canonical conjugate momentum $\Pi$ is defined
as $\Pi:=a^{3}\dot{\phi}$ and the Hamilton’s equations of motion are given by

$\dot{\phi}=a^{-3}\Pi\
,\qquad\dot{\Pi}=a\nabla^{2}\phi-a^{3}V^{\prime}(\phi)\,.$ (4)

###  2.2 Langevin equation

First, we divide the quantum fields $\phi$ and $\Pi$ into an infrared (IR)
part and an ultraviolet (UV) part respectively as

$\phi=\phi_{\text{IR}}+\phi_{\text{UV}}\
,\qquad\Pi=\Pi_{\text{IR}}+\Pi_{\text{UV}}\ .$ (5)

The UV part is defined as

$\displaystyle\phi_{\text{UV}}$
$\displaystyle:=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\theta(k-k_{c}(t))\bigg{[}a_{\bm{k}}u_{\bm{k}}(t)e^{i\bm{k}\cdot\bm{x}}+a_{\bm{k}}^{\dagger}u_{\bm{k}}^{*}(t)e^{-i\bm{k}\cdot\bm{x}}\bigg{]},$
$\displaystyle\Pi_{\text{UV}}$
$\displaystyle:=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\theta(k-k_{c}(t))a(t)^{3}\,\bigg{[}a_{\bm{k}}\dot{u}_{\bm{k}}(t)e^{i\bm{k}\cdot\bm{x}}+a_{\bm{k}}^{\dagger}\dot{u}_{\bm{k}}^{*}(t)e^{-i\bm{k}\cdot\bm{x}}\bigg{]},$
(6)

where $u_{\bm{k}}$ is a mode function that will be specified later, and
$a_{\bm{k}}$ and $a^{\dagger}_{\bm{k}}$ are annihilation and creation
operators, respectively. Here, $\theta(k-k_{c}(t))$ is a step function and
$k_{c}(t)$ is the cutoff scale defined as

$k_{c}(t):=\varepsilon a(t)H,$ (7)

where $\varepsilon$ is a constant. The value of $\varepsilon$ will be
discussed later. Substituting Eqs.(6) into Hamilton’s equations (4), we obtain
the following equations;

$\dot{\phi}_{\text{IR}}=a^{-3}\Pi_{\text{IR}}+\xi^{\phi}\
,\qquad\dot{\Pi}_{\text{IR}}=a\nabla^{2}\phi_{\text{IR}}-a^{3}V^{\prime}(\phi_{\text{IR}})+\xi^{\Pi}\
,$ (8)

where $\xi^{\phi}$ and $\xi^{\Pi}$ are defined as

$\displaystyle\xi^{\phi}$
$\displaystyle:=\dot{k}_{c}(t)\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\delta(k-k_{c}(t))\bigg{[}a_{\bm{k}}u_{\bm{k}}(t)e^{i\bm{k}\cdot\bm{x}}+a_{\bm{k}}^{\dagger}u_{\bm{k}}^{*}(t)e^{-i\bm{k}\cdot\bm{x}}\bigg{]},$
$\displaystyle\xi^{\Pi}$
$\displaystyle:=\dot{k}_{c}(t)\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\delta(k-k_{c}(t))\bigg{[}a_{\bm{k}}a(t)^{3}\dot{u}_{\bm{k}}(t)e^{i\bm{k}\cdot\bm{x}}+a_{\bm{k}}^{\dagger}a(t)^{3}\dot{u}_{\bm{k}}^{*}(t)e^{-i\bm{k}\cdot\bm{x}}\bigg{]}.$
(9)

To derive Eqs.(8) and (9), we assumed that the time evolution of UV modes can
be well approximated by the free theory that is defined around the false
vacuum. The mode function $u_{\bm{k}}(t)$ introduced in (9) then satisfies the
following equation

$\displaystyle\ddot{u}_{\bm{k}}+3H\dot{u}_{\bm{k}}+a^{-2}k^{2}u_{\bm{k}}+V^{\prime\prime}(\phi_{\text{false}})u_{\bm{k}}=0.$
(10)

From this equation, we see the mode function only depends on $k:=|\bm{k}|$.
Hence, we simply denote the mode function as $u_{k}$. Note that this mode
function is used for UV modes $k>k_{c}(t)$.

Let us consider statistical properties of $\xi^{\phi}$ and $\xi^{\Pi}$. The
annihilation and creation operators $a_{\bm{k}}$ and $a^{\dagger}_{\bm{k}}$
satisfy following commutation relations:

$\displaystyle[a_{\bm{k}},a^{\dagger}_{\bm{k}^{\prime}}]=(2\pi)^{3}\delta^{(3)}(\bm{k}-\bm{k}^{\prime})\
,\qquad[a_{\bm{k}},a_{\bm{k}^{\prime}}]=[a^{\dagger}_{\bm{k}},a^{\dagger}_{\bm{k}^{\prime}}]=0.$
(11)

Defining the vacuum state $\ket{0}$ as

$\displaystyle a_{\bm{k}}\ket{0}=0,\quad\forall\bm{k},$ (12)

we can calculate the one point and two point correlation functions of
$\xi^{\phi}$ and $\xi^{\Pi}$ as

$\displaystyle\braket{0}{\xi^{\alpha}(t,\bm{x})}{0}=0,\quad(\alpha,\beta=\phi,\Pi),$
$\displaystyle\braket{0}{\xi^{\alpha}(t,\bm{x})\xi^{\beta}(t^{\prime},\bm{x}^{\prime})}{0}=\frac{1}{2\pi^{2}}\dot{k}_{c}(t)k_{c}(t)^{2}\,\frac{\sin(k_{c}(t)r)}{k_{c}(t)r}g^{\alpha\beta}(t)\delta(t-t^{\prime}),$
(13)

where $r:=|\bm{x}-\bm{x}^{\prime}|$ and

$\displaystyle\begin{cases}g^{\phi\phi}(t)&:=|u_{k_{c}}(t)|^{2}\\\
g^{\Pi\Pi}(t)&:=a(t)^{6}|\dot{u}_{k_{c}}(t)|^{2}\\\
g^{\phi\Pi}(t)&=(g^{\Pi\phi})^{*}:=a(t)^{3}u_{k_{c}}(t)\dot{u}_{k_{c}}^{*}(t).\end{cases}.$
(14)

Because of the Wick theorem, all higher correlation functions can be
decomposed into two point functions. In the stochastic approach, we replace
the operator $\xi^{\alpha}$ by the real random field which has the following
statistical properties

$\displaystyle\braket{\xi^{\alpha}(t,\bm{x})}_{\xi}=0,\quad(\alpha,\beta=\phi,\Pi),$
$\displaystyle\braket{\xi^{\alpha}(t,\bm{x})\xi^{\beta}(t^{\prime},\bm{x}^{\prime})}_{\xi}=\frac{1}{2\pi^{2}}\dot{k}_{c}(t)k_{c}(t)^{2}\,\frac{\sin(k_{c}(t)r)}{k_{c}(t)r}\text{Re}[g^{\alpha\beta}(t)]\delta(t-t^{\prime}),$
(15)

where $\braket{\cdots}_{\xi}$ represents an expectation value with the
distribution function of $\xi^{\alpha}$. With this prescription, Eqs.(8) can
be interpreted as the Langevin equations with the noise $\xi^{\alpha}$
stemming from quantum fluctuations. Since $\xi^{\alpha}$ are now regarded as
classical variables, we ignored the imaginary parts of $g^{\phi\Pi}$ and
$g^{\Pi\phi}$ which come from non-commutativity. Accordingly, IR variables
$\phi_{\text{IR}}$ and $\Pi_{\text{IR}}$ are also regarded as classical
stochastic variables in this prescription. The validity of this classical
approximation will be discussed in the next section.

###  2.3 Bunch-Davies vacuum

Now we define the vacuum state $\ket{0}$ by specifying the mode function
$u_{k}$. Eq.(10) is the equation for the mode $k>k_{c}$. Let us assume that we
can ignore the term $V^{\prime\prime}$ in solving (10). This can be justified
when $|V^{\prime\prime}|$ is sufficiently small to satisfy
$\max\left[(k_{c}/a)^{2},H^{2}\right]\gg|V^{\prime\prime}|$. Under this
approximation, (10) can be solved as

$\displaystyle
u_{k}=(-k\eta)^{\frac{3}{2}}\bigg{[}C_{1}H^{(1)}_{\frac{3}{2}}(-k\eta)+C_{2}H^{(2)}_{\frac{3}{2}}(-k\eta)\bigg{]},$
(16)

where $C_{1},C_{2}$ are constants of integration and
$H^{(1)}_{\nu}(z),H^{(2)}_{\nu}(z)$ are Hankel functions. We choose the Bunch-
Davies vacuum, i.e., the vacuum coincides with Minkowski vacuum at past
infinity ($\eta\to-\infty$) so that

$\displaystyle
u_{k}\to\frac{-H\eta}{\sqrt{2k}}e^{-ik\eta},\quad(-k\eta\to\infty).$ (17)

This determines $C_{1},C_{2}$ as

$\displaystyle C_{1}=-\sqrt{\frac{\pi}{4k}}\frac{H}{k}\ ,\qquad C_{2}=0\ .$
(18)

Thus, we obtain the mode function

$\displaystyle
u_{k}=\frac{H\eta}{\sqrt{2k}}\bigg{(}\frac{1}{-ik\eta}-1\bigg{)}e^{-ik\eta}.$
(19)

To check (17), we used the following formulae

$\displaystyle H^{(1)}_{\frac{3}{2}}(z)=\sqrt{\frac{2}{\pi
z}}\bigg{(}\frac{1}{iz}-1\bigg{)}e^{iz}\ ,\qquad
H^{(2)}_{\frac{3}{2}}(z)=\sqrt{\frac{2}{\pi
z}}\bigg{(}\frac{1}{-iz}-1\bigg{)}e^{-iz}.$ (20)

Substituting (19) into (14) and using (7), we obtain

$\displaystyle\begin{cases}g^{\phi\phi}&=\frac{H^{2}\eta^{2}}{2k_{c}}\bigg{(}\frac{1}{k_{c}^{2}\eta^{2}}+1\bigg{)}=\frac{1}{2Ha^{3}}\bigg{(}\varepsilon^{-3}+\varepsilon^{-1}\bigg{)}\\\
g^{\Pi\Pi}&=\frac{k_{c}}{2H^{2}\eta^{2}}=\frac{Ha^{3}}{2}\varepsilon\\\
g^{\phi\Pi}&=g^{\Pi\phi*}=\frac{1}{2}\bigg{(}\frac{1}{k_{c}\eta}+i\bigg{)}=-\frac{1}{2}\varepsilon^{-1}+\frac{i}{2}\end{cases}.$
(21)

From (8), we have to scale $g^{\alpha\beta}$ like as $g^{\phi\phi}\to
g^{\phi\phi}$, $g^{\Pi\Pi}\to g^{\Pi\Pi}/(H^{2}a^{6})$ and $g^{\phi\Pi}\to
g^{\phi\Pi}/(Ha^{3})$ in order to compare these quantities. Then,
$g^{\phi\phi}$ and $g^{\Pi\Pi}$ become dominant for the case $\varepsilon\ll
1$ and $\varepsilon\gg 1$, respectively.

###  2.4 Reduced Langevin equation

Let us focus on physics at the super-horizon scales by taking $\varepsilon\ll
1$ and assume a shallow potential so that we can use the massless
approximation to evaluate the noise correlations. In this setup, we can use
(21) from which we find that $g^{\phi\phi}$ becomes dominant. We then ignore
the noise for momentum $\xi^{\Pi}$. We also ignore the gradient term in (8)
because we focus on the physics at super-horizon scales. Since the potential
is assumed to be shallow, the time variation of $\phi_{\text{IR}}(t)$ will be
then very small. Therefore, we may integrate the second equation of (8) as

$\displaystyle\Pi_{\text{IR}}\simeq-\frac{a^{3}}{3H}V^{\prime}(\phi_{\text{IR}}).$
(22)

Substituting this relation into the first equation of (8), we obtain

$\displaystyle\dot{\phi}_{\text{IR}}=-\frac{V^{\prime}(\phi_{\text{IR}})}{3H}+\xi^{\phi},$
(23)

where the two point correlation function of $\xi^{\phi}$ is given at the
leading order by

$\displaystyle\braket{\xi^{\phi}(t,\bm{x})\xi^{\phi}(t^{\prime},\bm{x}^{\prime})}_{\xi}=\frac{H^{3}}{4\pi^{2}}j_{0}(k_{c}r)\delta(t-t^{\prime})\,,\quad
j_{0}(k_{c}r):=\frac{\sin(k_{c}r)}{k_{c}r}\,.$ (24)

Eq.(23) is the Langevin equation with the noise correlation (24) derived in
Starobinsky:1994bd .

##  3 Path integral formalism for the stochastic approach

In this section, we introduce path integral approach for the Langevin equation
following altland_simons_2010 . From now on, we replace the IR field and its
conjugate momentum as
$(\phi_{\text{IR}},\Pi_{\text{IR}})\to(\phi_{c},\Pi_{c})$. As we see in
section 3.2, this notation corresponds to the Keldysh basis. In section 3.2,
we provide a first-principle derivation of the stochastic approach based on
the Schwinger-Keldysh formalism following Tokuda:2017fdh ; Tokuda:2018eqs
.111For earlier discussions of stochastic approach based on the Schwinger-
Keldysh formalism, see e.g., Morikawa:1989xz ; Tolley:2008qv . We briefly show
the sketch of the derivation in the main text, and details of derivations are
given in appendix A.

###  3.1 MSRJD functional integral representation

####  3.1.1 0+1 dimensional theories

Eq.(23) can be regarded as the equation determining the position of a particle
$\phi(t)$ with the white noise $\xi^{\phi}$ defined by Eq.(24). For $k_{c}r\ll
1$, we have $\sin(k_{c}r)/(k_{c}r)\simeq 1$ in a good approximation. Let us
discretize the field and noise as
$\phi_{c}(t)\to\phi_{i},\,\xi^{\phi}(t)\to\xi^{\phi}_{i},\,(i\in\mathbf{Z})$.
Then, Eq.(23) reads

$\displaystyle X_{i}:=\phi_{i}-\phi_{i-1}+\Delta
t\bigg{(}\frac{V^{\prime}(\phi_{i})}{3H}-\xi^{\phi}_{i}\bigg{)}=0,$ (25)

where $\Delta t$ is the temporal discretization interval. Note that the
discretization (25) is called Ito discretization.222Generally, we can choose
other discretization like as Stratonovich discretization. This ambiguity does
not affect the result since the amplitudes of the noise do not depend on
fields within the range of our approximation. Denoting a solution of (23) by
$\phi_{c}[\xi]$, the expectation value of an observable
$\braket{\mathcal{O}}_{\xi}$ can be formally expressed as

$\displaystyle\braket{\mathcal{O}[\phi_{c}(t)]}_{\xi}=\int\mathcal{D}\phi_{c}\,\mathcal{O}[\phi_{c}]\braket{\delta(\phi_{c}-\phi_{c}[\xi])}_{\xi}=\int\mathcal{D}\phi_{c}\,\mathcal{O}[\phi_{c}]\Braket{\bigg{missing}}{\frac{\delta
X}{\delta\phi_{c}}\bigg{missing}}{\,\delta(X)}_{\xi}.$ (26)

where $\mathcal{D}\phi_{c}:=\prod_{i}d\phi_{i}$ is the functional measure,
$\delta(\phi_{c}-\phi_{c}[\xi]):=\prod_{i}\delta(\phi_{i}-\phi_{i}[\xi])$, and
$|\delta X/\delta\phi_{c}|$ is the determinant of $\\{\delta
X_{i}/\delta\phi_{j}\\}$. When we choose the above discretization (25),
$\delta X/\delta\phi_{c}$ becomes triangular matrix with a unit in the
diagonal components, i.e., the functional determinant is unity. Thus,
substituting (25) into (26), we obtain

$\displaystyle\braket{\mathcal{O}[\phi_{c}(t)]}_{\xi}=\int\mathcal{D}\phi_{c}\,\mathcal{O}[\phi_{c}]\Braket{\prod_{i}\delta\bigg{(}\phi_{i}-\phi_{i-1}+\Delta
t\bigg{(}\frac{V^{\prime}(\phi_{i})}{3H}-\xi^{\phi}_{i}\bigg{)}\bigg{)}}_{\xi}.$
(27)

Representing the delta functions in terms of a Fourier integral and taking the
continuum limit, we obtain the following path integral representation

$\displaystyle\braket{\mathcal{O}[\phi_{c}(t)]}_{\xi}=\int\mathcal{D}(\phi_{c},\tilde{\phi})\,\mathcal{O}[\phi_{c}]\Braket{\exp\bigg{[}i\int\mathrm{d}t\tilde{\phi}\bigg{(}\dot{\phi}_{c}+\frac{V^{\prime}(\phi_{c})}{3H}-\xi^{\phi}\bigg{)}\bigg{]}}_{\xi}.$
(28)

After averaging over the noise assuming Gaussian statistics, we obtain the
Martin–Siggia–Rose –Janssen–de Dominicis (MSRJD) functional integral

$\displaystyle\braket{\mathcal{O}[\phi_{c}(t)]}_{\xi}=\int\mathcal{D}(\phi_{c},\tilde{\phi})\,\mathcal{O}[\phi_{c}]\exp\bigg{[}\int\mathrm{d}t\bigg{(}i\tilde{\phi}\bigg{(}\dot{\phi}_{c}+\frac{V^{\prime}(\phi_{c})}{3H}\bigg{)}-\frac{H^{3}}{8\pi^{2}}\tilde{\phi}^{2}\bigg{)}\bigg{]}.$
(29)

Putting $\mathcal{O}[\phi_{c}]=\delta(\phi_{c}-\phi_{c}(t))$ and taking the
boundary conditions $\phi_{c}(t^{\prime})=\phi_{c}^{\prime}$ and
$\phi_{c}(t)=\phi_{c}$, we obtain the transition probability
$p(\phi_{c},t|\phi_{c}^{\prime},t^{\prime})$

$\displaystyle
p(\phi_{c},t|\phi_{c}^{\prime},t^{\prime})=\int_{\phi_{c}(t^{\prime})=\phi_{c}^{\prime}}^{\phi_{c}(t)=\phi_{c}}\mathcal{D}(\phi_{c},\tilde{\phi})\,\exp\bigg{[}\int\mathrm{d}t\bigg{(}i\tilde{\phi}\bigg{(}\dot{\phi}_{c}+\frac{V^{\prime}(\phi_{c})}{3H}\bigg{)}-\frac{H^{3}}{8\pi^{2}}\tilde{\phi}^{2}\bigg{)}\bigg{]}\,.$
(30)

Finally, introducing new variables $\Pi_{\Delta}:=i\tilde{\phi}$, we obtain
the ”phase space” path integral

$\displaystyle
p(\phi_{c},t|\phi_{c}^{\prime},t^{\prime})=\int_{\phi_{c}(t^{\prime})=\phi_{c}^{\prime}}^{\phi_{c}(t)=\phi_{c}}\mathcal{D}(\phi_{c},\Pi_{\Delta})\,\exp\bigg{[}\int\mathrm{d}t(\Pi_{\Delta}\dot{\phi}_{c}-H(\phi_{c},\Pi_{\Delta}))\bigg{]}\
,$ (31)

where we defined the Hamiltonian as

$\displaystyle
H(\phi_{c},\Pi_{\Delta}):=-\frac{V^{\prime}(\phi_{c})}{3H}\Pi_{\Delta}-\frac{H^{3}}{8\pi^{2}}\Pi_{\Delta}^{2}\
.$ (32)

We emphasize that this is just a change of integration variable, not the Wick
rotation. A set of saddle point equations is defined as the Hamilton’s
equations for the Hamiltonian defined above.

####  3.1.2 3+1 dimensional theories

Let us consider the case of the full Langevin Eqs. (8), (13) and (21). When
the space is discretized, (8) can be seen as the dynamics of many body
particles. Thus, the procedure in the previous subsection can be utilized.
Taking the continuum limit, we obtain the following expression which is
analogous to (30),

$\displaystyle p(\phi_{c}(\bm{x}),t|\phi_{c}^{\prime}(\bm{x}),t^{\prime})$
$\displaystyle=\int_{\phi_{c}(t^{\prime},\bm{x})=\phi_{c}^{\prime}(\bm{x})}^{\phi_{c}(t,\bm{x})=\phi_{c}(\bm{x})}\mathcal{D}(\phi_{c},\Pi_{c},\phi_{\Delta},\Pi_{\Delta})$
$\displaystyle\times\exp\bigg{[}i\int\mathrm{d}^{4}x\bigg{(}\Pi_{\Delta}\dot{\phi}_{c}-\phi_{\Delta}\dot{\Pi}_{c}-H(\phi_{c},\Pi_{c},\phi_{\Delta},\Pi_{\Delta})\bigg{)}\bigg{]}\
.\quad$

We defined the Hamiltonian $H(\phi_{c},\Pi_{c},\phi_{\Delta},\Pi_{\Delta})$ as

$\displaystyle H(\phi_{c},\Pi_{c},\phi_{\Delta},\Pi_{\Delta})=$
$\displaystyle\frac{\Pi_{c}\Pi_{\Delta}}{a^{3}}-(a\nabla^{2}\phi_{c}-a^{3}V^{\prime}(\phi_{c}))\phi_{\Delta}$
$\displaystyle-\frac{i}{2}\sum_{\alpha,\beta}\int\mathrm{d}^{4}x^{\prime}X_{\alpha}(x)G^{\alpha\beta}(x,x^{\prime})X_{\beta}(x^{\prime})\,,$
(34)

where the Greek indices $\alpha,\beta=(\phi,\Pi)$ label the $\Delta$ fields as
$(X_{\phi},X_{\Pi})=(\Pi_{\Delta},-\phi_{\Delta})$ and $G^{\alpha\beta}$
denotes the correlations of noises:

$G^{\alpha\beta}(x,x^{\prime}):=\langle\xi^{\alpha}(x)\xi^{\beta}(x^{\prime})\rangle_{\xi}\,,$
(35)

where the RHS is given in (15). We use the formula (LABEL:eq:ConditionalFull)
in sections 4.2 and 5. Note that we have not changed the integration variables
as $(\phi_{\Delta},\Pi_{\Delta})\to(-i\phi_{\Delta},-i\Pi_{\Delta})$.

In sections 4 and 5, we take the initial state and the final state to be the
false vacuum and the field configurations after tunneling (true vacuum or
bubble), respectively.

###  3.2 Transition probability from the Schwinger-Keldysh formalism

In the above prescription, we treat IR fields
$(\phi_{\text{IR}},\Pi_{\text{IR}})$ as classical stochastic variables. To
quantify the validity of this approximation, we derive the stochastic approach
from the first principle based on the Schwinger-Keldysh formalism. Note that
we recover the label “IR” for IR fields in this subsection following the
notation adopted in appendix A.

Suppose that the system is in the false vacuum $\ket{\Psi_{\text{false}}}$ at
the past infinity $t=-\infty$ and it evolves into a certain quantum state
$|\\{\phi^{\text{IR}}(T,\bm{x})\\}_{\bm{x}\in\mathcal{D}}\rangle$ at the final
time $t=T$, where
$|\\{\phi^{\text{IR}}(T,\bm{x})\\}_{\bm{x}\in\mathcal{D}}\rangle$ is an
eigenstate of an IR field $\hat{\phi}^{\text{IR}}(\bm{x})$ at spatial points
$\bm{x}$ in the domain $\mathcal{D}$ with an eigenvalue
$\phi^{\text{IR}}(T,\bm{x})$. Here, $\hat{\phi}^{\text{IR}}(\bm{x})$ is an IR
field in the Schrödinger picture.333Precisely,
$\hat{\phi}^{\text{IR}}(\bm{x})$ is defined in terms of the the Schrödinger
picture field $\hat{\phi}({\bm{k}})$ in momentum space as
$\hat{\phi}^{\text{IR}}(\bm{x}):=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\hat{\phi}({\bm{k}})\,e^{i\bm{k}.\bm{x}}\theta(k_{c}(t)-k)$.
We suppressed the trivial time dependence stemming from the step function.
The transition probability $p$ for this process is given by

$p(\\{\phi^{\text{IR}}(T,\bm{x})\\}_{\bm{x}\in\mathcal{D}})=\left|\langle\\{\phi^{\text{IR}}(T,\bm{x})\\}_{\bm{x}\in\mathcal{D}}|\hat{U}(t,-\infty)|\Psi_{\text{false}}\rangle\right|^{2}\,,$
(36)

where $\hat{U}(t,t^{\prime})$ describes the unitary time evolution from
$t^{\prime}$ to $t$. The RHS is the Fourier component of the generating
functional $Z[J^{\text{IR}}]$ for IR fields, and we have

$\displaystyle
p(\\{\phi^{\text{IR}}(T,\bm{x})\\}_{\bm{x}\in\mathcal{D}})=\left.\prod_{\bm{x}\in\mathcal{D}}\int\mathrm{d}J^{\text{IR}}(T,\bm{x})Z[J^{\text{IR}}(T)]e^{-iJ^{\text{IR}}(T,\bm{x})\phi^{\text{IR}}(T,\bm{x})}\right|_{\\{J^{\text{IR}}(T,\bm{y})=0\\}_{\bm{y}\notin\mathcal{D}}}\,,$
(37)

where the generating functional $Z[J^{\text{IR}}(T)]$ is defined by

$\displaystyle
Z[J^{\text{IR}}(T)]:=\text{Tr}\left[\hat{U}(T,-\infty)\ket{\Psi_{\text{false}}}\bra{\Psi_{\text{false}}}\hat{U}^{\dagger}(T,-\infty)\prod_{\bm{x}}e^{iJ^{\text{IR}}(T,\bm{x})\phi^{\text{IR}}(T,\bm{x})}\right]\,.$
(38)

We can evaluate $Z[J^{\text{IR}}(T)]$ non-perturbatively in the IR sector
based on the method developed in Tokuda:2017fdh ; Tokuda:2018eqs . Our
strategy is that we integrate out UV sector $k>k_{c}(t)$ perturbatively in
nonlinear couplings to obtain $Z[J^{\text{IR}}(T)]$. This is analogous to
tracing out environmental degrees of freedom to evaluate the reduced density
matrix for the system under considerations. Each step we take can be
summarized as follows:

  1. 1.

First, we derive a path integral representation of $Z[J^{\text{IR}}(T)]$. We
then split the integration variables into UV fields and IR fields such that
the integration contour of UV variables of $\phi$ is closed: see discussions
around Eq.(94) for more details.444There is a subtlety that modes satisfying
$k\geq k_{c}(T)$ are initially regarded as the “UV” degrees of freedom (DoF)
while they become “IR” DoFs due to the accelerating expansion of the
spacetime. However, by adopting this splitting procedure, we can use the
Schwinger-Keldysh (or closed-time-path) formalism to evaluate the integration
over UV variables first as usual.

  2. 2.

We perform the integration over UV variables and evaluate an IR effective
action, the so-called Feynman-Vernon influence functional Feynman:1963fq
perturbatively.

  3. 3.

We substitute the obtained expression of $Z[J^{\text{IR}}(T)]$ into (37),
giving rise to the path integral expression for the transition probability
$p$.

Technical details are shown in appendix A. At the step 2, we assumed that the
quantum state for non-zero modes is given by the Bunch-Davies vacuum state for
a free field defined around the false vacuum, and that the exact zero mode
provides a non-fluctuating classical background field configuration
$\phi=\phi_{\text{false}}$.

In this way, we can obtain the path integral expression for the probability
$p$ as

$\displaystyle p(\\{\phi^{\text{IR}}(T,\bm{x})\\}_{\bm{x}\in\mathcal{D}})=$
$\displaystyle\int\mathcal{D}(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})\,\,e^{i\mathcal{S}}\,\delta\left[\mathcal{C}_{\text{fin}}\right]\delta\left[\mathcal{C}_{\text{ini}}\right]\,.$
(39)

Here, $\delta\left[\mathcal{C}_{\text{fin}}\right]$ and
$\delta\left[\mathcal{C}_{\text{ini}}\right]$ fix the boundary conditions of
path integral at the final time and the initial time, respectively:

$\displaystyle\delta\left[\mathcal{C}_{\text{fin}}\right]:=\prod_{\bm{x}\in\mathcal{D}}\delta(\phi^{\text{IR}}_{c}(T,\bm{x})-\phi^{\text{IR}}(T,\bm{x}))\prod_{\bm{y}}\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{y}))\,,$
(40a)
$\displaystyle\delta\left[\mathcal{C}_{\text{ini}}\right]:=\prod_{\bm{x}}\delta\left(\phi^{\text{IR}}_{c}(-\infty,\bm{x})-\phi_{\text{false}}\right)\delta\left(\Pi^{\text{IR}}_{c}(-\infty,\bm{x})\right)\delta[\mathcal{C_{\text{detail}}}]\,,$
(40b)

where $\delta[\mathcal{C}_{\text{detail}}]$ is not relevant for the main
points here, and we define it in appendix A. A term $\mathcal{S}$ may be
understood as an IR effective action,

$\displaystyle\mathcal{S}:=\int\mathrm{d}^{4}x\,\left[\Pi^{\text{IR}}_{\Delta}\dot{\phi}^{\text{IR}}_{c}-\phi^{\text{IR}}_{\Delta}\dot{\Pi}^{\text{IR}}_{c}-H(\phi^{\text{IR}}_{c},\Pi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{\Delta})-\delta
H\right]\,,$ (41a) $\displaystyle\delta
H:=a^{3}(t)\left[V(\phi^{\text{IR}}_{c}+(\phi^{\text{IR}}_{\Delta}/2))-V(\phi^{\text{IR}}_{c}-(\phi^{\text{IR}}_{\Delta}/2))-V^{\prime}(\phi^{\text{IR}}_{c})\phi^{\text{IR}}_{\Delta}\right]+\delta
H_{\text{higher}}\,,$ (41b)

where $H$ is given in (34). $\delta H_{\text{higher}}$ denotes the higher-
order terms in the coupling constant. Note that eq (39) allows us to
understand the transition probability $p$ in terms of the stochastic dynamics:
this point is discussed in appendix A.3.

##### Validity of Eq.(LABEL:eq:ConditionalFull).

Now we find that $\mathcal{S}$ coincides with the term in the exponent of
(LABEL:eq:ConditionalFull) up to the term $\delta H$. Hence, (39) shows that
the result (LABEL:eq:ConditionalFull) in the previous section can be justified
from the first-principle when the term $\delta H$ is negligible. Suppose that
the potential $V(\phi)$ is sufficiently flat in the regime of our interest
such that perturbation theory works when integrating out UV modes. In this
case, we can systematically calculate $\delta H_{\text{higher}}$
perturbatively and we have $\delta H_{\text{higher}}=0$ at the leading order.
We set $\delta H_{\text{higher}}=0$ in the present analysis. We should keep in
mind however that the non-perturbative physics at UV scales
$k>k_{c}(t)=\varepsilon a(t)H$ is lost by setting $\delta
H_{\text{higher}}=0$. From this viewpoint, we should choose $\varepsilon$ as
large as possible.

Even after setting $\delta H_{\text{higher}}=0$, we have nonzero $\delta H$.
In the classical approximation, we may regard the $\Delta$-variables
$(\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{\Delta})$ as tiny quantities and
keep only the terms which are linear in the $\Delta$-variables in
$\mathcal{S}$, leading to $\delta H\simeq 0$. In the case $\varepsilon\ll 1$,
this approximation would work thanks to the squeezing of quantum fluctuations
at super-horizon scales. However, it is more subtle if we can set $\delta H=0$
in the case $\varepsilon\gtrsim 1$. We will revisit this issue in section 5.

##  4 Hawking-Moss tunneling

In this section, we discuss the Hawking-Moss (HM) tunneling from the
perspective of MSRJD functional integral. We calculate the tunneling rate
defined by (31) and (LABEL:eq:ConditionalFull) in sections 4.1 and 4.2,
respectively.

Figure 2: The potential of (42) for various $\beta$. From top to bottom, the
curves correspond to $\beta=0,0.25,0.5,0.75,1$.

Hereafter, for the numerical calculations, we consider the potential
Weinberg:2012pjx  in FIG. 2;

$\displaystyle
V(\phi)=\frac{g^{2}}{4}\bigg{[}(\phi^{2}-\alpha^{2})^{2}+\frac{4\beta}{3}(\alpha\phi^{3}-3\alpha^{3}\phi-2\alpha^{4})\bigg{]},$
(42)

for which the derivative with respect to $\phi$ is given by a simple form

$\displaystyle
V^{\prime}(\phi)=g^{2}(\phi-\alpha)(\phi+\alpha)(\phi+\alpha\beta)\,.$ (43)

For this potential, we have $\phi_{\text{false}}=-\alpha$,
$\phi_{\text{true}}=\alpha$ and $\phi_{\text{top}}=-\beta\alpha$. The
dimensionless parameter $\beta$ must be $0<\beta<1$, otherwise $\phi=-\alpha$
is not a false vacuum. The potential at each point takes the following values:

$\displaystyle V(-\alpha)=0\,,\quad V(\alpha)=-\frac{4\beta
g^{2}\alpha^{4}}{3}\,,\quad
V(-\beta\alpha)=\frac{g^{2}\alpha^{4}}{12}(1-\beta)^{3}(3+\beta)\,.$ (44)

The height of potential barrier $\Delta V$ between the false vacuum and the
true vacuum is

$\displaystyle\Delta
V:=V(-\beta\alpha)-V(-\alpha)=\frac{g^{2}\alpha^{4}}{12}(1-\beta)^{3}(3+\beta)\,.$
(45)

The mass of fluctuations around a given point $\phi$ is

$\displaystyle
V^{\prime\prime}(\phi)=\alpha^{2}g^{2}\left[3(\phi/\alpha)^{2}+2\beta(\phi/\alpha)-1\right]\,.$
(46)

In our analysis, we work in the quantum field theory on the fixed de Sitter
background. This assumption will be valid when the condition
$\max\bigl{[}|V(-\alpha)|,|V(-\beta\alpha)|,|V(\alpha)|\bigr{]}\ll
3M_{\text{pl}}^{2}H^{2}$ is satisfied. This is always satisfied when the
following condition is imposed,

$H\ll\frac{3gM_{\text{pl}}}{2}\left(\frac{H}{g\alpha}\right)^{2}\approx
2.9\times 10^{12}\,\text{GeV}\left(\frac{g}{1.12\times
10^{-4}}\right)\left(\frac{\sqrt{140}}{g\alpha/H}\right)^{2}\,,$ (47)

where we use $0<\beta<1$ and choose parameters $(g,g\alpha/H)=(1.12\times
10^{-4},\sqrt{140})$ as a benchmark point for later convenience. This shows
that we can ignore the backreaction consistently with the weak coupling $g\ll
1$ provided that $H$ is well below the Planck scale $M_{\text{pl}}\approx
2.44\times 10^{18}$ GeV.

###  4.1 The case for the reduced Langevin equation

Let us start from the simpler formula (31). The Hamilton’s equations for the
Hamiltonian $H(\phi_{c},\Pi_{\Delta})$ defined in (32) are given as

$\displaystyle\dot{\phi}_{c}$
$\displaystyle=-\frac{V^{\prime}(\phi_{c})}{3H}-\frac{H^{3}}{4\pi^{2}}\Pi_{\Delta}$
$\displaystyle\dot{\Pi}_{\Delta}$
$\displaystyle=\frac{V^{\prime\prime}(\phi_{c})}{3H}\Pi_{\Delta}.$ (48)

There exist many Hamiltonian flow lines in phase space
$(\phi_{c},\Pi_{\Delta})$ that represent the solution of Hamilton’s equations
as shown in FIG.3. Each flow line can be specified by the value of Hamiltonian
since it is conserved on the given flow line. In the FIG.3, there are two
important flow lines corresponding to $H(\phi_{c},\Pi_{\Delta})=0$. Since the
Hamiltonian is the quadratic polynomial of $\Pi_{\Delta}$, there are two
$\Pi_{\Delta}$ satisfying $H(\phi_{c},\Pi_{\Delta})=0$:

$\displaystyle\Pi_{\Delta}=0,\,-\frac{8\pi^{2}}{3H^{4}}V^{\prime}\ .$ (49)

In the case of $\Pi_{\Delta}=0$, the equations (48) reduce to

$\displaystyle\dot{\phi}_{c}$ $\displaystyle=-\frac{V^{\prime}(\phi_{c})}{3H}\
.$ (50)

Similarly, in the case of
$\Pi_{\Delta}=-\frac{8\pi^{2}}{3H^{4}}V^{\prime}(\phi_{c})$, the equations
(48) reduce to

$\displaystyle\dot{\phi}_{c}=\frac{V^{\prime}(\phi_{c})}{3H}\ .$ (51)

These two flow lines have three intersections which correspond to the false
vacuum, the top of potential hill and the true vacuum from left to right. On
the flow line with $\Pi_{\Delta}=0$, the false and true vacuum are stable and
the top of potential is unstable. While on the flow line with
$\Pi_{\Delta}\neq 0$, the stability is reversed. The different signs of the
potential forces account for the reverse of the stability at the
intersections. We see that the tunneling configuration exists; starting from
the false vacuum $(\phi_{c},\Pi_{\Delta})=(-\alpha,0)$, going to the top of
the potential $(\phi_{c},\Pi_{\Delta})=(-\beta\alpha,0)$ through the flow line
with $\Pi_{\Delta}\neq 0$ and finally going to the true vacuum through the
flow line with $\Pi_{\Delta}=0$.

Figure 3: The Hamilton flow for (48) where we take $\alpha=H$, $g=0.4$ and
$\beta=0.1$. The solid lines correspond to $H(\phi_{c},\Pi_{\Delta})=0$. From
left to right, the intersections of the solid lines correspond to the false
vacuum, the top of potential hill and the true vacuum.

We numerically solve the second equation with the initial conditions
$\phi_{c}(t^{\prime})=-\alpha$ and switch to the first equation when the field
reaches the top of potential $\phi_{c}(t)=-\beta\alpha$. Thus, we obtain the
tunneling configuration in FIG.4.

Figure 4: The tunneling configuration for the homogeneous fields is plotted
for $\beta=0.1$ and $\alpha g/H=\sqrt{10}$. We choose the initial condition as
$\phi_{c}(10^{-15})/\alpha=-1+10^{-5}$ and solve the second equation of (50).
When $\phi_{c}/\alpha$ reached the top of potential $\phi_{c}/\alpha=-\beta$,
we switch to the first equation of (50) with the initial condition
$\phi_{c}/\alpha=-\beta+10^{-5}$.

Now, we evaluate the transition probability (31) by substituting the above
tunneling configurations. The action in (31) can be calculated without the
concrete solutions for $\phi_{c}$;

$\displaystyle
I=\int_{t^{\prime}}^{t}\mathrm{d}t\left[\Pi_{\Delta}\dot{\phi}_{c}-H(\phi_{c},\Pi_{\Delta})\right]=-\frac{8\pi^{2}}{3H^{4}}\int_{t^{\prime}}^{t_{*}}\mathrm{d}t\dot{\phi}_{c}V^{\prime}(\phi_{c})=-\frac{8\pi^{2}}{3H^{4}}\Delta
V,$ (52)

where $t_{*}$ is the time that $\phi_{c}$ reaches the top of potential and
$\Delta V$ is the difference of energy density between the top of the
potential and the false vacuum. The trajectory from the top of the potential
to the true vacuum does not contribute to the transition probability because
$H(\phi_{c},\Pi_{\Delta})=0$ and $\Pi_{\Delta}=0$ on its trajectory. Thus, the
transition probability of the tunneling can be evaluated as

$p(\alpha,t|-\alpha,t^{\prime})\sim\exp\bigg{(}-\frac{8\pi^{2}}{3H^{4}}\Delta
V\bigg{)}.$ (53)

This shows a complete agreement with the result of the HM instanton on the
fixed background Weinberg:2006pc .

Since the analysis in sec. 4.1 is based on (23) describing the stochastic
dynamics of the IR field $\phi_{c}$ at a single spatial point, eq. (53)
calculates the tunneling probability at a single spatial point, not at the
whole universe.555This is consistent with the observation made in
Weinberg:2006pc  that HM solution corresponds to the transition over a region
of a Hubble horizon volume. Hence, physically speaking, (53) provides the
tunneling probability of a coarse-grained patch with a physical radius
$(\varepsilon H)^{-1}$. This point becomes manifest also in sec. 4.2.

###  4.2 The case for the full Langevin equation

Next, we evaluate the tunneling rate based on the formula
(LABEL:eq:ConditionalFull). An advantage of (LABEL:eq:ConditionalFull) is that
one can discuss various non-trivial dynamics in the whole universe covered by
the flat chart in principle.

In the previous section, we focused on the dynamics at a single spatial point
and found the HM tunneling process as a non-trivial solution of the Hamilton’s
equations. In this section, we discuss the corresponding process in the global
picture based on (LABEL:eq:ConditionalFull) by identifying a non-trivial
configuration that satisfies the Hamilton’s equations in a good approximation.
We then reproduce the result (52). We also estimate the characteristic time
scale of the tunneling. In sec. 4.2.1, we remark some technical complications
that arise in the global picture.

We start with simplifying the expression (LABEL:eq:ConditionalFull). As
discussed in section 2, $G^{\phi\phi}$ becomes dominant and the others can be
neglected for $\varepsilon\ll 1$ in (34). Under this approximation, the
exponent of the integrand of (LABEL:eq:ConditionalFull) is linear in
$\phi_{\Delta}$. We can then perform the path integral
$\int\mathcal{D}\phi_{\Delta}$ in (LABEL:eq:ConditionalFull), yielding the
product of delta functions
$\prod_{\bm{x}}\delta\bigl{(}\dot{\Pi}_{c}-a\nabla^{2}\phi_{c}+a^{3}V^{\prime}(\phi_{c})\bigr{)}$.
These delta functions give an equation of motion for $\Pi_{c}$ and eliminate
the path integral $\int\mathcal{D}\Pi_{c}$. We solve this equation as
$\Pi_{c}\simeq-a^{3}V^{\prime}(\phi_{c})/3H$, ignoring the spatial gradient
and the time variation of $\phi_{c}$. Furthermore, we change the integration
variable as $\Pi_{\Delta}\to-i\Pi_{\Delta}$ in (LABEL:eq:ConditionalFull).
Consequently, (LABEL:eq:ConditionalFull) reduces to

$p(\phi_{c}(\bm{x}),t|\phi_{c}^{\prime}(\bm{x}),t^{\prime})\simeq\int_{\phi_{c}(t^{\prime},\bm{x})=\phi_{c}^{\prime}(\bm{x})}^{\phi_{c}(t,\bm{x})=\phi_{c}(\bm{x})}\mathcal{D}(\phi_{c},\Pi_{\Delta})\exp\left[\int\mathrm{d}^{4}x\left(\Pi_{\Delta}\dot{\phi}_{c}-H_{\text{HM}}(\phi_{c},\Pi_{\Delta})\right)\right]\,,$
(54)

where the Hamiltonian $H_{\text{HM}}$ is defined as

$\displaystyle H_{\text{HM}}(\phi_{c},\Pi_{\Delta})$
$\displaystyle:=-\frac{1}{3H}V^{\prime}(\phi_{c})\Pi_{\Delta}-\frac{H^{3}}{8\pi^{2}}\Pi_{\Delta}\overline{\Pi}_{\Delta}\,,$
(55)

and $\overline{\Pi}_{\Delta}$ is given by

$\displaystyle\overline{\Pi}_{\Delta}(t,\bm{x})$
$\displaystyle:=\int\mathrm{d}^{3}x^{\prime}\,j_{0}\left(k_{c}(t)|\bm{x}-\bm{x}^{\prime}|\right)\Pi_{\Delta}(t,\bm{x}^{\prime})=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\frac{4\pi^{2}}{H^{3}}\frac{H^{2}}{2k^{3}}\delta(t-t_{k})\Pi_{\Delta}(t,\bm{k})e^{i\bm{k}\cdot\bm{x}}\,.$
(56)

Here we define $t_{k}$ by the condition $k_{c}(t_{k})=k$. The Hamilton’s
equations are

$\displaystyle\dot{\phi}_{c}=-\frac{1}{3H}V^{\prime}(\phi_{c})-\frac{H^{3}}{4\pi^{2}}\overline{\Pi}_{\Delta}\,,$
(57a)
$\displaystyle\dot{\Pi}_{\Delta}=\frac{1}{3H}V^{\prime\prime}(\phi_{c})\Pi_{\Delta}\,.$
(57b)

Here, the term $\dot{\Pi}_{\Delta}(t,\bm{x})$ on the LHS of (57b) is defined
as the Fourier transform of $\dot{\Pi}_{\Delta}(t,\bm{k})$ i.e.,
$\dot{\Pi}_{\Delta}(t,\bm{x}):=(2\pi)^{-3}\int\mathrm{d}^{3}k\,\dot{\Pi}_{\Delta}(t,\bm{k})e^{i\bm{k}\cdot\bm{x}}$.
The same rule applies to other IR variables in the real space as in eq. (115).
Intuitively, this is because the stochastic formalism is formulated in the
momentum space and its Fourier transform provides the stochastic formalism in
the real-space: see appendix A for more details.

Eqs. (57) admit a trivial configuration $\Pi_{\Delta}(t,\bm{k})=0$ under which
we have $H_{\text{HM}}=0$ and (57a) becomes
$\dot{\phi}_{c}=-V^{\prime}(\phi_{c})/(3H)$ describing the standard classical
time evolution. Now we are interested in the non-trivial configuration which
satisfies (57). In the previous section, we focus on the non-trivial dynamics
at the single spatial point $\bm{x}=\bm{x}_{0}$ and it is found that the
tunneling configuration is obtained by
$\Pi_{\Delta}(t,\bm{x}_{0})=-\frac{8\pi^{2}}{3H^{4}}V^{\prime}(\phi_{c}(t,\bm{x}_{0}))$.
This configuration can be naturally extended to the one in the global picture
by considering the following configuration

$\displaystyle\Pi_{\Delta}(t,\bm{k})=-\frac{8\pi^{2}}{3H^{4}}V^{\prime}(\phi_{c}(t,\bm{x}_{0}))e^{-i\bm{k}\cdot\bm{x}_{0}}\,.$
(58)

From now on, we show that this configuration satisfies a set of Hamilton
equations (57) in a good approximation. For this configuration, we have

$\displaystyle\Pi_{\Delta}(t,\bm{x})=-\frac{8\pi^{2}}{3H^{4}}V^{\prime}(\phi_{c}(t,\bm{x}_{0}))f(r;t)\,,\quad$
(59a)
$\displaystyle\dot{\Pi}_{\Delta}(t,\bm{x})=-\frac{8\pi^{2}}{3H^{4}}V^{\prime\prime}(\phi_{c}(t,\bm{x}_{0}))\dot{\phi}_{c}(t,\bm{x}_{0})f(r;t)\,,$
(59b)
$\displaystyle\overline{\Pi}_{\Delta}(t,\bm{x})=-\frac{8\pi^{2}}{3H^{4}}V^{\prime}(\phi_{c}(t,\bm{x}_{0}))j_{0}\left(k_{c}(t)r\right)\,,$
(59c)

where $r:=|\bm{x}-\bm{x}_{0}|$ and
$f(r;t):=(2\pi)^{-3}\int\mathrm{d}^{3}k\,\theta(k_{c}(t)-k)\,e^{i\bm{k}\cdot(\bm{x}-\bm{x}_{0})}$.
The functions $j_{0}(k_{c}(t)r)$ and $f(r;t)$ rapidly oscillate for $r\gg
k_{c}^{-1}(t)$. Hence, they exponentially decay to zero after they are smeared
over the Hubble time: for instance, $j_{0}$ and $f$ becomes the smooth
function after the suitable coarse-graining as 666 Note that this behavior may
motivate one to replace the term $j_{0}(k_{c}(t)r)$ in (55) by the step
function  $\displaystyle j_{0}(k_{c}(t)r)\rightarrow\theta(1-k_{c}(t)r)\,.$
(60) This is the approximation adopted in Starobinsky:1994bd .

$\displaystyle j_{0}(k_{c}(t)r)$ $\displaystyle\xrightarrow{\text{time
c.g.}}W(r;t):=\int\mathrm{d}t^{\prime}w(t^{\prime},t)j_{0}\left(k_{c}(t^{\prime})r\right)\approx\begin{cases}1&(r\ll
k_{c}^{-1}(t))\\\ 0\quad&(r\gg k_{c}^{-1}(t))\end{cases}\,,$ $\displaystyle
f(r;t)$ $\displaystyle\xrightarrow{\text{time
c.g.}}W_{f}(r;t):=\int\mathrm{d}t^{\prime}\,w(t^{\prime},t)f(r;t^{\prime})\approx\begin{cases}k_{c}^{3}(t)/(6\pi^{2})&(r\ll
k_{c}^{-1}(t))\\\ 0\quad&(r\gg k_{c}^{-1}(t))\end{cases}\,.$ (61)

Here, we perform the smearing by using a window function $w(t^{\prime},t)$
which is approximately constant in the domain $|t-t^{\prime}|\ll H^{-1}$ while
it exponentially decays to zero at $|t-t^{\prime}|\gg H^{-1}$. We also impose
the normalization condition $\int\mathrm{d}t^{\prime}w(t^{\prime},t)=1$. We
can choose $w(t^{\prime},t)=H\pi^{-1/2}\exp[-H^{2}(t-t^{\prime})^{2}]$ for
example. Note that the property (61) is based on the behavior
$j_{0}(k_{c}r)\simeq 1$ and $f(r;t)\simeq k_{c}^{3}/(6\pi^{2})$ for $r\ll
k_{c}^{-1}$ and the rapid oscillations of $j_{0}(k_{c}r)$ and $f_{0}(r;t)$ for
$r\gg k_{c}^{-1}$. Hence, the property (61) will be robust against the choice
of $w(t^{\prime},t)$.

Therefore, substituting Eqs. (59c) and (61) into Eq. (57a) and performing the
smearing in time, we obtain

$\displaystyle\dot{\phi}_{c}(t,\bm{x})=$
$\displaystyle-\frac{1}{3H}\left[V^{\prime}(\phi_{c}(t,\bm{x}))-2V^{\prime}(\phi_{c}(t,\bm{x}_{0}))j_{0}\left(k_{c}(t)r\right)\right]$
(62a) $\displaystyle\xrightarrow{\text{time
c.g.}}\begin{cases}-V^{\prime}(\phi_{c}(t,\bm{x}))/(3H)&(r\gg
k_{c}^{-1}(t))\\\ V^{\prime}(\phi_{c}(t,\bm{x}_{0}))/(3H)&(r\ll
k_{c}^{-1}(t))\end{cases}\,.$ (62b)

This exhibits the non-trivial dynamics only in the domain $r\lesssim
k_{c}^{-1}$, i.e., a coarse-grained patch centered at a point $\bm{x}_{0}$
with a physical radius $(\varepsilon H)^{-1}$. Here, we used
$\phi_{c}(t,\bm{x})\simeq\phi_{c}(t,\bm{x}_{0})$ for $r\ll k_{c}^{-1}(t)$. In
the second line, it is assumed that $V^{\prime}(\phi_{c}(t,\bm{x}_{0}))$ does
not cancel the rapid oscillations of $j_{0}(k_{c}(t)r)$ so that their product
$V^{\prime}j_{0}$ decays exponentially at $r\gg k_{c}^{-1}$ after the suitable
smearing. This assumption will be valid thanks to the approximate constancy of
super-Horizon fluctuations $\phi_{c}(t,\bm{k})$ over the Hubble time:
$\phi_{c}(t,\bm{k})\simeq\phi_{c}(t^{\prime},\bm{k})$ for
$|t-t^{\prime}|\lesssim H^{-1}$. Adopting the same assumption, from (59a) and
(59b) we also obtain the smeared expressions of $\Pi_{\Delta}$ and
$\dot{\Pi}_{\Delta}$ which turn out to satisfy (57b) under the condition
(57a). Hence, we conclude that (59) describes a non-trivial saddle-point
solution in a good approximation. Furthermore, after the smearing we can find
$H_{\text{HM}}\approx 0$ for this configuration.

Now we use the configuration (58) or (59) to obtain the tunneling
configuration shown in FIG.4 again. We can calculate the action for this
tunneling configuration similarly to (52). By substituting the exact
expressions (59) and (62a), we obtain 777Note that we can also estimate the
action $I$ by performing the coarse-graining in time suitably and substituting
the smeared expression of $\Pi_{\Delta}$ into the action:  $\displaystyle I$
$\displaystyle\simeq-\frac{8\pi^{2}}{3H^{4}}\int_{t^{\prime}}^{t_{*}}\mathrm{d}t\,V^{\prime}(\phi_{c}(t,\bm{x}_{0}))\int\mathrm{d}^{3}x\,\dot{\phi}_{c}(t,\bm{x})W_{f}(r;t)\simeq-\frac{8\pi^{2}}{3H^{4}}\int_{t^{\prime}}^{t_{*}}\mathrm{d}t\,\dot{\phi}_{c}(t,\bm{x}_{0})V^{\prime}(\phi_{c}(t,\bm{x}_{0}))=-\frac{8\pi^{2}}{3H^{4}}\Delta
V\,,$ reproducing the previous result (52). In the first approximate equality,
we used $H_{\text{HM}}=0$ which is well satisfied after the smearing. In the
second one, we use $\phi_{c}(t,\bm{x})\simeq\phi_{c}(t,\bm{x}_{0})$ for $r\ll
k_{c}^{-1}(t)$ and the condition $\int\mathrm{d}^{3}x\,W_{f}(r;t)=1$ which
follows from the definition of $W_{f}$. This estimate may be more analogous to
the calculation of the action in sec. 4.1 since the approximate locality of
the dynamics is restored after the smearing.

$\displaystyle I$
$\displaystyle=-\frac{H^{3}}{8\pi^{2}}\int_{t^{\prime}}^{t_{*}}\mathrm{d}t\,\int\mathrm{d}^{3}x\,\Pi_{\Delta}(t,\bm{x})\overline{\Pi}_{\Delta}(t,\bm{x})$
$\displaystyle=-\frac{8\pi^{2}}{3H^{4}}\int_{t^{\prime}}^{t_{*}}\mathrm{d}t\,V^{\prime}(\phi_{c}(t,\bm{x}_{0}))\frac{V^{\prime}(\phi_{c}(t,\bm{x}_{0}))}{3H}\int\mathrm{d}^{3}x\,j_{0}\left(k_{c}(t)r\right)f(r;t)$
$\displaystyle=-\frac{8\pi^{2}}{3H^{4}}\int_{t^{\prime}}^{t_{*}}\mathrm{d}t\,\dot{\phi}_{c}(t,\bm{x}_{0})V^{\prime}(\phi_{c}(t,\bm{x}_{0}))=-\frac{8\pi^{2}}{3H^{4}}\Delta
V\,.$ (63)

In the first line, we used (57a) to eliminate $\dot{\phi}_{c}$ from the
integrand. In the second line, we substituted the concrete configurations
(59). In the third line, we used (62a) and performed the spatial integral as
$\int\mathrm{d}^{3}x\,j_{0}\left(k_{c}(t)r\right)f(r;t)=1$. Thus, we reproduce
the previous result (52). Our analysis confirms that (63) gives the tunneling
probability of a coarse-grained patch with a physical radius $(\varepsilon
H)^{-1}$.

It is worth mentioning the time scale of the HM tunneling. We may naively
calculate the typical time scale of the HM tunneling as
$t_{*}-t^{\prime}=\int^{\phi_{\text{top}}}_{\phi_{\text{false}}}\mathrm{d}\phi_{c}\,\frac{3H}{V^{\prime}(\phi_{c})}$.
Here, the space-time argument of the field is suppressed. However, this
integral is in general divergent since $V^{\prime}=0$ at the both ends of the
integral. In the vicinity of the domain where $V^{\prime}=0$, we expect that
the quantum fluctuations of the field would play an important role. Since the
typical size of quantum fluctuations accumulated over the Hubble time is
$H/(2\pi)$, we may define the regulated quantity $t_{\text{HM}}$ as

$\displaystyle
t_{\text{HM}}:=\int^{\phi_{\text{top}}-\frac{H}{2\pi}}_{\phi_{\text{false}}+\frac{H}{2\pi}}\mathrm{d}\phi_{c}\,\frac{3H}{V^{\prime}(\phi_{c})}\,,$
(64)

and we expect that $t_{\text{HM}}$ would correctly characterize the typical
time scale of the HM tunneling. In particular, for our potential (42) we have

$\displaystyle t_{\text{HM}}$
$\displaystyle=\frac{3H}{2g^{2}\alpha^{2}\left(1+\beta\right)}\left[\left(\frac{3+\beta}{1-\beta}\right)\log\left|1-\frac{1-\beta}{H/2\pi\alpha}\right|-\log\left|\frac{1-\frac{2}{H/2\pi\alpha}}{1+\frac{1+\beta}{H/2\pi\alpha}}\right|\right]\sim\frac{1}{H}\cdot\mathcal{O}\left((H/g\alpha)^{2}\right)\,.$
(65)

Since we have
$(H/g\alpha)^{2}\sim(H^{2}/V^{\prime\prime})_{\phi_{\text{false}}\lesssim\phi_{c}\lesssim\phi_{\text{top}}}\gg
1$ for the shallow potential, we conclude that the time scale of HM tunneling
is much longer than the Hubble time.

####  4.2.1 Remark

So far it is found that, starting from (LABEL:eq:ConditionalFull), the
configuration (58) approximately solves the Hamilton’s equations (57) and
reproduce the previous result (52) when it is substituted into the action.
There is actually an important reason behind why we should choose the
configuration (58) to evaluate the HM tunneling process.

To see this, it is important to realize an important difference between the
current Hamilton’s equations (57) and the previous one (48); the equation for
$\phi_{c}$ now contains $\overline{\Pi}_{\Delta}$ rather than $\Pi_{\Delta}$.
Due to this difference, the structures of Hamilton’s equations (57) are
understood as follows;

  * •

the equation for $\phi_{c}$ (57a) is determined once $\overline{\Pi}_{\Delta}$
is specified,

  * •

$\overline{\Pi}_{\Delta}(t,\bm{x})$ is understood as initial conditions for
(57b) since $\overline{\Pi}_{\Delta}(t,\bm{x})$ contains only the boundary
modes satisfying $k=k_{c}(t)$. Hence, $\Pi_{\Delta}(t,\bm{x})$ is obtained by
solving (57b) for given $\overline{\Pi}_{\Delta}$ under the condition (57a).

On top of that, substituting the Hamilton’s equation for $\phi_{c}$ into the
action, we have the first line of (63) which can be rewritten as

$\displaystyle
I=-\frac{1}{2}\int^{t_{*}}_{t^{\prime}}\mathrm{d}t\,\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\Pi_{\Delta}(t,\bm{k})\Pi_{\Delta}(t,-\bm{k})\frac{H^{2}}{2k^{3}}\delta(t-t_{k})\,.$
(66)

The RHS depends only on the value of $\Pi_{\Delta}(t,\bm{k})|_{k=k_{c}(t)}$.
Hence, we can evaluate the tunneling action once $\overline{\Pi}_{\Delta}$ is
specified, provided that (66) or equivalently (LABEL:eq:ConditionalFull) is
reliable.

We claim that (LABEL:eq:ConditionalFull) will be valid when
$\overline{\Pi}_{\Delta}$ is chosen so that $\Pi_{\Delta}(t>t_{k},\bm{k})$
becomes a smooth function in time. Otherwise, it will not be justified to
ignore the higher-order corrections $\delta H_{\text{higher}}$ which are
neglected in (LABEL:eq:ConditionalFull). Intuitively, this states that
perturbation theory tend to be broken down around the exotic configurations.
When $\Pi_{\Delta}$ is a smooth function, we can relate
$\overline{\Pi}_{\Delta}$ to $\Pi_{\Delta}$ after the suitable smearing in
time:

$\displaystyle\overline{\Pi}_{\Delta}(t,\bm{x})\xrightarrow{\text{time
c.g.}}\int\mathrm{d}^{3}x^{\prime}\Pi_{\Delta}(t,\bm{x}^{\prime})W(|\bm{x}-\bm{x}^{\prime}|;t)\sim\mathcal{V}\Pi_{\Delta}(t,\bm{x})\,.$
(67)

Here, we also used $\Pi_{\Delta}(t,\bm{x})\sim\Pi_{\Delta}(t,\bm{x}^{\prime})$
for $|\bm{x}-\bm{x}^{\prime}|\lesssim k_{c}^{-1}$ and defined
$\mathcal{V}:=(4\pi/3)k_{c}^{-3}(t)\sim\int\mathrm{d}^{3}x^{\prime}\,W(|\bm{x}-\bm{x}^{\prime}|;t)$.
We then require that the Hamilton’s equation for $\Pi_{\Delta}$ (57b) is
satisfied in a good approximation to ensure the validity of
(LABEL:eq:ConditionalFull). This constrains the choice of
$\overline{\Pi}_{\Delta}$. Indeed, the configuration (58) approximately solves
(57b) and satisfies the smoothness and henceforth (67) after the smearing.
This would be the reason why our analysis with (58) based on
(LABEL:eq:ConditionalFull) can reproduce the correct result.

###  4.3 On the $\varepsilon$-independence of the Hawking-Moss tunneling

In sections 4.1 and 4.2, the tunneling probability of a coarse-grained patch
with a physical radius $(\varepsilon H)^{-1}$ is calculated. Our results
coincide with the HM tunneling and are independent of $\varepsilon$.

The $\varepsilon$-independence of the results would be a consequence of the
scale-independence of the dynamics of light scalar fields at super-horizon
scales. To see this, it is useful to notice that a coarse-grained patch of
physical radius $(\varepsilon_{1}H)^{-1}$ at a time $t=T$ expands to a patch
of larger physical radius $(\varepsilon_{2}H)^{-1}$ at a later time
$t=T+\delta t>T$ with $\varepsilon_{2}=\varepsilon_{1}\exp[-H\delta
t]<\varepsilon_{1}\ll 1$. The value of IR field does not evolve from $t=T$ to
$T+\delta t$ because the fluctuations of light scalar field at super-horizon
scales are approximately time-independent,

$\displaystyle\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\phi_{c}(t,\bm{k})e^{i\bm{k}.\bm{x}}\theta(k_{c}(t)-k)\simeq\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\phi_{c}(t+\delta
t,\bm{k})e^{i\bm{k}.\bm{x}}\theta(k_{c}(t)-k)\,,$ (68)

unless $\delta t$ is much longer than $H^{-1}$. Hence, we can relate the
coarse-grained dynamics of different $\varepsilon\ll 1$ by considering the
time shift $t\to t+\delta t$ with the value of IR field being kept fixed. This
means that the tunneling probability from $\phi=\phi_{\text{false}}$ to
$\phi_{\text{true}}$ of a patch with physical radius $(\varepsilon H)^{-1}$
should be independent of $\varepsilon$.

##  5 Coleman-de Luccia tunneling

In the previous section, the result of the HM instanton is reproduced by using
(LABEL:eq:ConditionalFull). We expect that the formula
(LABEL:eq:ConditionalFull) can describe not only the HM tunneling but also the
CDL tunneling. In other words, we expect that there will be a flow line
starting from the false vacuum to the bubble configurations as illustrated in
FIG. 5. In this section, we concretely show an interesting configuration by
following the prescription in the previous section.

Figure 5: A schematic picture of Hamiltonian flows in the ”field” phase space.
We expect that there is a flow line which starts from the false vacuum (left
intersection) and reaches the bubble configuration (right intersection)
through a non-zero $\phi_{\Delta}(\bm{x})$ path.

###  5.1 Coleman-de Luccia bubble solution

In Euclidean method, the CDL instanton is dominant rather than the HM
instanton when the potential barrier is steep Jensen:1983ac ; Jensen:1988zx ;
Hackworth:2004xb ; Batra:2006rz . By taking $\varepsilon\gg 1$, we can study
nonperturbative physics at sub-horizon scales such as the formation of bubble.
In this case, $G^{\Pi\Pi}$ becomes dominant and the other noises can be
ignored in (34). Under this approximation, the exponent of the integrand of
(LABEL:eq:ConditionalFull) is linear in $\Pi_{\Delta}$. We can then perform
the path integral $\int\mathcal{D}\Pi_{\Delta}$ in (LABEL:eq:ConditionalFull),
yielding the product of delta functions
$\prod_{\bm{x}}\delta\bigl{(}\dot{\phi}_{c}-a^{-3}\Pi_{c}\bigr{)}$. These
delta functions eliminate the path integral $\int\mathcal{D}\Pi_{c}$.888This
is the reason why we illustrate flow lines in the
$(\phi_{c},\phi_{\Delta})$-plane in FIG. 5. Furthermore, we change the
integration variable as $\phi_{\Delta}\to-i\phi_{\Delta}$ in
(LABEL:eq:ConditionalFull), leading to

$p\simeq\int\mathcal{D}(\phi_{c},\phi_{\Delta})\exp\left[\int\mathrm{d}^{4}x\,\left(a^{3}\dot{\phi}_{\Delta}\dot{\phi}_{c}-H_{\text{CDL}}(\phi_{c},\phi_{\Delta})\right)\right]$
(69)

with appropriate boundary conditions; we consider bubble configurations as
boundary conditions later. Here, the Hamiltonian $H_{\text{CDL}}$ is defined
by

$H_{\text{CDL}}(\phi_{c},\phi_{\Delta}):=-\left(a\nabla^{2}\phi_{c}-a^{3}V^{\prime}(\phi_{c})\right)\phi_{\Delta}-\frac{a^{3}}{2}\phi_{\Delta}(x)\overline{\phi}_{\Delta}(x)\,,$
(70)

with

$\displaystyle\overline{\phi}_{\Delta}(t,\bm{x}):=\frac{Hk_{c}^{4}(t)}{4\pi^{2}a(t)}\int\mathrm{d}^{3}x^{\prime}\,j_{0}\left(k_{c}(t)|\bm{x}-\bm{x}^{\prime}|\right)\phi_{\Delta}(t,\bm{x}^{\prime})\,.$
(71)

We would like to find the appropriate configuration $\overline{\phi}_{\Delta}$
which describes the tunneling process. Following the choice made in sec. 4,
let us suppose that the appropriate choice is given by the configuration for
which the Hamiltonian (70) vanishes:

$\overline{\phi}_{\Delta}(x)=0\,,\,-2(a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c}))\,.$
(72)

Note that this condition is imposed up to the suitable smearing in time
because $\overline{\phi}_{\Delta}(t,\bm{x})$ includes only the boundary
Fourier mode $k=k_{c}(t)$. As discussed in sec. 4.2.1, $\phi_{\Delta}$ becomes
sufficiently smooth for the appropriate $\overline{\phi}_{\Delta}$, and hence
we also suppose

$\displaystyle\overline{\phi}_{\Delta}\sim\frac{H^{2}\varepsilon}{3\pi}\phi_{\Delta}(x)$
(73)

after the suitable averaging over the Hubble time, similarly to (67). We use
the relation (73) to evaluate the action later. This would work at least for
the purpose of estimating the action.

In principle, we do not need to use the estimate (73) in evaluating the action
since it is determined once $\overline{\phi}_{\Delta}$ is specified based on
the analogous logic discussed around (66). For this purpose, we need to
specify the configurations $\\{\phi_{\Delta}(t,\bm{k})|_{k=k_{c}(t)}\\}$ in
momentum space which result in the sufficiently smooth $\phi_{\Delta}$ that
solves the Hamilton equations. It is not easy to find such appropriate
configurations precisely, however. Hence, we leave more careful analysis on
the choice of $\overline{\phi}_{\Delta}$ for future work. In this study,
instead, we focus on how we can proceed the analysis for given configuration
(72) with adopting the estimate (73) and how the bubble nucleation process
could be described in the stochastic approach.

Using the relation (73), the Hamiltonian (70) and the configuration (72)
become

$\displaystyle
H_{\text{CDL}}\sim-\left(a\nabla^{2}\phi_{c}-a^{3}V^{\prime}(\phi_{c})\right)\phi_{\Delta}-\frac{H^{2}\varepsilon}{6\pi}a^{3}\phi_{\Delta}^{2}\,,$
(74)
$\displaystyle\phi_{\Delta}(x)=0\,,\,-\frac{6\pi}{H^{2}\varepsilon}(a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c}))\,.$
(75)

The Hamilton’s equations under the constraints (75) give the following
equation of motions for $\phi_{c}$:

$\displaystyle\ddot{\phi}_{c}+3H\dot{\phi}_{c}$
$\displaystyle=-(V^{\prime}(\phi_{c})-a^{-2}\nabla^{2}\phi_{c}),\quad(\phi_{\Delta}=0),$
$\displaystyle\ddot{\phi}_{c}+3H\dot{\phi}_{c}$
$\displaystyle=V^{\prime}(\phi_{c})-a^{-2}\nabla^{2}\phi_{c},\qquad\,\,\,(\phi_{\Delta}=-\frac{6\pi}{H^{2}\varepsilon}(a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c}))).$
(76)

Similar to the previous case (50), the signs of the potential term and the
gradient term are flipped in the equation of motion with the non-trivial
$\phi_{\Delta}\neq 0$. Hence, we can expect that the tunneling process is
realized. From (69), we find that only the solutions satisfying
$\phi_{\Delta}\neq 0$ contribute to the action. Then we focus on the second
equation in (76).

Interestingly, this equation is the classical equation of motion in the
Euclidean anti-de Sitter (AdS) space which is defined by the embedding
equation

$\displaystyle-X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}+X_{4}^{2}=-H^{-2}$ (77)

in five-dimensional Minkowski spacetime

$\displaystyle\mathrm{d}s^{2}=-\mathrm{d}X_{0}^{2}+\mathrm{d}X_{1}^{2}+\mathrm{d}X_{2}^{2}+\mathrm{d}X_{3}^{2}+\mathrm{d}X_{4}^{2}\,.$
(78)

In fact, the d’Alembert operator
$\partial_{t}^{2}+3H\partial_{t}+a^{-2}\nabla^{2}$ can be obtained from the
following induced metrics;

$\displaystyle\mathrm{d}s^{2}=\mathrm{d}t^{2}+e^{2Ht}\mathrm{d}\bm{x}^{2}=\frac{1}{H^{2}\eta^{2}}(\mathrm{d}\eta^{2}+\mathrm{d}\bm{x}^{2})=H^{-2}(\mathrm{d}\rho^{2}+\sinh^{2}\rho\,\mathrm{d}\Omega^{2})\
,$ (79)

where
$\mathrm{d}\Omega^{2}:=\mathrm{d}\theta_{1}^{2}+\sin^{2}\theta_{1}\mathrm{d}\theta_{2}^{2}+\sin^{2}\theta_{1}\sin^{2}\theta_{2}\mathrm{d}\theta_{3}^{2}$
and $0<\rho<\infty$, $0\leq\theta_{1}\leq\pi$, $0\leq\theta_{2}\leq\pi$ and
$0\leq\theta_{3}\leq 2\pi$. The second metric is that in the Poincaré
coordinates and the third one is that in the global coordinates. The explicit
coordinate transformations are given by

$\displaystyle X_{0}$
$\displaystyle=\frac{1}{2}\frac{\eta^{2}+\bm{x}^{2}+H^{-2}}{-\eta}=H^{-1}\cosh\rho,$
$\displaystyle X_{1}$
$\displaystyle=\frac{x_{1}}{-H\eta}=H^{-1}\sinh\rho\sin\theta_{1}\cos\theta_{2},$
$\displaystyle X_{2}$
$\displaystyle=\frac{x_{2}}{-H\eta}=H^{-1}\sinh\rho\sin\theta_{1}\sin\theta_{2}\cos\theta_{3},$
$\displaystyle X_{3}$
$\displaystyle=\frac{x_{3}}{-H\eta}=H^{-1}\sinh\rho\sin\theta_{1}\sin\theta_{2}\sin\theta_{3},$
$\displaystyle X_{4}$
$\displaystyle=\frac{1}{2}\frac{\eta^{2}+\bm{x}^{2}-H^{-2}}{-\eta}=H^{-1}\sinh\rho\cos\theta_{1}.$
(80)

For later convenience, we derive the relation between $(\eta,r:=|\bm{x}|)$ and
$(\rho,\sigma_{1})$ as

$-H\eta=\frac{1}{\cosh\rho-\sinh\rho\cos\theta_{1}}\,,\qquad
r=H^{-1}\frac{\sinh\rho\sin\theta_{1}}{\cosh\rho-\sinh\rho\cos\theta_{1}}\,.$
(81)

From this, we can check that a point $(\eta,r)=(-H^{-1},0)$ is mapped to
$\rho=0$ for any $\theta_{1}\in[0,\pi]$. Except for this, there is one-to-one
correspondence between points in the $(\eta,r)$-plane with $\eta\leq 0$ and
$r\geq 0$ and points in the $(\rho,\theta_{1})$-plane with $\rho>0$ and
$0\leq\theta_{1}\leq\pi$. On a constant-$\eta$ slice in the $(\eta,r)$-plane,
the value of $\rho$ for given spatial point $r$ is given by

$\rho=\text{arccosh}\bigg{(}\frac{1}{2}\frac{H^{2}(\eta^{2}+r^{2})+1}{-H\eta}\bigg{)}.$
(82)

This function increases monotonically in $r$ for given $\eta$ in the region
$r>0$.

From the global coordinate (79), we see the Euclidean AdS spacetime has $O(4)$
symmetry. Hence, we assume that the field $\phi_{c}$ depends only on $\rho$.
Thus the second equation of (76) reads

$\displaystyle\frac{\mathrm{d}^{2}\phi_{c}}{\mathrm{d}\rho^{2}}+\frac{3}{\tanh\rho}\frac{\mathrm{d}\phi_{c}}{\mathrm{d}\rho}=\frac{V^{\prime}(\phi_{c})}{H^{2}}.$
(83)

Note that this is the same equation utilized in the Euclidean method
Coleman:1980aw ; Rubakov:1999ir . Imposing the boundary conditions
Coleman:1977py ; Coleman:1980aw

$\displaystyle\lim_{\rho\to\infty}\phi_{c}=-\alpha,\quad\frac{d\phi_{c}}{d\rho}\bigg{|}_{\rho=0}=0,$
(84)

we obtain the bubble solutions as is shown in FIG. 6. Note that the latter
condition ensures the continuity of $\partial_{\eta}\phi_{c}(\eta,r)$ and
$\partial_{\eta}^{2}\phi_{c}(\eta,r)$ at $(H\eta,Hr)=(-1,0)$.

It is useful to fit the bubble configurations by the following fitting
function (FIG. 6);

$\displaystyle\phi_{c}(\rho)=-\alpha\tanh\mu(\rho-\bar{\rho}),$ (85)

where $\mu$ and $\bar{\rho}$ represent the thickness and the position of the
bubble wall, respectively. Note that, for $\beta\gtrsim 0.6$, the deviation
from the true vacuum at the $\rho=0$ becomes significant Weinberg:2012pjx  and
the fitting (85) becomes bad.

Figure 6: The tunneling configuration for the inhomogeneous field is plotted.
The solid curve is the numerical solution for $\beta=0.5$ and $\alpha
g/H=\sqrt{140}$. We take the initial conditions as
$\phi_{c}(10^{-15})/\alpha=1-4.98284\times 10^{-2}$ and
$\phi_{c}^{\prime}(10^{-15})/\alpha=0$. The dashed curve is the fitting
function (85) where $\mu=7.32515$ and $\bar{\rho}=0.34397$.

We also have to examine the boundary conditions in the Hamilton flow. As in
the case of HM tunneling, we expect that there are several intersections where
$\phi_{\Delta}=0$ (FIG. 5). Since we have the concrete $\phi_{\Delta}$ (75)
and the bubble configurations (85), it is possible to obtain the curves in
$(r,\eta)$-plane on which $\phi_{\Delta}=0$. Using (75) and (76), such curves
are given as

$F(\eta,r):=\partial_{\eta}^{2}\phi_{c}-\frac{2}{\eta}\partial_{\eta}\phi_{c}=0.$
(86)

Note that this is equivalent to solve
$a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c})=0$. Substituting (75) into (86)
and using the fitting formula (85), we find four solutions which are expressed
as four lines in the $(\eta,r)$-plane; two of them are placed at
$\eta=-\infty$ with fixed $r$ and $r=\infty$ with fixed $\eta$. These
correspond to $\rho=\infty$ where $\phi=-\alpha$ (FIG. 6). Also, we
numerically find other two non-trivial hypersurfaces (FIG. 7(a)). It can be
seen that one of the curves is totally spacelike but another is partially
timelike. Thus, it seems natural to take $\eta=-\infty$ curve as an initial
time slice on which $\phi$ is false vacuum and totally spacelike non-trivial
curve as a final time slice on which the bubble is nucleated. We denote the
region between the two spacelike curves as $\Sigma$.

As a matter of fact, the configurations on the final time slice has a bubble.
It can be checked as follows; the value of $\phi_{c}$ at the origin exceeds
the top of potential hill. In the case of FIG. 7(a) where we set
$(\beta,\alpha g/H)=(0.5,\sqrt{140})$, we have
$\phi_{c}/\alpha\sim-0.1>-0.5=-\beta$ at $Hr=0,H\eta\sim-0.7$.

(a) |  (b)
---|---
(c) |  (d)
Figure 7: (a): The two non-trivial curves on which (86) is satisfied. We
insert the mesh to confirm whether the curves are spacelike or timelike. It
can be seen that the upper line is spacelike but the lower one is partially
timelike. (b): The integration region for (87). There are another spacelike
curve at $\eta=-\infty$. (c): The two non-trivial curves in
$(\rho,\theta_{1})$ plane is plotted. (d): The integration region for (87) in
$(\rho,\theta_{1})$ plane. All plots are the case for $\beta=0.5$ and $\alpha
g/H=\sqrt{140}.$

Now, we can evaluate the action for the bubble configurations for $\Sigma$
(FIG. 7(b) or (d)). We use the fitting function (85) for the numerical
integration.

$\displaystyle I$
$\displaystyle\simeq\int_{\Sigma}\mathrm{d}^{4}x\left[-a^{3}\phi_{\Delta}\left(\ddot{\phi}_{c}+3H\dot{\phi}_{c}\right)-H_{\text{CDL}}\right]$
$\displaystyle=-\frac{6\pi}{H^{6}\varepsilon}\int_{\Sigma}\mathrm{d}\rho\mathrm{d}\theta_{1}\mathrm{d}\theta_{2}\mathrm{d}\theta_{3}\frac{\sin^{2}\theta_{1}\sin\theta_{2}\sinh^{3}\rho}{(\cosh\rho-\sinh\rho\cos\theta_{1})^{4}}\bigg{(}\partial_{\eta}^{2}\phi_{c}-\frac{2}{\eta}\partial_{\eta}\phi_{c}\bigg{)}^{2}$
$\displaystyle=-\frac{24\pi^{2}}{H^{2}\varepsilon}\int_{\Sigma}\mathrm{d}\rho\mathrm{d}\theta_{1}\sin^{2}\theta_{1}\sinh^{3}\rho\bigg{[}\frac{(\sinh\rho-\cosh\rho\cos\theta_{1})^{2}}{(\cosh\rho-\sinh\rho\cos\theta_{1})^{2}}\frac{\partial^{2}\phi_{c}}{\partial\rho^{2}}$
$\displaystyle\quad\,+\bigg{\\{}3\frac{\sinh\rho-\cosh\rho\cos\theta_{1}}{\cosh\rho-\sinh\rho\cos\theta_{1}}+\frac{\sin^{2}\theta_{1}}{\tanh\rho(\cosh\rho-\sinh\rho\cos\theta_{1})^{2}}\bigg{\\}}\frac{\partial\phi_{c}}{\partial\rho}\bigg{]}^{2}$
$\displaystyle:=-\frac{24\pi^{2}\alpha^{2}}{H^{2}\varepsilon}\tilde{I}(\mu,\bar{\rho}).$
(87)

In the second line, we used $H_{\text{CDL}}=0$, (75) and (76). Also we
performed a coordinate transformation $(t,r)\to(\rho,\theta_{1})$. In the
third line, the following relations are used

$\displaystyle-\frac{2}{\eta}\frac{\partial\phi_{c}(\rho)}{\partial\eta}=-\frac{2}{\eta}\frac{\partial\rho}{\partial\eta}\frac{\partial\phi_{c}(\rho)}{\partial\rho}=2H^{2}(\cosh\rho-\sinh\rho\cos\theta_{1})(\sinh\rho-\cosh\rho\cos\theta_{1})\frac{\partial\phi_{c}(\rho)}{\partial\rho},$

and

$\displaystyle\frac{\partial^{2}\phi_{c}(\rho)}{\partial\eta^{2}}$
$\displaystyle=$ $\displaystyle
H^{2}\bigg{[}(\sinh\rho-\cosh\rho\cos\theta_{1})^{2}\frac{\partial^{2}}{\partial\rho^{2}}$
$\displaystyle+\bigg{\\{}(\sinh\rho-\cosh\rho\cos\theta_{1})(\cosh\rho-\sinh\rho\cos\theta_{1})+\frac{\sin^{2}\theta_{1}}{\tanh\rho}\bigg{\\}}\frac{\partial}{\partial\rho}\bigg{]}\phi_{c}(\rho).$

Here, the partial differentiation with respect to $\eta$ is taken while
keeping $r$ constant.

###  5.2 Discussions

####  5.2.1 Appropriate choice of $\varepsilon$

Since our results (87) depend on $\varepsilon$, we need to choose some
specific value of $\varepsilon$ to predict the tunneling rate. We emphasize
that the $\varepsilon$ dependence does not imply the pathology of our result.
Rather our results should depend on $\varepsilon$ because the non-perturbative
effects from UV modes are missed in our formalism as discussed in section 3.2.
We then choose $\varepsilon$ as large as possible to evaluate the tunneling
rate.

Our analysis will be valid only when the value of $\varepsilon$ lies in a
certain range. Below, we briefly discuss necessary conditions for
$\varepsilon$ to justify our analysis.

##### Lower bound on $\varepsilon$.

We need to choose sufficiently large $\varepsilon$ so that IR fields can
describe the bubble configuration. This imposes that $(\varepsilon H)^{-1}$
should be smaller than the typical physical size of the bubble or the bubble
wall. This condition will be satisfied by taking $\varepsilon\gtrsim\mu$ for
our parameter choices.

##### Upper bound on $\varepsilon$.

Our analysis is based on Eq.(LABEL:eq:ConditionalFull) which will be invalid
when quantum fluctuations of IR fields are too large. This imposes an upper
bound on $\varepsilon$ because the size of fluctuations of $\phi_{c}$ at the
scale $k\sim k_{c}(t)$ is proportional to $\varepsilon$:

$\delta\phi_{c}|_{k\sim
k_{c}}:=\left[\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\delta\left(\ln(k/k_{c}(t))\right)\langle|\hat{\phi}(t,\bm{k})|^{2}\rangle\right]^{1/2}\approx\frac{k_{c}/a}{2\pi}=\frac{\varepsilon
H}{2\pi}\,,$ (88)

where we assumed $(k_{c}/a)^{2}\gg V^{\prime\prime}(\phi_{c})$ 999This is
compatible with the condition $\varepsilon<\alpha/H$ when the weak coupling
$g\ll 1$ is considered. and approximated $\phi$ as a massless scalar field.

As discussed in section 3.2, (LABEL:eq:ConditionalFull) will be justified when
the term $\delta H$ is negligible. Then, we define the ratio $\mathcal{Q}$ of
the term $\delta H$ to the final term on the RHS of (74) as 101010We can also
compare $\delta H$ with the second and the third term on the RHS of (74). Such
considerations do not change our estimate of $\varepsilon_{\text{max}}$ in
(89).

$\mathcal{Q}:=\left|\frac{\delta
H}{H^{2}\varepsilon\,a^{3}\phi_{\Delta}^{2}/(6\pi)}\right|_{\phi_{\Delta}=-\frac{6\pi}{H^{2}\varepsilon}(a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c}))}=\frac{3\pi^{2}}{2\varepsilon^{2}H^{4}}\left|V^{\prime\prime\prime}(\phi_{c})\left(a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c})\right)\right|\,,$

and we impose $\mathcal{Q}<1$. In the second equality, we used $\delta
H=(a^{3}/24)V^{\prime\prime\prime}(\phi_{c})\phi_{\Delta}^{3}$. Similarly to
(88), we can estimate the size of quantum fluctuations of
$a^{-2}\nabla^{2}\phi$. Then, we estimate $\mathcal{Q}$ at large $\varepsilon$
as $\mathcal{Q}\sim g^{2}\alpha\varepsilon/H$ where we also used
$|V^{\prime\prime\prime}(\phi_{c})|\sim g^{2}\alpha$ for
$|\phi_{c}|\lesssim\alpha$. The condition $\varepsilon<H/(g^{2}\alpha)$
follows from $\mathcal{Q}<1$.

##### Summary.

In summary, it is necessary to choose $\varepsilon$ satisfying the following
condition to justify our analysis,

$\mu\lesssim\varepsilon\lesssim\varepsilon_{\text{max}}:=H/(g^{2}\alpha)\approx
755\times\left(\frac{1.12\times
10^{-4}}{g}\right)\left(\frac{\sqrt{140}}{g\alpha/H}\right)\,.$ (89)

This suggests that, at least when $g$ is sufficiently tiny, we can reliably
choose large $\varepsilon$. Note that we cannot make the action $I$ given in
(87) arbitrarily small even if we take $\varepsilon_{\text{max}}\to\infty$ in
the free-theory limit $g\to 0$; for $\varepsilon=\varepsilon_{\text{max}}$, we
have
$I|_{\varepsilon=\varepsilon_{\text{max}}}\sim-g^{-1}(g\alpha/H)^{3}\tilde{I}(\mu,\bar{\rho})$.
Roughly speaking, the small $g\alpha/H$ corresponds to the bubble radius being
larger than the Hubble radius. Since we are interested in the bubble smaller
than the Hubble radius, $g\alpha/H$ cannot be taken small arbitrarily.

A few more comments on the $\varepsilon$-dependence of our results are in
order. Our result (87) shows that the value of $I$ becomes larger for smaller
choice of $\varepsilon$. This is because the size of quantum fluctuations of
modes $k\sim\varepsilon a(t)H$ is proportional to $\varepsilon$, while the
nonperturbative physics caused by UV modes $k>\varepsilon a(t)H$ is truncated
in our formalism. This interpretation however implicitly assumes that the
modes $k\sim\varepsilon_{\text{max}}a(t)H$ are relevant for the tunneling
process. We expect that this is the case because a kinetic energy of
fluctuations of modes $k\sim\varepsilon_{\text{max}}a(t)H$ is estimated as
$(\varepsilon_{\text{max}}H)^{4}$ which is much smaller than the height of
potential barrier $\Delta V\sim g^{2}\alpha^{4}$ for our parameter choices.

By contrast, very short-scale physics which is insensitive to the detailed
structure of $V(\phi)$ near its origin might be irrelevant for the tunneling
process. Hence, we expect that the tunneling rate would converge to some
finite value in the limit $\varepsilon\to\infty$. It would be interesting to
see whether the tunneling rate converges to a finite value as $\varepsilon$
increases, and if so, from which value of $\varepsilon$ this convergence
begins.

####  5.2.2 Hawking-Moss vs. Coleman-de Luccia

From the formula (LABEL:eq:ConditionalFull), we derived the two configurations
which describe the tunneling from the false vacuum to the true vacuum. One is
HM configuration and the other is CDL-like configuration. Using these
configurations, we evaluate the probability of the tunneling. Here, we compare
these probabilities. Denoting the action (87) and (52) as $I_{\text{bubble}}$
and $I_{\text{HM}}$, the ratio of the two actions is given by

$\displaystyle\gamma:=\frac{I_{\text{bubble}}}{I_{\text{HM}}}=\frac{9\alpha^{2}H^{2}}{\Delta
V\varepsilon}\tilde{I}(\mu,\bar{\rho})=\frac{1}{\varepsilon}\frac{108}{(1-\beta)^{3}(\beta+3)}\frac{H^{2}}{\alpha^{2}g^{2}}\tilde{I}(\mu,\bar{\rho})$
(90)

For example, if we choose $\beta=0.5$ and $\alpha g/H=\sqrt{140}$ (see FIG.
6), the fitting parameters become $\mu=7.32515$ and $\bar{\rho}=0.34397$ and
the ratio becomes $\gamma=12.9840/\varepsilon$. For these parameters, the
upper bound of $\epsilon$ is given as (89) and then $\gamma$ can be smaller
than one for appropriately small g. This means that the CDL bubble
configuration is dominant rather than HM configuration. Results for other
parameters are also shown in Table 1.

####  5.2.3 Comparison with the Euclidean method

We also compute the CDL action in Euclidean method (Appendix B) and compare
the ratios of the CDL action to the HM action (Table 1). For
$\varepsilon>\varepsilon_{*}$, our tunneling process becomes more probable
than the one predicted by the Euclidean method. As discussed in section 5.2.1,
we can consistently choose $\varepsilon>\varepsilon_{*}$ for certain choices
of parameters $(g,\alpha,\beta)$. Perhaps, our results may indicate the
presence of tunneling process which is more probable than the Euclidean
method.

$\beta$ | $g^{2}\alpha^{2}/H^{2}$ | $1-\phi_{c}(10^{-15})/\alpha$ | $\mu$ | $\bar{\rho}$ | $\varepsilon\gamma$ | $\gamma^{E}$ | $\epsilon_{*}$
---|---|---|---|---|---|---|---
0.3 | 80 | 4.87210$\times$ $10^{-5}$ | 6.22563 | 1.05197 | 355.089 | 0.247004 | 1437.58
0.4 | 80 | 5.20273$\times$ $10^{-3}$ | 5.99645 | 0.65257 | 64.1185 | 0.0909796 | 704.758
0.5 | 80 | 3.89106$\times$ $10^{-2}$ | 5.63444 | 0.475247 | 33.9526 | 0.0599470 | 566.376
0.3 | 140 | 3.50945$\times$ $10^{-4}$ | 8.13990 | 0.673305 | 52.2515 | 0.0486678 | 1073.64
0.4 | 140 | 8.96003$\times$ $10^{-3}$ | 7.82419 | 0.459659 | 19.2912 | 0.0239903 | 804.123
0.5 | 140 | 4.98284$\times$ $10^{-2}$ | 7.32515 | 0.34397 | 12.9840 | 0.0172063 | 754.606
Table 1: The values of $\gamma$ for several parameter sets. First and second
ones are potential parameters. Third one is the initial condition for the
equation of motion (83). Forth and fifth ones are fitting parameters for (85).
Sixth and seventh ones are the numerical results of (90) and (139). The last
one is the cut-off parameter $\epsilon$ for which (90) coincides with (139).

####  5.2.4  Bubble nucleation hypersurface and the subsequent evolution

We find the non-trivial spacelike hypersurface on which the bubble is
nucleated (FIG. 7(a)), where the hypersurface is given by the condition
$F(\eta,r)=0$ or equivalently
$a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c})=0$. The subsequent evolution is
then described by the one shown in the first line of (76) which is the
standard classical equation of motion. The configuration on the hypersurface
gives the initial data for the classical dynamics after the bubble nucleation.

The field value on the spacelike hypersurface does not reach the true vacuum
but rather it lies between the true vacuum and the top of the potential as
mentioned in the subsection 5.1. This is possible because the location of
hypersurface is defined by the condition
$a^{-2}\nabla^{2}\phi_{c}-V^{\prime}(\phi_{c})=0$ where the gradient force is
balanced with the potential force. Though it is non-trivial to solve the
evolution starting from the general spacelike hypersurfaces, it is desirable
to solve it to fully understand the formation of the true-vacuum bubble in our
scenario.

We conclude this section by pointing out that our formalism naturally predicts
that the condition $\overline{\phi}_{\Delta}=0$ (or $\phi_{\Delta}=0$ via
(73)) defines the hypersurface on which the quantum dynamics is switched to
the classical dynamics and the non-trivial field configuration is nucleated.
This would be the generic prediction of our formalism that would hold once the
appropriate configuration $\overline{\phi}_{\Delta}$ is specified.

##  6 Conclusion

We studied the tunneling processes on de Sitter background by using the
stochastic approach. A novel point is that we applied the MSRJD functional
integral to the problem. In this formalism, the tunneling rate can be
evaluated with the saddle approximation as in the Euclidean method. Using this
method, we investigated both the HM and the CDL tunnelings.

For the HM case, we first analyzed (31) which is the MSRJD functional integral
of the stochastic equation (23) on a single spatial point. We succeeded in
deriving the tunneling solution in the “phase space” (FIG. 3) which represents
the tunneling process from the false vacuum, through the top of the potential
hill, and finally to the true vacuum (FIG. 4). This solution has natural
tunneling boundary conditions which cannot be obtained from the Euclidean
method. The tunneling rate for this configuration coincides with that of the
HM instanton at the leading order. In sec. 4.2, we succeeded in re-deriving
this result starting from (LABEL:eq:ConditionalFull) which describes the
stochastic dynamics in the global region covered by the flat chart. We also
estimated the time scale of the HM tunneling as (64) which becomes much longer
than the Hubble time scale. Our analysis clarifies the physical picture of the
HM instanton, i.e., the HM transition probability represents the transition
probability of a coarse-grained patch with a physical radius $(\varepsilon
H)^{-1}$. Some technical complications that arise in dealing with
(LABEL:eq:ConditionalFull) and the way of handling them are remarked in sec.
4.2.1. Physically, these complications are due to the spatial correlations of
stochastic noises.

For the CDL case, we found the configurations which describe the bubble
nucleation process. The tunneling rates for the bubble configurations depend
on the cutoff scale dividing IR and UV fields. We argued that this dependence
comes from the truncation of the non-perturbative effects from UV modes. We
also discussed the valid choice of the cutoff scale for which the MSRJD
functional integral will be reliable. This consideration is based on the
first-principle derivation of stochastic approach from the Schwinger-Keldysh
formalism. With an appropriate cutoff scale, it turned out that the CDL
tunneling rate is larger than the HM one for the steep potential barrier.
However, as mentioned above (74), it has not yet been investigated if our
configuration can be really used for estimating the tunneling action. This
aspect would be important for evaluating the tunneling action, being left for
future work. Nonetheless, we believe that our study clarifies how we can
proceed the analysis and how the bubble nucleation process could be described
in the stochastic approach; for instance, our formalism can naturally define
the location of hypersurface on which the quantum dynamics is switched to the
classical dynamics and the non-trivial field configuration is nucleated.

It is interesting to understand the relation between the stochastic approach
and the Euclidean method. We compared the CDL tunneling rate in our method
with that in Euclidean method and found the former becomes larger than the
latter for the potential considered with a certain cutoff scale. It would be
worth investigating the meaning of this result. Especially, we need to know a
precise relation between the CDL instanton and our configuration.
Intriguingly, our method is also related to that discussed in Braden:2018tky ;
Blanco-Pillado:2019xny ; Hertzberg:2020tqa ; Tranberg:2022noe ;
Hertzberg:2019wgx , where the bubble accidentally appears from the initial
quantum fluctuations. In our method, however, we assume that we can neglect
the non-perturbative effects from UV modes, the quantum component of the
potential of the form $V^{\prime\prime\prime}(\phi_{c})\phi_{\Delta}^{3}$, and
that the configuration (75) can be used to estimate the tunneling rate. The
first two are summarized into the term $\delta H$. We also assume the fixed
background spacetime. The relaxation of these assumptions would be important
to seek the relations among the three methods. It would also be useful to
apply our path-integral method to the tunneling in flat space and make a
comparison with the Euclidean method.

The application of our method to the inflation models such as the chain
inflation Freese:2004vs  and warm inflation Berera:1995ie  is also intriguing.
Using our method, we can study the tunneling process in the inflationary
background under dissipation and fluctuations coming from circumference. We
leave these issues for future work.

###### Acknowledgements.

T. M. was supported by JST SPRING, Grant Number JPMJSP2148 and JSPS KAKENHI
Grant Number JP23KJ1543. J. S. was in part supported by JSPS KAKENHI Grant
Numbers JP17H02894, JP17K18778, JP20H01902, JP22H01220. J.T. is supported by
IBS under the project code, IBS-R018-D1.

##  Appendix A Stochastic approach from the first principle

In this section, we evaluate the generating functional for IR fields
$Z[J^{\text{IR}}(T)]$ defined in Eq.(38) to obtain (39). Note that each step
we need to take for evaluating $Z[J^{\text{IR}}(T)]$ is summarized in section
3.2.

###  A.1 Path integral representation

A path integral representation of $Z[J^{\text{IR}}(T)]$ takes the following
standard form,

$\displaystyle Z[J^{\text{IR}}(T)]=$
$\displaystyle\int\mathcal{D}(\phi_{+},\phi_{-},\Pi_{+},\Pi_{-})\,e^{i\bm{J}^{\text{IR}}\cdot(\bm{\phi}^{\text{IR}}_{+}+\bm{\phi}^{\text{IR}}_{-})/2}$
$\displaystyle\quad\times
e^{i(S^{+}_{H}-S^{-}_{H})}\prod_{\bm{x}}\delta(\phi_{+}(T,\bm{x})-\phi_{-}(T,\bm{x}))\Psi_{0}[\phi_{+}]\Psi^{*}_{0}[\phi_{-}]\,,$
(91)

where $\Psi_{0}[\phi]$ is an initial wave functional. We specify it more
concretely in section A.2.1.
$S_{H}^{\pm}=\int^{t=T}\mathrm{d}^{4}x(\Pi^{\pm}\dot{\phi}^{\pm}-H[\phi^{\pm},\Pi^{\pm}])$
is a Hamiltonian action. After the rotation of the basis
$(X_{+},X_{-})\to(X_{c},X_{\Delta}):=(\frac{X_{+}+X_{-}}{2},X_{+}-X_{-})$ with
$X=\phi,\Pi$, (91) is written as

$\displaystyle Z[J^{\text{IR}}(T)]=$
$\displaystyle\int\mathcal{D}(\phi_{c},\phi_{\Delta},\Pi_{c},\Pi_{\Delta})\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}e^{iS_{H}[\phi_{c},\phi_{\Delta},\Pi_{c},\Pi_{\Delta}]}\prod_{\bm{x}}\delta(\phi_{\Delta}(T,\bm{x}))\rho_{0}[\phi_{c},\phi_{\Delta}]\,.$
(92)

Here, we defined
$S_{H}[\phi_{c},\phi_{\Delta},\Pi_{c},\Pi_{\Delta}]:=(S_{H}^{+}-S_{H}^{-})|_{(X_{+},X_{-})\to(X_{c},X_{\Delta})}$
with $X=\phi,\Pi$. We also defined
$\rho_{0}[\phi_{c},\phi_{\Delta}]:=\Psi_{0}[\phi_{c}+(\phi_{\Delta}/2)]\Psi^{*}_{0}[\phi_{c}-(\phi_{\Delta}/2)]$.

###  A.2 Nonperturbative generating functional for IR sector

Now we explain how to calculate $Z[J^{\text{IR}}(T)]$ non-perturbatively in
the IR sector. This allows us to capture the non-perturbative physics of the
sector $k\leq k_{c}(t)$. Our strategy is to split the integration variables
into UV modes $k>k_{c}(t)$ and IR modes $k\leq k_{c}(t)$ for each time step
$t$, and perform the integration over UV variables to get
$Z[J^{\text{IR}}(T)]$. One may simply split the integration variables
$X_{a}(t,\bm{k})$,111111In the main text, we use the notation $X_{\bm{k}}(t)$
to write variables in momentum space. We however adopt the notation
$X(t,\bm{k})$ in this appendix since we have many subscripts such as $c$ and
$\Delta$. where $X=(\phi,\Pi)$ and $a=(c,\Delta)$, by the following
replacement in (92):

$X_{a}(t,\bm{k})\to\begin{cases}X_{a}^{\text{IR}}(t,\bm{k})&(t\geq t_{k})\\\
X_{a}^{\text{UV}}(t,\bm{k})&(t<t_{k})\,,\end{cases}$ (93)

where $t_{k}$ is defined by the condition $k_{c}(t_{k})=k$. It is then tempted
to perform the integration over UV variables $X^{\text{UV}}_{a}$ by using the
Schwinger-Keldysh (or closed-time-path) formalism. However, the expression
obtained from (92) after the replacement (93) does not have the product of
delta functions
$\prod_{\bm{k}}\delta(\phi^{\text{UV}}_{\Delta}(t_{k}-0^{+},\bm{k}))$ at the
final time $t=t_{k}-0^{+}$.121212Here, $0^{+}$ is the infinitesimal time step
which is introduced to obtain path integral representation of unitary time
evolution as usual. That is, the time contour for each UV $\phi$-variable with
modes $0<k\leq k_{c}(T)$ is not closed at the final time. Hence, the usual
Schwinger-Keldysh formalism does not apply for evaluating the integration over
UV variables. To resolve this issue, Refs.Tokuda:2017fdh ; Tokuda:2018eqs
proposed a new way of splitting integration variables. The splitting is given
by (93) for $X=\Pi$, while for $\phi$-variable it is defined by the following
rule rather than (93):

$\begin{array}[]{lr}\phi_{c}(t,\bm{k})\to\begin{cases}\phi_{c}^{\text{IR}}(t,\bm{k})&(t>t_{k})\\\
\phi_{c}^{\text{UV}}(t,\bm{k})&(t\leq
t_{k})\,,\end{cases}&\quad\phi_{\Delta}(t,\bm{k})\rightarrow\begin{cases}\phi_{\Delta}^{\text{IR}}(t,\bm{k})&\quad(t\geq
t_{k})\\\
\phi_{\Delta}^{\text{UV}}(t,\bm{k})&\quad(t<t_{k})\,.\end{cases}\end{array}$
(94)

In this replacement (94), the variables
$\phi^{\text{UV}}_{\Delta}(t_{k},\bm{k})$ are absent while the variables
$\phi^{\text{UV}}_{c}(t_{k},\bm{k})$ are present. Therefore, in the expression
obtained from (92) after the new replacement, the time contour for every UV
$\phi$-variable including those for modes $\bm{k}$ with $0<k\leq k_{c}(T)$ is
closed. This allows us to perform the integration over UV variables based on
the Schwinger-Keldysh formalism as we see below. Note that there is no
subtlety in the replacement of integration variables $X_{a}(t,\bm{k})$ for a
zero mode $k=0$ and those for deep UV modes $k>k_{c}(T)$: in both replacement
rules mentioned above, they are simply replaced by IR variables
$X_{a}^{\text{IR}}(t,\bm{k})$ and UV variables $X_{a}^{\text{UV}}(t,\bm{k})$,
respectively. The time contours for them are closed thanks to the product of
delta functions in the original expression (92).

After the new replacement, the term
$S_{H}[\phi_{c},\phi_{\Delta},\Pi_{c},\Pi_{\Delta}]$ in the exponent of (92)
is decomposed into three pieces: purely IR terms
$S_{H}^{\text{IR}}:=S_{H}[\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta}]$,
purely UV terms
$S_{H}^{\text{UV}}:=S_{H}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta},\Pi^{\text{UV}}_{c},\Pi^{\text{UV}}_{\Delta}]$,
and the remaining IR-UV mixing terms $S_{\text{mixed}}$:

$S_{H}[\phi_{c},\phi_{\Delta},\Pi_{c},\Pi_{\Delta}]\to
S_{H}^{\text{IR}}+S_{H}^{\text{UV}}+S_{\text{mixed}}\,.$ (95)

In terms of these quantities, $Z[J^{\text{IR}}(T)]$ can be formally written as
Tokuda:2017fdh ; Tokuda:2018eqs

$\displaystyle Z[J^{\text{IR}}(T)]=$
$\displaystyle\int\mathcal{D}(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}e^{iS_{H}^{\text{IR}}}e^{i\Gamma}$
$\displaystyle\times\prod_{k\leq
k_{c}(T)}\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\prod_{0<k\leq
k_{c}(T)}\delta(\phi^{\text{IR}}_{c}(t_{k},\bm{k})),$ (96)

where $i\Gamma$ is the effective action for IR fields,

$e^{i\Gamma}:=\int\mathcal{D}(\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta},\Pi^{\text{UV}}_{c},\Pi^{\text{UV}}_{\Delta})\,e^{i(S_{H}^{\text{UV}}+S_{\text{mixed}})}\prod_{k>0}\delta(\phi^{\text{IR}}_{\Delta}(t_{f}(k),\bm{k}))\rho_{0}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta};\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta}]\,,$
(97)

with $t_{f}(k):=\min[T,t_{k}]$. Here,
$\rho_{0}[\phi_{c},\phi_{\Delta}]\to\rho_{0}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta};\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta}]$
due to the replacement of variables. The IR variables
$(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta})$ in the argument of
$\rho_{0}$ contain only a zero mode. Since the time path for each UV
$\phi$-variable is closed in (97), we can calculate $i\Gamma$ perturbatively
as usual by writing the connected diagrams with $n$ external IR fields that
are connected by UV propagators. The diagramatic rules such as vertex factors
and symmetry factors follow the standard rules of Scheinger-Keldysh formalism
for given vertexes in $S_{H}^{\text{UV}}+S_{\text{mixed}}$.

Interestingly, due to the non-trivial replacement rule (94), the
$\Pi\dot{\phi}$ terms in $S_{H}$ also contribute to $S_{\text{mixed}}$
Morikawa:1989xz ; Tolley:2008qv ; Tokuda:2017fdh ; Tokuda:2018eqs . Referring
to such contributions and the remaining terms in $S_{\text{mixed}}$ as
$S_{\text{tr}}$ and $S_{\text{mix-int}}$, respectively, we have
$S_{\text{mixed}}=S_{\text{tr}}+S_{\text{mix-int}}$ with 131313Precisely
speaking, we need to perform the UV-IR splitting in the path integral with
infinitesimal discrete time step for deriving $S_{\text{tr}}$ correctly. After
taking the continuum limit, we obtain (99); see also Tokuda:2018eqs .
Furthermore, the form of $S_{\text{mix-int}}$ depends on the model. In our
case, we have  $\displaystyle S_{\text{mix-int}}=\int\mathrm{d}^{4}x$
$\displaystyle\bigl{[}V(\phi^{\text{UV}}_{+}+\phi^{\text{IR}}_{+})-V(\phi^{\text{UV}}_{+})-V(\phi^{\text{IR}}_{+})$
$\displaystyle-V(\phi^{\text{UV}}_{-}+\phi^{\text{IR}}_{-})+V(\phi^{\text{UV}}_{-})+V(\phi^{\text{IR}}_{-})\bigr{]}_{\phi^{\text{UV}}_{\pm}=\phi^{\text{UV}}_{c}\pm(\phi^{\text{UV}}_{\Delta}/2),\,\phi^{\text{IR}}_{\pm}=\phi^{\text{IR}}_{c}\pm(\phi^{\text{IR}}_{\Delta}/2)}\,.$
(98)

$\displaystyle
S_{\text{tr}}:=\int^{T}_{-\infty}\mathrm{d}t\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\delta(t-t_{k})\left[\phi^{\text{IR}}_{\Delta}(t,-\bm{k})\Pi^{\text{UV}}_{c}(t,\bm{k})-\Pi^{\text{IR}}_{\Delta}(-\bm{k},t)\phi^{\text{UV}}_{c}(t,\bm{k})\right]\,.$
(99)

The bi-linear vertexes exist only at the UV$\to$IR transition time $t_{k}$ for
given modes with $0<k\leq k_{c}(T)$. Physically, these vertexes set the
initial fluctuations for such modes in (96).

####  A.2.1 Integrate out short-wavelength modes

Now we calculate $i\Gamma$ perturbatively by specifying the setup more
concretely. We organize the perturbation theory around the false vacuum state
$\rho_{0}$ for which the expectation value of the zero mode is
$\phi_{\text{false}}$. We assume that the fluctuations of zero modes are tiny
enough so that the couplings between zero-mode fluctuations and those of non-
zero modes $k>0$ are switched off in the past infinity $t=-\infty$. We then
take the initial state of non-zero modes to be the Bunch-Davies vacuum state
for a free field, where the time evolution of a free field is defined by the
quadratic action expanded around the homogeneous background
$\phi_{\text{false}}$. These assumptions can be implemented by writing the
initial quantum state as

$\rho_{0}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta};\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta}]=\rho_{\text{BD}}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta};\phi_{\text{false}}]\,\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]$
(100)

which is properly normalized as

$\displaystyle 1$
$\displaystyle=\prod_{k>0}\int\mathrm{d}\phi^{\text{UV}}_{c}(-\infty,\bm{k})\int\mathrm{d}\phi^{\text{UV}}_{\Delta}(-\infty,\bm{k})\,\rho_{\text{BD}}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta};\phi_{\text{false}}]$
$\displaystyle=\int\mathrm{d}\phi^{\text{IR}}_{c}(-\infty,\bm{0})\int\mathrm{d}\phi^{\text{IR}}_{\Delta}(-\infty,\bm{0})\,\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]\,.$
(101)

$\rho_{\text{BD}}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta};\phi_{\text{false}}]$
is the density matrix for non-zero modes. An initial state for the zero mode
is given by
$\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]$
for which we have

$\bra{\Psi(-\infty)}\hat{\phi}(\bm{k}=\bm{0})\ket{\Psi(-\infty)}=\int\mathrm{d}\phi^{\text{IR}}_{c}\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]\phi^{\text{IR}}_{c}=\phi_{\text{false}}(2\pi)^{3}\delta(\bm{0})\,.$
(102)

Combining this with
$\bra{\Psi(-\infty)}\hat{\phi}(\bm{k})\ket{\Psi(-\infty)}=0$ for non-zero
modes $k>0$, we have

$\bra{\Psi(-\infty)}\hat{\phi}(\bm{k})\ket{\Psi(-\infty)}=\phi_{\text{false}}(2\pi)^{3}\delta(\bm{k})\,.$
(103)

Hence, the VEV of the IR field $\hat{\phi}^{\text{IR}}(t,\bm{x})$ in the past
infinity $t=-\infty$ is given by $\phi_{\text{false}}$.

The interaction vertexes for calculating $i\Gamma$ are the bi-linear vertexes
(99) and non-linear vertexes coming from interacting potential $V(\phi)$. We
assume that the potential is sufficiently flat under the region we are
interested in. It is then natural to separate $i\Gamma$ into the leading order
pieces and higher-order terms in the coupling constant as follows,

$\displaystyle
e^{i\Gamma}=\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]\,e^{i\Gamma_{\text{LO}}^{k>0}}e^{-i\int\mathrm{d}^{4}x\,\delta
H_{\text{higher}}}\,,$ $\displaystyle
e^{i\Gamma^{k>0}_{\text{LO}}}:=\int\mathcal{D}(\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta},\Pi^{\text{UV}}_{c},\Pi^{\text{UV}}_{\Delta})\,e^{iS^{\text{UV}}_{H,\text{free}}+S_{\text{tr}}}\prod_{k>0}\delta(\phi^{\text{IR}}_{\Delta}(t_{f}(k),\bm{k}))\rho_{\text{BD}}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta};\phi_{\text{false}}]\,.$
(104)

Here,
$S^{\text{UV}}_{H,\text{free}}[\phi^{\text{UV}}_{c},\phi^{\text{UV}}_{\Delta},\Pi^{\text{UV}}_{c},\Pi^{\text{UV}}_{\Delta};\phi_{\text{false}}]$
is the free part of the Hamiltonian action defined around the false vacuum as
explained above. Higher-order corrections to $\Gamma$ in the coupling constant
are represented by the term $\delta H_{\text{higher}}$ which can be evaluated
perturbatively. We have $\delta H_{\text{higher}}=0$ at the leading order.

To calculate $i\Gamma^{k>0}_{\text{LO}}$ by writing down the connected
diagrams with external IR fields, we only need to consider the bi-linear
vertexes $S_{\text{tr}}$ given in (99). The number of diagrams are only four,
and $i\Gamma_{\text{LO}}^{k>0}$ can be calculated as Tokuda:2017fdh ;
Tokuda:2018eqs

$\displaystyle i\Gamma_{\text{LO}}^{k>0}$
$\displaystyle=\frac{-1}{2}\int\mathrm{d}t\int\mathrm{d}t^{\prime}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,X_{\alpha}(t,\bm{k})g^{\alpha\beta}(t,t^{\prime},k)X_{\beta}(t^{\prime},-\bm{k})$
$\displaystyle=\frac{-1}{2}\int\mathrm{d}^{4}x\int\mathrm{d}^{4}x^{\prime}\,X_{\alpha}(x)G^{\alpha\beta}(x,x^{\prime})X_{\beta}(x^{\prime})\,.$
(105)

Here, the Greek indices $\alpha,\beta=(\phi,\Pi)$ label the IR-$\Delta$
fields:
$(X_{\phi},X_{\Pi})=(\Pi^{\text{IR}}_{\Delta},-\phi^{\text{IR}}_{\Delta})$.
Substituting Eq.(104) into (96), we find

$\displaystyle Z[J^{\text{IR}}(T)]=$
$\displaystyle\int\mathcal{D}(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]\,e^{i(S_{H}^{\text{IR}}+\Gamma_{\text{LO}}^{k>0})}e^{-i\int\mathrm{d}^{4}x\,\delta
H_{\text{higher}}}$ $\displaystyle\times\prod_{k\leq
k_{c}(T)}\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\prod_{0<k\leq
k_{c}(T)}\delta(\phi^{\text{IR}}_{c}(t_{k},\bm{k}))\,.$ (106)

Now we introduce $i\Gamma_{k=0}$ as

$\displaystyle
e^{i\Gamma_{k=0}[\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{\Delta}]}:=\int\mathrm{d}\phi^{\text{IR}}_{c}\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]e^{-i\Pi^{\text{IR}}_{\Delta}(\phi^{\text{IR}}_{c}-\phi_{\text{false}}(2\pi)^{3}\delta(\bm{0}))}\,.$
(107)

By definition, we have

$\displaystyle\int\mathrm{d}\phi^{\text{IR}}_{c}\rho_{0}[\phi^{\text{IR}}_{c}-\phi_{\text{false}},\phi^{\text{IR}}_{\Delta}]\,e^{-i\Pi^{\text{IR}}_{\Delta}\left(\phi^{\text{IR}}_{c}-\phi_{\text{false}}(2\pi)^{3}\delta(\bm{0})\right)}$
$\displaystyle=\int\mathrm{d}\phi^{\text{IR}}_{c}\,e^{-i\Pi^{\text{IR}}_{\Delta}\phi^{\text{IR}}_{c}}e^{i\Gamma_{k=0}[\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{\Delta}]}\delta\left(\phi^{\text{IR}}_{c}-\phi_{\text{false}}(2\pi)^{3}\delta(\bm{0})\right)\,.$
(108)

Substituting (108) into (106), we find

$\displaystyle Z[J^{\text{IR}}(T)]=$
$\displaystyle\int\mathcal{D}(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}\,e^{i(S_{H}^{\text{IR}}+\Gamma_{\text{LO}}^{k>0}+\Gamma_{k=0})}e^{-i\int\mathrm{d}^{4}x\,\delta
H_{\text{higher}}}$ $\displaystyle\times\prod_{k\leq
k_{c}(T)}\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\delta(\phi^{\text{IR}}_{c}(t_{k},\bm{k})-\phi_{\text{false}}(2\pi)^{3}\delta(\bm{k}))\,,$
(109)

with defining $t_{k}|_{k=0}=-\infty$.

####  A.2.2 Long-wavelength sector

For later convenience, let us focus on the term
$\int\mathrm{d}t\,\Pi^{\text{IR}}_{c}(t,\bm{k})\dot{\phi}^{\text{IR}}_{\Delta}(t,-\bm{k})$
in $S_{H}^{\text{IR}}$. We use the following trick to perform the integration
by parts:

$\displaystyle\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\,e^{i\int^{T}_{t_{k}}\mathrm{d}t\,\Pi^{\text{IR}}_{c}(t,\bm{k})\dot{\phi}^{\text{IR}}_{\Delta}(t,-\bm{k})}$
$\displaystyle=\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\int\mathrm{d}\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k})\,e^{i\int^{T}_{t_{k}-0^{+}}\mathrm{d}t\,\Pi^{\text{IR}}_{c}(t,\bm{k})\dot{\phi}^{\text{IR}}_{\Delta}(t,-\bm{k})}\delta(\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k}))$
$\displaystyle=\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\int\mathrm{d}\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k})\,e^{-i\int^{T}_{t_{k}-0^{+}}\mathrm{d}t\,\dot{\Pi}^{\text{IR}}_{c}(t,\bm{k})\phi^{\text{IR}}_{\Delta}(t,-\bm{k})}\delta(\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k}))\,.$
(110)

Thanks to this trick, (109) can be written as

$\displaystyle Z[J^{\text{IR}}(T)]$
$\displaystyle=\int\mathcal{D}(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})\int\widetilde{\mathcal{D}}\Pi^{\text{IR}}_{c}\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}\,e^{i(S_{H,(d)}^{\text{IR}}+S_{H,(s)}^{\text{IR}}+\Gamma_{\text{LO}}^{k>0}+\Gamma_{k=0})}e^{-i\int\mathrm{d}^{4}x\,\delta
H_{\text{higher}}}$ $\displaystyle\quad\times\prod_{k\leq
k_{c}(T)}\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\delta(\phi^{\text{IR}}_{c}(t_{k},\bm{k})-\phi_{\text{false}}(2\pi)^{3}\delta(\bm{k}))\delta(\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k}))\,,$
(111)

where, $\displaystyle\widetilde{\mathcal{D}}\Pi_{c}^{\text{IR}}:=\prod_{k\leq
k_{c}(T)}\mathrm{d}\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k})$. Below, we simply
omit this integration measure just for simplifying the notation. We also
decomposed $S_{H}^{\text{IR}}$ as
$S_{H}^{\text{IR}}=S_{H,(d)}^{\text{IR}}+S_{H,(s)}^{\text{IR}}$ up to boundary
terms which vanish thanks to the trick (110) as

$\displaystyle S^{\text{IR}}_{H,(d)}$
$\displaystyle=\int\mathrm{d}^{4}x\,\left[\Pi^{\text{IR}}_{\Delta}\left(\dot{\phi}^{\text{IR}}_{c}-a^{-3}\Pi^{\text{IR}}_{c}\right)-\phi^{\text{IR}}_{\Delta}\left(\dot{\Pi}^{\text{IR}}_{c}-a(t)\nabla^{2}\phi^{\text{IR}}_{c}+a^{3}(t)\,V^{\prime}(\phi^{\text{IR}}_{c})\right)\right]\,,$
(112) $\displaystyle S^{\text{IR}}_{H,(s)}$
$\displaystyle=-\int\mathrm{d}^{4}x\,a^{3}(t)\,\left[V(\phi^{\text{IR}}_{c}+(\phi^{\text{IR}}_{\Delta}/2))-V(\phi^{\text{IR}}_{c}-(\phi^{\text{IR}}_{\Delta}/2))-V^{\prime}(\phi^{\text{IR}}_{c})\phi^{\text{IR}}_{\Delta}\right]\,.$
(113)

Here, the space-time argument of IR variables are omitted. We introduced the
IR variables and their time derivatives as

$\displaystyle
X^{\text{IR}}(t,\bm{x}):=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\theta(k_{c}(t)-k)\,X^{\text{IR}}(t,\bm{k})\,e^{i\bm{k}.\bm{x}}\,,$
(114)
$\displaystyle\dot{X}^{\text{IR}}(t,\bm{x}):=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\theta(k_{c}(t)-k)\,\dot{X}^{\text{IR}}(t,\bm{k})\,e^{i\bm{k}.\bm{x}}\,,$
(115)

where $X^{\text{IR}}$ represents the IR variables:
$X^{\text{IR}}=(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})$.
Precisely speaking, for $X^{\text{IR}}=\Pi^{\text{IR}}_{c}$, we should replace
$k_{c}(t)$ by $k_{c}(t+0^{+})$ so that we have $k_{c}(t)=k$ at
$t=t_{k}-0^{+}$. One may concern about the subtlety of the time derivatives
$\dot{\phi}^{\text{IR}}_{c}(t,\bm{x})$ and
$\dot{\Pi}^{\text{IR}}_{c}(t,\bm{x})$ in (112): We in general have
$\dot{X}^{\text{IR}}(t,\bm{x})\neq\frac{\partial}{\partial
t}(X^{\text{IR}}(t,\bm{x}))$, where the LHS is defined by (115), due to the
presence of time-dependent step function. However, there is no subtlety thanks
to the initial conditions for all IR modes $k\leq k_{c}(T)$ of the form

$\displaystyle\phi^{\text{IR}}_{c}(t_{k},\bm{k})=\phi_{\text{false}}(2\pi)^{3}\delta(\bm{k})\,,\qquad\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k})=0\,,$
(116)

which are imposed by the delta functions in the final line in (111). The
corresponding conditions in the real space for all $\bm{x}$ are

$\displaystyle\dot{X}^{\text{IR}}(t,\bm{x})=\partial_{t}(X^{\text{IR}}(t,\bm{x}))\quad\text{for}\quad
X^{\text{IR}}=(\phi^{\text{IR}}_{c},\,\Pi^{\text{IR}}_{c})\,,$ (117a)
$\displaystyle\phi^{\text{IR}}_{c}(-\infty,\bm{x})=\phi_{\text{false}}\,,\quad\Pi^{\text{IR}}_{c}(-\infty,\bm{x})=0\,.$
(117b)

To implement these conditions in the path integral, we define
$\delta\left[\mathcal{C}_{\text{ini}}\right]$ by (40b) with

$\delta\left[\mathcal{C}_{\text{detail}}\right]:=\prod_{-\infty\leq t\leq
T}\prod_{X^{\text{IR}}=(\phi^{\text{IR}}_{c},\Pi^{\text{IR}}_{c})}\prod_{\bm{x}}\delta\left(\dot{X}^{\text{IR}}(t,\bm{x})-\partial_{t}(X^{\text{IR}}(t,\bm{x}))\right)\,.$
(118)

We find from eqs. (37) and (111) that the transition probability can be
written in terms of path integral as

$\displaystyle p(\\{\phi^{\text{IR}}(T,\bm{x})\\}_{T,\bm{x}\in\mathcal{D}})=$
$\displaystyle\int\mathcal{D}(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})\,\,e^{i\mathcal{S}}e^{i\Gamma_{k=0}}\delta\left[\mathcal{C}_{\text{fin}}\right]\delta\left[\mathcal{C}_{\text{ini}}\right]\,,$
(119)

where $\delta\left[\mathcal{C}_{\text{fin}}\right]$ is defined in (40a), and
the term $\mathcal{S}$ in the exponent is given by

$\mathcal{S}=(S_{H,(d)}^{\text{IR}}+S_{H,(s)}^{\text{IR}}+\Gamma_{\text{LO}}^{k>0})-\int\mathrm{d}^{4}x\,\delta
H_{\text{higher}}$ (120)

which agrees with the definition of $\mathcal{S}$ given in (41). When
fluctuations of zero mode are negligible, we can set $\Gamma_{k=0}=0$. This is
the setup adopted in the main text. Then, (119) coincides with (39).

###  A.3 Stochastic interpretation of the tunneling

It is useful to see how the stochastic equations emerge as a consequence of
performing the path integral over IR variables non-perturbatively. This leads
to the stochastic interpretation of the tunneling probability. A final result
of this section is (130).

For simplicity, let us ignore higher-order corrections $\delta
H_{\text{higher}}$ for a while. We start with introducing the auxiliary fields
$(\xi_{\phi},\xi_{\Pi})$ as

$\displaystyle
e^{i\left(\Gamma^{k>0}_{\text{LO}}+\Gamma_{k=0}+S^{\text{IR}}_{H,(s)}\right)}=\int\mathcal{D}(\xi_{\phi},\xi_{\Pi})\,P[\xi_{\phi},\xi_{\Pi}]e^{i\int\mathrm{d}^{4}x\left(\xi_{\Pi}(x)\phi^{\text{IR}}_{\Delta}(x)-\xi_{\phi}(x)\Pi^{\text{IR}}_{\Delta}(x)\right)}\,.$
(121)

We substitute (121) into (111). Then the exponent becomes linear in
$\phi^{\text{IR}}_{\Delta}(t,\bm{k})$ or $\Pi^{\text{IR}}_{\Delta}(t,\bm{k})$
with $t_{0}\leq t<T$:

$\displaystyle Z[J^{\text{IR}}(T)]$
$\displaystyle\simeq\int\mathcal{D}(\phi^{\text{IR}}_{c},\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{c},\Pi^{\text{IR}}_{\Delta})\int\widetilde{\mathcal{D}}\Pi^{\text{IR}}_{c}\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}\int\mathcal{D}(\xi_{\phi},\xi_{\Pi})\,P[\xi_{\phi},\xi_{\Pi}]$
$\displaystyle\times\exp\left\\{i\int^{T}_{-\infty}\mathrm{d}t\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\,\left[\Pi^{\text{IR}}_{\Delta}(t,-\bm{k})\mathcal{L}_{\phi}(t,\bm{k})-\phi^{\text{IR}}_{\Delta}(t,-\bm{k})\mathcal{L}_{\Pi}(t,\bm{k})\right]\Theta(k_{c}(t)-k)\right\\}$
$\displaystyle\times\prod_{k\leq
k_{c}(T)}\delta(\phi^{\text{IR}}_{\Delta}(T,\bm{k}))\delta(\phi^{\text{IR}}_{c}(t_{k},\bm{k})-\phi_{\text{false}}(2\pi)^{3}\delta(\bm{k}))\delta(\Pi^{\text{IR}}_{c}(t_{k}-0^{+},\bm{k}))\,,$
(122)

where

$\displaystyle\mathcal{L}_{\phi}(t,\bm{k}):=\dot{\phi}^{\text{IR}}_{c}(t,\bm{k})-a^{-3}(t)\Pi^{\text{IR}}_{c}(t,\bm{k})-\xi_{\phi}(t,\bm{k})\,,$
(123a)
$\displaystyle\mathcal{L}_{\pi}(t,\bm{k}):=\dot{\Pi}^{\text{IR}}_{c}(t,\bm{k})+a(t)k^{2}\phi^{\text{IR}}_{c}(t,\bm{k})+a^{3}(t)\,V^{\prime}(\phi^{\text{IR}}_{c})(t,\bm{k})-\xi_{\Pi}(t,\bm{k})\,.$
(123b)

Consequently, we can perform the path integral
$\int\mathcal{D}(\phi^{\text{IR}}_{\Delta},\Pi^{\text{IR}}_{\Delta})$, leading
to

$\displaystyle Z[J^{\text{IR}}(T)]\simeq\prod_{k\leq
k_{c}(T)}\left.\int\mathrm{d}\phi^{\text{IR}}_{c}(T,\bm{k})\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}\int\mathcal{D}(\xi_{\phi},\xi_{\Pi})\,P[\xi_{\phi},\xi_{\Pi}]\right|_{\text{$\mathcal{L}_{\phi}=\mathcal{L}_{\Pi}=0$
\& \eqref{initial}}}\,.$ (124)

Now we regard auxiliary fields $(\xi_{\phi},\xi_{\Pi})$ as noise variables
whose correlations are given by

$\displaystyle\langle\xi_{\alpha}(t,\bm{k})\xi_{\beta}(t^{\prime},\bm{k}^{\prime})\rangle$
$\displaystyle:=\int\mathcal{D}(\xi_{\phi},\xi_{\Pi})\,P[\xi_{\phi},\xi_{\Pi}]\,\xi_{\alpha}(t,\bm{k})\xi_{\beta}(t^{\prime},\bm{k}^{\prime})$
$\displaystyle=\left.\left[\frac{i\delta}{\delta
X_{\alpha}(t,\bm{k})}\frac{i\delta}{\delta
X_{\beta}(t^{\prime},\bm{k}^{\prime})}\right]e^{i\left(\Gamma^{k>0}_{\text{LO}}+\Gamma_{k=0}+S^{\text{IR}}_{H,(s)}\right)}\right|_{\phi^{\text{IR}}_{\Delta}=\Pi^{\text{IR}}_{\Delta}=0}\,,$
(125)

where $\alpha,\beta=(\phi,\Pi)$ and
$(X_{\phi},X_{\Pi})=(\Pi^{\text{IR}}_{\Delta},-\phi^{\text{IR}}_{\Delta})$
following the notation in (105). We also have the corresponding expression in
the momentum space. If we ignore $S_{H,(s)}$ and $i\Gamma_{k=0}$, we have

$\langle\xi_{\alpha}(t,\bm{k})\xi_{\beta}(t^{\prime},\bm{k}^{\prime})\rangle=(2\pi)^{3}\delta\left(\bm{k}+\bm{k}^{\prime}\right)\text{Re}\left[g^{\alpha\beta}(t)\right]\delta(t-t^{\prime})\delta(t-t_{k})\,.$
(126)

Eq.(124) shows that the effective dynamics of IR fields are described by the
set of Langevin equations $\mathcal{L}_{\phi}=\mathcal{L}_{\Pi}=0$ with the
initial condition (116) and the noise correlations (126).

In the real space, the dynamics is described by the set of Langevin equations,

$\displaystyle\dot{\phi}^{\text{IR}}_{c}(x)=a^{-3}(t)\Pi^{\text{IR}}_{c}(x)+\xi_{\phi}(x)\,,$
(127a)
$\displaystyle\dot{\Pi}^{\text{IR}}_{c}(x)=a(t)\nabla^{2}\phi^{\text{IR}}_{c}(x)-a^{3}V^{\prime}(\phi^{\text{IR}}_{c}(x))+\xi_{\Pi}(x)\,,$
(127b)

with the initial conditions (117) and the noise correlations

$\langle\xi_{\alpha}(x)\xi_{\beta}(y)\rangle=G^{\alpha\beta}(x,y)\,.$ (128)

Eq.(124) takes the following form in the real space:

$\displaystyle
Z[J^{\text{IR}}(T)]\simeq\prod_{\bm{x}}\left.\int\mathrm{d}\phi^{\text{IR}}_{c}(T,\bm{x})\,e^{i\bm{J}^{\text{IR}}\cdot\bm{\phi}^{\text{IR}}_{c}}\int\mathcal{D}(\xi_{\phi},\xi_{\Pi})\,P[\xi_{\phi},\xi_{\Pi}]\right|_{\text{\eqref{initial_real}
\& \eqref{langevin_real1}}}\,.$ (129)

Substituting (129) into (37), we have

$\displaystyle
p(\\{\phi^{\text{IR}}(T,\bm{x})\\}_{T,\bm{x}\in\mathcal{D}})\simeq$
$\displaystyle\prod_{\bm{y}}\int\mathrm{d}\phi^{\text{IR}}_{c}(T,\bm{y})\,\prod_{\bm{x}\in\mathcal{D}}\delta(\phi^{\text{IR}}_{c}(T,\bm{x})-\phi^{\text{IR}}(T,\bm{x}))$
$\displaystyle\quad\times\left.\int\mathcal{D}(\xi_{\phi},\xi_{\Pi})\,P[\xi_{\phi},\xi_{\Pi}]\right|_{\text{\eqref{initial_real}
\& \eqref{langevin_real1}}}\,.$ (130)

This expression provides a stochastic interpretation of the transition
probability; the dynamics of $\phi^{\text{IR}}_{c}$ is described by the set of
Langevin equations and the transition probability is simply understood as the
probability to realize the field configuration
$\phi^{\text{IR}}_{c}(T,\bm{x})=\phi^{\text{IR}}(T,\bm{x})$ in the domain
$\bm{x}\in\mathcal{D}$ at the final time $T$ starting from the initial
configuration $\phi^{\text{IR}}_{c}(-\infty,\bm{x})=\phi_{\text{false}}$. In
particular, the set of Langevin equations is given by (8) with the noise
correlations (15) when we ignore the terms $\delta H_{\text{higher}}$,
$S_{H,(s)}^{\text{IR}}$, and $\Gamma_{k=0}$. Note that such terms simply
correct the Langevin equations141414For instance, leading-order corrections to
the stochastic dynamics at the super-horizon scales are calculated in
Tokuda:2017fdh . and do not invalidate the above stochastic interpretation.

##  Appendix B Coleman - de Luccia tunneling in Euclidean method

In this appendix, we provide the CDL action in de Sitter background with
Euclidean method.

Generally, the ways of Euclideanization for de Sitter spacetime depend on the
coordinates. In this paper, we discuss the field theory on the metric (1), and
then we choose the following coordinate again 151515The analytic continuations
of this coordinate is examined in Rubakov:1999ir .

$\displaystyle\mathrm{d}s^{2}=\frac{1}{H^{2}\eta^{2}}(-\mathrm{d}\eta^{2}+\mathrm{d}\bm{x}^{2}).$
(131)

By taking the wick rotation as $\eta\to-i\tau$, the coordinate becomes

$\displaystyle-\mathrm{d}s^{2}=\frac{1}{H^{2}\tau^{2}}(\mathrm{d}\tau^{2}+\mathrm{d}\bm{x}^{2}).$
(132)

This is the Euclidean AdS space discussed in section 5 and we can take the
global coordinate;

$\displaystyle-\mathrm{d}s^{2}=H^{-2}(\mathrm{d}\rho^{2}+\sinh^{2}\rho\mathrm{d}\Omega^{2}).$
(133)

We also impose that the field $\phi$ only depends on $\rho$. Therefore, the
Euclidean action $I^{E}$ and the equation of motion becomes

$\displaystyle
I^{E}=2\pi^{2}\int_{0}^{\infty}\mathrm{d}\rho\sinh^{3}\rho\bigg{[}\frac{1}{2H^{2}}(\partial_{\rho}\phi)^{2}+\frac{V(\phi)}{H^{4}}\bigg{]},$
(134)
$\displaystyle\frac{\mathrm{d}^{2}\phi}{\mathrm{d}\rho^{2}}+\frac{3}{\tanh\rho}\frac{\mathrm{d}\phi}{\mathrm{d}\rho}=\frac{V^{\prime}(\phi)}{H^{2}}.$
(135)

The second equation is the same as (83) and we can obtain the same bounce
solutions discussed in section 5. For numerical calculations, we consider the
potential (42) and fitting function (85). Also we take the following non-
dimensionalizations;

$\displaystyle\phi$ $\displaystyle=\alpha\tilde{\phi},$ (136) $\displaystyle
V(\phi)$ $\displaystyle=g^{2}\alpha^{4}\tilde{V}(\tilde{\phi}).$ (137)

Therefore, the Euclidean action (134) becomes

$\displaystyle I^{E}$
$\displaystyle=2\pi^{2}\frac{\alpha^{2}}{H^{2}}\int_{0}^{\infty}\mathrm{d}\rho\sinh^{3}\rho\bigg{[}\frac{1}{2}(\partial_{\rho}\tilde{\phi})^{2}+\frac{g^{2}\alpha^{2}}{H^{2}}\tilde{V}(\tilde{\phi})\bigg{]}$
$\displaystyle:=2\pi^{2}\frac{\alpha^{2}}{H^{2}}\tilde{I}^{E}.$ (138)

The ratio of the Coleman-de Luccia Euclidean action $I^{E}_{\text{bounce}}$ to
the Hawking-Moss action $I_{\text{HM}}$ is given as

$\displaystyle\gamma_{E}=\frac{I^{E}_{\text{bounce}}}{I_{\text{HM}}}=\frac{H^{2}}{g^{2}\alpha^{2}}\frac{9\tilde{I}^{E}_{\text{bounce}}}{(1-\beta)^{3}(\beta+3)}.$
(139)

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