text:
\section{Introduction}
False vacuum decay in quantum field theory is a fascinating phenomenon of key importance for cosmology, both in the Standard Model and its extensions. A giant step in the theoretical analysis of such decays (in particular for the calculation of the tunneling action that controls the exponential suppression of vacuum decay) was due to Coleman \cite{Coleman} who pioneered an elegant and powerful approach
based on an Euclidean formulation of the problem. This formulation is well known, as it became the standard approach, it proved particularly convenient to go beyond the semi-classical approximation including quantum corrections \cite{CC}, and was also extended, by Coleman and De Luccia, to include gravitational corrections \cite{CdL}.

Recently, an alternative approach to the calculation of tunneling actions was proposed \cite{E}. It reformulates the problem without reference to Euclidean space or quantities, as a variational problem in field space, using an auxiliary function, $V_t$, dubbed tunneling potential, that contains all the information needed to calculate the decay action. It is always useful to have alternative formulations for important problems, like false vacuum decay certainly is, and the $V_t$ approach has proven quite useful in many respects, see \cite{Eg,EK,EnoB,ESM,Estab,RM}.

It was also possible to extend the $V_t$ formalism to include in a very compact way gravitational corrections \cite{Eg}.  In this paper we  make extensive use of this general formulation, reviewed in section~\ref{sec:Vt} and push forward in several directions the results obtained already in \cite{Eg}. One particularly interesting type of vacuum decay is that of de Sitter (dS) vacua. While in \cite{Eg} it was already discussed how the $V_t$ formalism recovered the Hawking-Moss rate \cite{HM} in the appropriate limit, we enlarge that discussion here, explaining in more detail how this new formalism reproduces the detailed balance for transitions between dS vacua (section~\ref{sec:dStodS}), some basic properties of the dS decay action under several rescalings of parameters (section~\ref{sec:basics}) and different ways in which
the Hawking-Moss (HM) regime takes over the usual Coleman-De Luccia (CdL) one (section~\ref{sec:HM}), for which the $V_t$ approach is ideally suited.

While the CdL to HM transition is discussed in an approximate way in section~\ref{sec:HM}, one of the useful applications of the $V_t$ approach is to find potentials, $V$, for which the problem of finding the decay configuration is exactly solvable. In \cite{E} this was done in a generic way formally integrating an equation for $V$ in terms of $V_t$, that can be explicitly solved for simple $V_t$ choices. With gravity included, this problem is harder and \cite{Eg} did not provide a general solution, although it did
manage to give an exactly solvable $V$ for a particular $V_t$. This  solvable model, which had some adjustable parameters, could provide analytic examples for decays of Minkowski and anti-de Sitter (AdS) false vacua. No example was given for dS false vacua and the possibility of analytically examining the switch from a CdL-dominated decay to a HM-dominated one was not achieved.
In this paper we fix both shortcomings of \cite{Eg}: in section~\ref{sec:exact} we develop a general method to find exactly solvable potentials if $V_t$ is given, including gravity. The method is applicable to any false vacua, Minkowski, dS or AdS. In section~\ref{sec:ex} we collect examples of solvable potentials with false dS vacua decaying to other dS  or  AdS vacua. This serves, in particular, to study in an analytically controllable way how CdL-dominated decays turn to HM-dominated transitions.  We conclude in section~\ref{sec:conclusions}.

\section{Brief Review of the Tunneling Potential Approach\label{sec:Vt}}

The so-called tunneling potential approach \cite{E,Eg} is an alternative formulation of the problem of finding the tunneling action for the decay of a false vacuum at $\phi_{\scriptscriptstyle +}$ of some potential $V(\phi)$. It does not rely on the Euclidean
formulation of Coleman \cite{Coleman} and transforms the problem into a simple variational problem in field space. In the case with gravity, this variational problem is the following: find the (tunneling potential) function $V_t(\phi)$ which interpolates between $\phi_{\scriptscriptstyle +}$ and some unknown $\phi_0$ on the basin of the true vacuum\footnote{If we call $\phi_{\scriptscriptstyle -}$ the true vacuum, we use the convention $\phi_{\scriptscriptstyle -}>\phi_{\scriptscriptstyle +}$. Then, $\phi_{\scriptscriptstyle +}<\phi_0< \phi_{\scriptscriptstyle -}$.} and minimizes the action functional
\begin{equation}
 S[V_t]=\frac{6\pi^2}{\kappa^2}\int_{\phi_{\scriptscriptstyle +}}^{\phi_0}d\phi\ \frac{(D+V_t')^2}{V_t^2 D}\ ,
\label{SVt}
\end{equation}
with
\begin{equation}
D^2\equiv V_t'{}^2+6\kappa (V-V_t)V_t\ ,
\label{D2}
\end{equation}
where primes denote field derivatives, and $\kappa=1/m_P^2$  ($m_P$ is the reduced Planck mass). This method reproduces the Euclidean result of Coleman and De Luccia \cite{CdL} and has a number of advantages discussed elsewhere \cite{E,Eg,Estab,EK}.
The Euler-Lagrange equation $\delta S/\delta V_t=0$  gives
\begin{equation}
\frac{\delta S}{\delta V_t} = -108 \pi^2 \frac{(V-V_t)}{D^5}\  {\rm{EoM}}=0\ ,
\label{EoM}
\end{equation}
where the  ``equation of motion'' (EoM) for $V_t$ is:
\begin{equation}
{\rm{EoM}}\equiv
6(V-V_t)\left[V_t''+\kappa\left(
3V-2V_t\right)\right]
+ V_t'\left(4V_t'-3V'\right)=0
\ ,
\label{EoMVt}
\end{equation}
or, in terms of $V_t'/D$,
\begin{equation}
\frac{d}{d\phi}\left(\frac{V_t'}{D}\right)= \kappa\frac{\left(2V_t-3V  \right)}{D}\ .
\label{EoMVtD}
\end{equation}

The tunneling potential $V_t$ satisfies the boundary conditions
\begin{equation}
V_t(\phi_+)=V(\phi_+)\equiv V_{\scriptscriptstyle +}\ ,\quad\quad  V_t(\phi_0)=V(\phi_0)\equiv V_0\ ,
\label{BCsVt}
\end{equation}
where $\phi_0$ must be determined by minimizing (\ref{SVt})
and equals the central value of the bounce, $\phi(0)$, in the Euclidean approach.

$V_t$ is qualitatively different depending on the false vacuum nature \cite{Eg} :
\begin{itemize}

\item For decays of Minkowski or AdS false vacua, $V_t$ is monotonic with $V_t'\leq 0$. The EoM for $V_t$ also fixes
$V_t'(\phi_+)=V'(\phi_+)=0,$ and $V_t'(\phi_0)=3 V'(\phi_0)/4.$
As known from \cite{CdL}, for this type of false vacua, gravity can forbid decay (gravitational quenching). In the $V_t$ formulation, to have
a real tunneling action, $V_t$ must  give $D^2>0$.
Gravitational quenching occurs when this condition cannot be satisfied for any $V_t$ \cite{Eg}. For these vacua the second term in (\ref{D2}) is negative and in some cases it can be impossible to satisfy  $D^2>0$ for any $V_t$ and the potential is stabilized \cite{Estab}.

\item For decays of dS  vacua, $V_t$ is not monotonic, see figure~\ref{fig:dStodS}. From $\phi_+$ to $\phi_{0+}$ one has $V_t=V$
[notice that this also solves $\delta S/\delta\phi=0$, see (\ref{EoM})].  From $\phi_{0+}$ to some $\phi_{0-}$, $V_t<V$  and this range corresponds to the field range for the CdL bounce, with
$V_t'(\phi_{0\pm})=3V'(\phi_{0\pm})/4$ and $\phi_{0-}$ corresponding to $\phi(0)$ of the bounce. Finally, from $\phi_{0-}$ to the second minimum $\phi_-$, one has again $V_t=V$. If the overall energy scale of the potential is increased, the range of the CdL interval shrinks to zero and the action tends to the HM one \cite{Eg}.

\end{itemize}

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.5\textwidth]{dStodS.pdf}
\end{center}
\vspace{-0.5cm}
\caption{Potential $V(\phi)$ with two dS mimima at $\phi_\pm$ and
tunneling potential $V_t(\phi)$ describing the transitions between them. The interval $(\phi_{0+},\phi_{0-})$ is the field range of the  CdL bounce.
\label{fig:dStodS}
}
\end{figure}

For later reference we collect here several formulas that connect the tunneling potential formalism to the Euclidean approach and allow to translate the results between the two. For details on their derivation see \cite{Eg}. In the Euclidean formulation, vacuum decay is described by a bounce configuration $\phi(\xi)$, an $O(4)$-symmetric  extremal of the Euclidean action, and a metric function, $\rho(\xi)$, for the $O(4)$-symmetric  Euclidean space-time metric
\begin{equation}
ds^2= d\xi^2 +\rho(\xi)^2 d\Omega_3^2\ ,
\end{equation}
where $\xi$ is a radial coordinate and $d\Omega_3^2$ is the line element on a unit three-sphere.

The key link between both formulations is
\begin{equation}
V_t (\phi)= V(\phi) -\frac12 \dot\phi^2\ ,
\label{Vtdef}
\end{equation}
where $\dot x\equiv dx/d\xi$, and $\dot\phi$ on the right hand side is assumed to be expressed in terms of the field, using the bounce profile $\phi(\xi)$. Using (\ref{Vtdef}) and the differential equations satisfied by $\phi$ and $\rho$ one can establish the following dictionary, where the left hand side is a Euclidean quantity and the right hand side does not depend at all on Euclidean quantities. We have
\begin{equation}
\dot\phi = - \sqrt{2(V-V_t)}\ ,\quad \ddot\phi=V'-V_t'\ ,
\label{dphi}
\end{equation}
where the minus sign in the equation for $\dot\phi$ applies for our convention $\phi_{\scriptscriptstyle +}<\phi_{\scriptscriptstyle -}$, and
\begin{equation}
\rho=\frac{3\sqrt{2(V-V_t)}}{D}\ ,\quad
\dot\rho =-\frac{V_t'}{D}\ ,\quad
\frac{\ddot\rho}{\rho} = -\frac{\kappa}{3}\ (3V-2V_t)\ .
\label{rho}
\end{equation}

\section{dS to dS Transitions\label{sec:dStodS}}

In the decay of a dS vacuum, only the finite space inside the horizon is required to transition (see e.g. \cite{Brown}). This makes the rate non zero generically and allows upward transitions, from a dS vacuum to another with higher cosmological constant. There is a simple relation between the actions for transitions between two dS vacua and it is instructive to derive such relation in the tunneling potential formalism.

The tunneling action $S_{{\scriptscriptstyle +}\to{\scriptscriptstyle -}}$ for the decay from a dS vacuum at $\phi_{\scriptscriptstyle +}$ to a dS vacuum at $\phi_{\scriptscriptstyle -}$ can be obtained as the action integral
(\ref{SVt}). For the discussion in this section it is convenient to extend the integration interval to the full interval from $\phi_{\scriptscriptstyle +}$ to $\phi_{\scriptscriptstyle -}$. The integral has three different pieces \cite{Eg} according to the three different ranges described in the previous section and illustrated by figure~\ref{fig:dStodS}: In the first, from $\phi_{\scriptscriptstyle +}$ to $\phi_{0{\scriptscriptstyle +}}$, where one has $V_t\equiv V$, with $V_t'=V'\geq 0$, one gets $D=V_t'$ and the action density is simply
\begin{equation}
{\it s}=\frac{24\pi^2 V'}{\kappa^2 V^2}
\label{strivial}
\ ,
\end{equation}
which can be integrated exactly.
 In the CdL range, $(\phi_{0{\scriptscriptstyle +}},\phi_{0{\scriptscriptstyle -}})$, one has $V_t\leq V$. Finally, from $\phi_{0{\scriptscriptstyle -}}$ to $\phi_{{\scriptscriptstyle -}}$ with $V_t\equiv V$ again  but with $V_t'=V'\leq 0$, the action density is $s=0$.\footnote{ One also sees that there are no transitions from AdS to dS: the action density would diverge if $V_{\scriptscriptstyle +}<0$ as $V_t$ would need to cross zero with positive slope.}  The total decay action can then be written as the sum of two nonzero pieces, one that we call (with an slight abuse of notation) the ``Hawking-Moss''  part and the other the CdL contribution\footnote{The mixed nature of the dS to dS transitions, with a ``HM'' piece and a CdL part has been given a thermal interpretation in \cite{BW}.}:
\begin{equation}
S_{{\scriptscriptstyle +}\to{\scriptscriptstyle -}}= \Delta S_{HM}+\Delta S_{CdL}=\frac{24\pi^2}{\kappa^2}\left(\frac{1}{V_{\scriptscriptstyle +}}-\frac{1}{V_{0{\scriptscriptstyle +}}}\right)+\frac{6\pi^2}{\kappa^2}\int_{\phi_{0{\scriptscriptstyle +}}}^{\phi_{0{\scriptscriptstyle -}}}\frac{(D+V_t')^2}{D V_t^2}\ d\phi\ ,
\label{SHMCdL}
\end{equation}
with $V_{0{\scriptscriptstyle +}}\equiv V(\phi_{0{\scriptscriptstyle +}})$. When the overall mass scale of the potential (say $V_{\scriptscriptstyle +}$) is sufficiently large, the CdL bounce part disappears and the transition is purely of Hawking-Moss type, as discussed in section~\ref{sec:HM}.

The decay in the opposite direction, from $\phi_{\scriptscriptstyle -}$ to $\phi_{\scriptscriptstyle +}$ proceeds in a similar manner, in fact with the same $V_t$ function, but now taken as starting from $\phi_{\scriptscriptstyle -}$, so that its derivative flips sign. This implies that now there is a simple non-zero contribution from the interval $\phi_{\scriptscriptstyle -}$  to $\phi_{0{\scriptscriptstyle -}}$ and a zero contribution from the interval from $\phi_{0{\scriptscriptstyle +}}$ to $\phi_{\scriptscriptstyle +}$.
The difference between the two tunneling actions, $\Delta S\equiv S_{{\scriptscriptstyle +}\to{\scriptscriptstyle -}}-S_{{\scriptscriptstyle -}\to{\scriptscriptstyle +}}$, takes a very simple form, as only the term linear in $V_t'$ in the action density, the only one that flips sign, contributes. This term can be integrated exactly and one gets
\begin{equation}
\Delta S =\frac{24\pi^2}{\kappa^2}\left(\frac{1}{V_{\scriptscriptstyle +}}-\frac{1}{V_{\scriptscriptstyle -}}\right)\ .
\end{equation}
This can be rewritten simply as $\Delta S = S_{\scriptscriptstyle +} - S_{\scriptscriptstyle -}$,
where $S_\pm$ is the Gibbons-Hawking entropy of a dS vacuum with cosmological constant $V_\pm$. Indeed, this entropy is one fourth of the horizon's area in Planck units
$
S_\pm=A_\pm/(4l_P^{2})$,
where the area is given by
$
A_\pm=4\pi/ H_\pm^{2}\ ,
$
with $H_\pm^2=\kappa V_\pm/3$.
In the formulas above one has $l_P=1/M_P$ and $1/M_P^2=G$, $8\pi G=\kappa$.

\section{Some Basic Properties of dS Decays\label{sec:basics}}
The expression for the tunneling action in terms of $V_t$ can be used to derive in a simple way the following basic properties for dS vacuum decays under several rescalings:  (1) Stretching of the potential; (2) stretching of the field; (3) changing the strength of gravity and (4) a combination of the latter two.

(1) If the potential is rescaled by a constant, $V\to a V$, then the tunneling potential that minimizes the action is $a V_t$ (as is clear from the EoM for $V_t$) and the tunneling action gets rescaled as
$S\to S/a$. Therefore, for $a>1$ the rescaling of $V$ increases the energy scale of the false vacuum and makes it more unstable, even though the height of the potential barrier is also increased.  This conclusion holds whether the transition happens via a CdL bounce or via a HM instanton.

(2) The effect of gravity can be examined by studying how the action is affected by a change in $\kappa$ [in the understanding that a larger (smaller) $\kappa$ means all mass scales of the potential are smaller (larger) compared to $m_P$]. This is best analyzed by looking at
\begin{equation}
\frac{dS[V_t]}{d\kappa}= \frac{d}{d\kappa}\int_{\phi_{\scriptscriptstyle +}}^{\phi_{\scriptscriptstyle -}}s(V,V_t,V_t',\kappa)d\phi =\int_{\phi_{\scriptscriptstyle +}}^{\phi_{\scriptscriptstyle -}}\left[\left(\frac{\partial s}{\partial V_t}-\frac{d}{d\phi}\frac{\partial s}{\partial V_t'}\right)\frac{dV_t}{d\kappa}+\frac{\partial s}{\partial \kappa}\right]d\phi
\ ,
\end{equation}
where $s(V,V_t,V_t',\kappa)$ is the tunneling action density and we have written the total action as an integral in the full interval $(\phi_{\scriptscriptstyle +},\phi_{\scriptscriptstyle -})$, as in the previous section.
The term inside round brackets above cancels due to the EoM for $V_t$, eqs.~(\ref{EoM}) and (\ref{EoMVt}) in the whole integration interval,
and the rest gives
\begin{equation}
\frac{dS[V_t]}{d\kappa}=\frac{3\pi^2}{\kappa^3} \int_{\phi_{\scriptscriptstyle +}}^{\phi_{\scriptscriptstyle -}}\frac{D}{V_t^2}\left(1+\frac{V_t'}{D}\right)^2\left[\left(1-\frac{V_t'}{D}\right)^2-4\right]d\phi\ .
\label{dSdk}
\end{equation}
The sign of this derivative depends on the type of transition considered. For dS to dS transitions, defined as those for which $V(\phi_{0{\scriptscriptstyle -}})>0$ even if $V_{\scriptscriptstyle -}$ might be negative, we have $V_t>0$ for $\phi<\phi_{0{\scriptscriptstyle -}}$ (and zero integrand for $\phi>\phi_{0{\scriptscriptstyle -}}$). Then, it follows that $-1\leq V_t'/D \leq 1$ and
$dS[V_t]/d\kappa<0$ and so, one concludes that larger $\kappa$ (stronger gravitational effects) lowers the tunneling action, making dS vacua more unstable.

However, for dS to AdS transitions [i.e. those with $V(\phi_{0{\scriptscriptstyle -}})<0$], $V_t$ gets negative at some point resulting in $V_t'/D<-1$ and the sign of
$dS[V_t]/d\kappa$ depends on the shape of the potential.  If the region with negative $V_t$ dominates in the integral (\ref{dSdk}) one would get a positive slope and stronger gravity makes the false vacuum more stable (as always happens for the decays of Minkowski or AdS false vacua). In subsection~\ref{7.5} we show an example of potential that can realize both signs of $dS[V_t]/d\kappa$ depending on its parameters.

(3) If we rescale the field as $\phi\to \phi/a$, the tunneling potential
for $V_a(\phi)\equiv V(\phi/a)$ is not simply $V_t(\phi/a)$ but we can still see how the action changes depending on whether $a<1$ (thinner barrier) or $a>1$ (wider barrier). As the HM part of the action does not change under a field rescaling, we simply look at the CdL part. Consider the CdL part of the action for a rescaled $V_{ta}(\phi)\equiv V_t(\phi/a)$ (for any arbitrary $V_t$):
\begin{equation}
\Delta S_{CdL}[V_{ta}]=\frac{6\pi^2}{\kappa^2}\int_{a\phi_{0{\scriptscriptstyle +}}}^{a\phi_{0{\scriptscriptstyle -}}}\frac{[D^a_\kappa+V_{ta}']^2}{D^a_\kappa V_{ta}^2}d\phi\ ,
\end{equation}
where
\begin{equation}
D_\kappa^a\equiv\left[V_{ta}'{}^2+6\kappa(V_a-V_{ta})V_{ta}\right]^{1/2}\ .
\end{equation}
Changing the integration variable $\phi\to \phi/a$ we have
\begin{equation}
\Delta S_{CdL}[V_{ta}]=\frac{6\pi^2}{\kappa^2}\int_{\phi_{0{\scriptscriptstyle +}}}^{\phi_{0{\scriptscriptstyle -}}}\frac{[D_{\kappa a^2}+V_t']^2}{D_{\kappa a^2} V_{t}^2}d\phi\ ,
\label{Srescaled}
\end{equation}
where
\begin{equation}
D_{\kappa a^2}\equiv\left[V_{t}'{}^2+6\kappa a^2(V-V_{t})V_{t}\right]^{1/2}\ .
\end{equation}
Now, for $a>1$ it is easy to show that $[D_{\kappa a^2}+V_t']^2/(D_{\kappa a^2}V_t^2) \geq [D_{\kappa}+V_t']^2/(D_{\kappa}V_t^2)$ so that the action integrand is larger
for $a>1$ than for $a=1$.\footnote{Taking $x=6\kappa(V-V_t)V_t/V_t'{}^2$ reduces the problem to the inequality $f(x a^2)\geq f(x)$
involving the function $f(x)\equiv \sqrt{1+x}+1/\sqrt{1+x}$ for $x\in (-1,\infty)$,  as $D^2>0$ . The equality holds only for $x=0$, which requires $V=V_t$, i.e. a HM transition} This proves that, for $a>1$,  $\Delta S_{CdL}[V_{ta}]>\Delta S_{CdL}[V_t]$ for any $V_t$. Therefore, the minimum of
the functional $S[V_{ta}]$ is larger (or equal\footnote{If the original decay was already a HM one, then the rescaling has no effect on the action and that is why we also include the possible equality here.}) than the minimum of  $S[V_{t}]$ and the tunneling action must increase (or stay constant) when the barrier is made wider ($a>1$).  Eventually, the action for tunneling via a CdL instanton will become larger than the HM one (unchanged by the field rescaling) and for barriers wider than a critical value it is the HM action that controls the decay. Finally, for the opposite case of $a<1$ all inequalities are reversed, so that a  thinner barrier leads to a higher decay rate (lower tunneling action).

(4) From (\ref{Srescaled}) and the previous discussion of the $\kappa$ dependence of the action, we see that the tunneling action after a field
rescaling $\phi\to \phi/a$ is related to the action after $\kappa\to\kappa a^2$ by $S_{\phi\to\phi/a} = a^4  S_{\kappa\to\kappa a^2}$.
We can combine this inequality with $S_{\phi\to\phi/a}\geq S$ for $a>1$ from (3) above (wider barriers increase the action). For dS to dS transitions or dS to AdS transitions with $dS/d\kappa<0$, we also know from (2) above that $S_{\kappa\to\kappa a^2}<S$ for $a>1$ (stronger gravity makes such dS vacua more unstable). Combining
all this we get $S\leq S_{\phi\to\phi/a} = a^4  S_{\kappa\to\kappa a^2}<a^4 S$. Therefore we arrive at the following two inequalities
for $a>1$
\begin{eqnarray}
S\leq S_{\phi\to\phi/a}<a^4 S\ , \nonumber\\
a^{-4} S\leq S_{\kappa\to\kappa a^2}< S\ ,
\label{scalingrange}
\end{eqnarray}
that set upper and lower bounds for these rescalings. Notice that this behaviour is quite different from the one for Minkowski or AdS vacuum decay. In these two cases, making the barrier wide enough or increasing sufficiently the strength of gravity could quench completely the decay (leading to $S\to\infty$). The bounds above prevent this from happening in the dS case. After all, a dS vacuum
can always decay via the HM instanton.

\section{Onset of Hawking-Moss Transitions \label{sec:HM}}

It is well known that, if the vacuum energy of a false dS vacuum is sufficiently large, the CdL instanton disappears and vacuum decay  proceeds via the Hawking-Moss \cite{HM} instanton instead.\footnote{For the possible cosmological signatures of a HM transition, see e.g. \cite{BK}.} In this section we
use the tunneling potential approach to examine this phenomenon.

The decay of a false dS vacuum minimizes the tunneling action (\ref{SVt}) which is the sum of two nonzero pieces, as written in (\ref{SHMCdL}). The first action contribution,  $\Delta S_{HM}$, pulls $V_{0{\scriptscriptstyle +}}$ towards $V_{\scriptscriptstyle +}$  (and thus $\phi_{0{\scriptscriptstyle +}}\to\phi_{\scriptscriptstyle +}$) to become smaller, while  $\Delta S_{CdL}$ wants the interval $(\phi_{0{\scriptscriptstyle +}},\phi_{0{\scriptscriptstyle -}})$ to shrink so as to lower the value of the CdL integral. The final solution comes from the interplay between these two opposing demands \cite{Eg}.
To get a simple qualitative understanding of this trade-off it is useful to
make the rough approximation of taking $V_t$ to be a constant.
In that case \cite{Eg}
\begin{equation}
S(V_t)=\frac{24\pi^2}{\kappa^2}\left(\frac{1}{V_{\scriptscriptstyle +}}-\frac{1}{V_t}\right)+\frac{6\pi^2\sqrt{3}}{(\kappa V_t)^{3/2}}\int_{\phi_{0{\scriptscriptstyle +}}}^{\phi_{0{\scriptscriptstyle -}}}\sqrt{2(V-V_t)}\ d\phi\ ,
\end{equation}
where $\phi_{0{\scriptscriptstyle{\pm}}}$ are just the two solutions of $V=V_t$.
If one keeps the shape of $V$ fixed but raises the overall energy scale $V_{\scriptscriptstyle +}$ then   $\Delta S_{HM}\sim 1/V_{\scriptscriptstyle +}^2$ and $\Delta S_{CdL}\sim 1/V_{\scriptscriptstyle +}^{3/2}$ and the minimization of the total tunneling action requires $\Delta S_{CdL}\to 0$ with $\phi_{0{\scriptscriptstyle +}}\to\phi_{0{\scriptscriptstyle -}}\to\phi_B$, the field value for the top of the barrier. In that case, vacuum decay proceeds via the Hawking-Moss instanton with rate
\begin{equation}
S=S_{HM}=\frac{24\pi^2}{\kappa^2}\left(\frac{1}{V_{\scriptscriptstyle +}}-\frac{1}{V_B}\right)\ ,
\end{equation}
[where $V_B\equiv V(\phi_B)$] as obtained from (\ref{SVt}) when the CdL interval shrinks to zero \cite{Eg}.\footnote{For a derivation of this rate in the context of a stochastic approach to tunneling, see \cite{Vennin}.}

Although the previous behaviour is generic and will happen for potentials of any shape, the particular way in which the transition between CdL-mediated and HM-mediated decay happens depends on the shape of the potential barrier.

Consider first the case of a barrier with an inverse quadratic top
\begin{equation}
V= V_B - \frac12 m^2\phi^2+...
\label{Vquadratic}
\end{equation}
In the constant-$V_t$ approximation $S(V_t)$ can be calculated (and minimized) analytically. One gets
\begin{equation}
S(V_t)=\frac{24\pi^2}{\kappa^2}\left(\frac{1}{V_{\scriptscriptstyle +}}-\frac{1}{V_t}\right)+\frac{6\sqrt{3}\pi^3}{m(\kappa V_t)^{3/2}}(V_B-V_t)\ .
\end{equation}
Figure~\ref{fig:top}, left plot, shows this action (blue continuous curves) as a function of $V_t$ for different values of $V_B$ (the end point of each curve corresponds to $V_t=V_B$). The black dashed line marks the location of the minimum of $S(V_t)$ while the horizontal dashed lines give the value of $S_{HM}$ for each case.
As the scale of the potential $V_B$ increases, the location of the minimum of
$S(V_t)$ gets closer to the top of the barrier $V_B$ and the action tends to the Hawking-Moss one. Above a certain critical value
the minimum of $S(V_t)$ is at $V_B$ and the CdL instanton has disappeared being replaced by the Hawking-Moss one. While we vary $V_B$ we keep $V_{\scriptscriptstyle +}$ fixed and this is why the action increases with $V_B$.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.45\textwidth]{QuadraticTop.pdf}\hspace{0.2cm}
\includegraphics[width=0.45\textwidth]{QuarticTop.pdf}
\end{center}
\caption{Tunneling action $S$ (continuous blue lines) in the constant $V_t$ approximation
for $V=V_B-m^2\phi^2/2$ (left) and $V=V_B-\lambda\phi^4/4$ (right) for different values of $V_B$ ($V_B$ corresponds to the $V_t$ value of the right end of each curve), with the dashed black lines tracking the extrema. The horizontal dashed lines give the corresponding action for HM transitions. We take $\kappa=1$, $V_{\scriptscriptstyle +}=0.1\ (0.001)$ for the left (right) plot, $m^2=30$ and $\lambda=1$.
\label{fig:top}
}
\end{figure}

The value of the critical $V_B$ calculated in the constant $V_t$ approximation is obtained as $V_{Bc}=16 m^2/(3\kappa \pi^2)\simeq 0.54 m^2/\kappa$ which gives right its order of magnitude.
The exact value can be obtained easily by solving the EoM  for the exact $V_t(\phi)$ in the limit of CdL disappearance, when $V_t$ gets very close to the top of the barrier.  In that limit, a quadratic expansion should be enough to study the behaviour of $V$ and $V_t$. Writing $V=V_B-m^2\phi^2/2+...$ and $V_t=V_B-\delta -m_t^2\phi^2/2+...$, and imposing the boundary condition $V_t'=3V'/4$ at the $\phi_0\simeq \sqrt{2\delta/(m^2-m_t^2)}$ for which $V_t=V$ one gets
\begin{equation}
m_t^2 = \frac34 m^2\ .
\end{equation}
Using this in the EoM for $V_t$ it follows that $V_{tB}''+\kappa(3V_B-2V_{tB})=0$, from which we get
the criticality condition
\begin{equation}
V_{Bc}=\frac{3m^2}{4\kappa}\ .
\label{VBc}
\end{equation}
For $V_B\geq V_{Bc}$ the Hawking-Moss instanton is the relevant one to describe dS vacuum decay. In the analytical examples
we present in section~\ref{sec:ex} it can be checked explicitly how this critical relation is satisfied.
We can also interpret the critical relation (\ref{VBc}) as a limit on how large the curvature at the top should be for the CdL instanton to exist:
\begin{equation}
m^2> \frac43 \kappa V_B\ .
\label{mbound}
\end{equation}
In the Euclidean formalism, this bound is understood as a necessary condition for the CdL compact bounce to fit inside the horizon at $\phi_B$. Substituting the expressions  for $V$ and $V_t$ above in eq.~(\ref{dphi}) for $\dot\phi$ and integrating that equation
\begin{equation}
\int_0^{\xi_{CdL}}d\xi = \int_{-\phi_0}^{\phi_0}\frac{d\phi}{\sqrt{2(V-V_t)}}\simeq \frac{1}{\sqrt{m^2-m_t^2}}\int_{-\phi_0}^{\phi_0}\frac{d\phi/\phi_0}{\sqrt{1-\phi^2/\phi_0^2}}\ ,
\end{equation}
we find that the size of the CdL instanton is $\xi_{CdL}=\pi/\sqrt{m^2-m_t^2}=2\pi/m$. The horizon radius at the top of the barrier, $\phi_B$, is $\xi_{dS,B}=\pi\sqrt{3/(\kappa V_B)}$ and imposing $\xi_{CdL}<\xi_{dS,B}$ gives (\ref{mbound}).

We can then examine what would happen if the top of the potential barrier is flatter, say like an inverted quartic:
\begin{equation}
V=V_B-\frac14\lambda\phi^4+...
\end{equation}
Again, the constant-$V_t$ approximation can give the rough qualitative behaviour. One gets
\begin{equation}
S(V_t)=\frac{24\pi^2}{\kappa^2}\left(\frac{1}{V_{\scriptscriptstyle +}}-\frac{1}{V_t}\right)+\frac{16\sqrt{3}\pi^2 K(-1)}{\lambda^{1/4}(\kappa V_t)^{3/2}}(V_B-V_t)^{3/4}\ ,
\end{equation}
where $K(m)$ is the complete elliptic integral of the first kind, with $K(-1)\simeq 1.3$.  Figure~\ref{fig:top}, right plot, shows
$S(V_t)$ for different values of $V_B$ (again keeping $V_{\scriptscriptstyle +}$ fixed). For low values of $V_B$, $S(V_t)$ has a minimum with action lower than the HM action (corresponding to the CdL instanton decay) and a maximum with higher action. The location of these minima and maxima is given by the black dashed lines. As $V_B$ increases, the minimum is lifted, until a critical value is reached for which the minimum is degenerate
with the HM one. For $V_B$ higher than that critical value, the action is minimized by the HM instanton.  For $V_B$ higher than the critical value, the minimum and maximum of $S(V_t)$ get closer until they merge into a saddle point and disappear for even higher $V_B$.

Note that this case is qualitatively different from the previous one with quadratic barrier top for which the CdL-mediated minimum merges with the HM one at the critical value, while now they are still separate. When the barrier top is flat, the CdL integral decreases more slowly as $V_B$ increases and at some point it is advantageous to switch to the HM instanton. The picture obtained above from the simple constant-$V_t$ approximation is confirmed by a numerical calculation of $V_t$.

Potentials with a flat-top barrier have been studied before using the Euclidean approach \cite{JS,BD,HW,BLL},  which emphasized the potential obstruction to the existence of the CdL bounce  when condition (\ref{mbound}) is violated, as well as possible exceptions to that bound.\footnote{Some of these works also discuss the so-called oscillating bounces, which cross the top of the barrier more than once. We do not discuss them here  (although the $V_t$ formalism can be applied to them as well) as they do not seem to be relevant for vacuum decay.}  Although one can also apply the $V_t$ formalism to examine in more detail flat-top potentials, we do not pursue this goal further. We tried to obtain exactly solvable examples of this type of flat-top potentials to include in section~\ref{sec:ex} but we failed to find any.

\section{Exactly Solvable Models: Getting  ${\bma V}$ from ${\bma{V_t}}$ \label{sec:exact}}
\noindent

The tunneling potential formulation can be quite useful to generate potentials with exactly solvable false vacuum decay.\footnote{Exactly solvable models with gravity included have been obtained using other methods
before, see \cite{exact0,exact1,exact2} for an incomplete list.
} The idea is to postulate a simple enough $V_t$ and solve its EoM for $V$, which is a much simpler task that solving for $V_t$ given $V$. Without gravity, the integration for $V$ can be done formally, and explicitly for certain $V_t$'s. This was done in \cite{E} to generate some exactly solvable potentials. In the case with gravity the formal integration for $V$ has not been done but \cite{Eg} presented one
exactly solvable case for a particular choice of $V_t$ leading to
analytic examples of Minkowski or AdS decays. In this section we perform the formal integration for $V$ in the presence of gravity and in the next section we will make use of it to generate a number of examples for dS decays.

Given the expression for $D^2$ in (\ref{D2}) we can formally write $V(\phi)$ in terms of $V_t(\phi)$ and $D(\phi)$
as
\begin{equation}
V(\phi)=V_t+\frac{D^2-V_t'{}^2}{6\kappa V_t}\ .
\label{VfromD}
\end{equation}
When $D\equiv 0$ the action $S[V_t]$ becomes infinite and the decay is forbidden by gravity. In that case the potential $V$ reproduces the generic form of a critical potential \cite{Estab}. In that sense, $D$ measures deviations of $V$ from criticality.

Using (\ref{VfromD}) the equation of motion (\ref{EoMVt}), given in terms of $V_t$ and $V$, can be rewritten in terms of $D$ and $V_t$ and takes the form
\begin{equation}
 V_t''+\kappa V_t+\frac{D^2-V_t'{}^2}{2V_t}- V_t'\frac{D'}{D}=0\ .
\label{EoMD}
\end{equation}
This differential equation can be integrated formally to obtain
$D^2$ in terms of $V_t$ as
\begin{equation}
D^2(\phi)= \frac{V_t'{}^2}{1-V_{t}F}\ ,
\label{D2F}
\end{equation}
where
\begin{equation}
F(\phi)\equiv \frac{2\kappa}{E(\phi)} \int_{\phi_0}^{\phi}\frac{E(\tilde\phi)}{V_t'(\tilde\phi)}d\tilde\phi\ ,\quad
E(\phi)\equiv \exp\left[2\kappa\int_{\phi_0}^{\phi}\frac{V_t(\tilde\phi)}{V_t'(\tilde\phi)}d\tilde\phi\right]\
\label{FE}
\end{equation}
and $\phi_0$ is some reference field value, taken here to be one of the two contact points between $V$ and $V_t$, so that $D^2(\phi_0)=V_t'{}^2(\phi_0)$.

The procedure to find an analytical $V$ under control is then to find a $V_t$ simple enough that the integrals $E$ and $F$ can be performed analytically. Once $D^2$ is found, it can be plugged in (\ref{VfromD}) to obtain $V$ as
\begin{equation}
\boxed{
V(\phi)=V_t+\frac{V_t'{}^2}{6\kappa(1/F-V_t)}}
\label{formalV}
\end{equation}
The function $F$ plays a key role in what follows and it is instructive to rewrite (\ref{EoMD}) as a
differential equation for $F$, which takes the very simple form
\begin{equation}
F'V_t'=2\kappa(1-FV_t)\ .
\label{EoMF}
\end{equation}
 As is evident from (\ref{VfromD}) and (\ref{D2F}), the zeros of $F$ determine the field values at which $V_t$ touches $V$. In (\ref{formalV}) we see that at such points $1/F$ diverges and the second term drops, giving precisely $V=V_t$.

The previous results are general: they apply to decays from Minkowski, AdS or dS vacua, and the formalism can be used to obtain analytical potentials in all those cases. In what follows we focus on dS decays.
In order to get sensible results, $V_t$ must satisfy the proper boundary conditions and constraints.  In the case of dS decays,
the field range of the CdL bounce is $(\phi_{0+},\phi_{0-})$, with  $\phi_{0\pm}$  being different from the minima of the potential. The values of $\phi_{0\pm}$ will correspond to the zeros of $F$:
\begin{equation}
F(\phi_{0\pm})=0\ .
\label{phi0fix}
\end{equation}
This condition is crucial as it fixes the value of $\phi_{0\pm}$. We show explicit examples of this in the next section.

In the dS case, $V_t$ is not monotonic but rather it should have a maximum at some intermediate field value $\phi_{T}$ (which corresponds to the maximum value of the metric function $\rho$ in the Euclidean formulation, see \cite{Eg})  and at this maximum the top of $V_t$ should be an inverted quadratic:
 From the EoM (\ref{EoMVt}) it follows that
 \begin{equation}
 V_{tT}''=-\kappa(3V_T-2V_{tT})<0\ .
 \end{equation}
 (The subindex $T$ on a  function indicates that it is evaluated at $\phi_T$.)
 Eq.~(\ref{formalV}) shows that  $V_t'=0$ would naively lead to $V_T\equiv V(\phi_T)=V_{tT}\equiv V_t(\phi_T)$ while the contact points between $V$ and $V_t$ should be just $\phi_{0\pm}$. The condition $V_T\neq V_{tT}$ requires
\begin{equation}
F(\phi_T)V_t(\phi_T)=1\ ,
\label{FVt1}
\end{equation}
so that the ratio in (\ref{formalV}) gives a finite value. This means
$1-FV_t$ goes to zero at $\phi_T$ as $V_t'{}^2$ does. From (\ref{EoMF}) we then learn that $F'$ also vanishes at $\phi_T$.
Using (\ref{EoMF}), (\ref{FVt1}) and L'H\^opital's rule we get
\begin{equation}
V_T=V_{tT}+\frac{V_{tT}''}{3V_{tT}F_T''}\ .
\label{VTexp}
\end{equation}
Moreover, (\ref{D2F}) shows that $FV_t<1$ and therefore (\ref{EoMF}) forces $F'$ to have the same sign as $V_t'$: $F$ is also a nonmonotonic function with a maximum and $V_{tT}''/F_T''>0$, with $F_T''<0$ and $V_{tT}''<0$.

In the next section we put this formalism to work and obtain several examples of analytical potentials describing dS to dS transitions  (as well as one transition from dS to AdS).  We relegate to the  appendix a simple case of Minkowski false vacuum decay, that generalizes the scale invariant potential $V(\phi)=-\lambda\phi^4/4$ to the case with gravity.

\section{Examples of Solvable  dS Decays\label{sec:ex}}

To use the formalism described in the previous section we can postulate a simple $V_t$ (with a maximum at some point $\phi_T$) that allows to integrate explicitly (\ref{EoMF}) for $F$. The result will depend on an integration constant $C$ that is fixed in the following way.  A Taylor expansion of $F$ around $\phi_T$ takes the generic form
\begin{equation}
F(\phi_T+\epsilon) = \frac{1}{V_{tT}}+\frac{g^2}{\kappa-\kappa_c}\epsilon^2+{\cal O}(\epsilon^3)+(C-C_\kappa){\cal O}(\epsilon^{2\kappa/\kappa_c})\ ,
\label{Fexp}
\end{equation}
with $g^2,\kappa_c>0$,  $C_\kappa$ is some model dependent constant and
\begin{equation}
\kappa_c=-V_{tT}''/V_{tT}\ .
\end{equation}
As explained in the discussion at the end of the previous section, around eq.~(\ref{VTexp}), we need $F''(\phi_T)<0$ and finite to have $V_t$ below $V$ at $\phi_T$. From (\ref{Fexp}) we immediately see that to satisfy that requirement we need to respect the limit $\kappa<\kappa_c$ (if this is violated the decay occurs via the HM instanton rather than the CdL bounce) and fix the integration constant as $C=C_\kappa$.

Although most of the examples we present below follow this method, there is an alternative way to generate solvable potentials:  postulate a simple enough $F$, with two zeros and a maximum between them, and integrate equation (\ref{EoMF}) for $V_t$. Example~\ref{7.4} illustrates this route.  Besides the examples we present below we have found many others and the interested reader should be able to produce new ones easily.

\subsection{$\bma{V_t(\phi) =A-1+ \cos\phi}$\label{7.1}}
For simplicity, in this $V_t$ (and some other examples below) we suppress mass scales (e.g. in the normalization of the field and the prefactor of the cosine, etc.). These scales can be recovered trivially if needed. As $V_t$ is symmetric under $\phi\to-\phi$, the potential $V$ enjoys the same symmetry. The CdL field range will simply be $(\phi_{0+},\phi_{0-})=(-\phi_0,\phi_0)$ with $0<\phi_0<\pi$ and the maximum of $V_t$ occurs at $\phi_T=0$.

Equation~(\ref{EoMF}) can be integrated analytically and $F$ expressed in terms of a hypergeometric function as
\begin{equation}
F(\phi)=\frac{[\cos(\phi/2)]^{4\kappa}}{A}\, _2F_1[-2\kappa,-A\kappa;1-A\kappa;-\tan^2(\phi/2)]+C \frac{(\sin^2\phi)^\kappa}{[\tan^2(\phi/2)]^{(1-A)\kappa}}\ ,
\end{equation}
where $C$ is an integration constant. Expanding $F(\phi)$ around $\phi_T=0$ we have
\begin{equation}
F(\epsilon)=\frac{1}{A} + \frac{\kappa}{2A(A\kappa-1)}\epsilon^2+{\cal O}(\epsilon^3)+
 C\, \epsilon^{2A\kappa}\left[4^{(1-A)\kappa}+{\cal O}(\epsilon^2)\right]\ ,
\end{equation}
which conforms to the generic expression (\ref{Fexp}). We immediately conclude that $C=0$ and the condition to have a CdL bounce is $\kappa<\kappa_c=1/A$. If that bound is not satisfied the transition occurs via the Hawking-Moss instanton.
Having fixed $F$, the potential is then obtained by plugging $F$ above on (\ref{formalV})
and $\phi_0$ is obtained from $F(\phi_0)=0$.

For numerics, let us examine first the case with fixed $A$ (we take $A=2$) and let $\kappa$ vary. Figure~\ref{fig:71b} shows $\phi_0$ as a function of $\kappa$ and different $V$'s for several choices of $\kappa$, as indicated.
The numerical analysis confirms  the critical value $\kappa_{c}=1/A=1/2$, at which the CdL range shrinks to zero ($\phi_0\to 0$). Above that critical value the transition occurs via the Hawking-Moss instanton.

It is also instructive to fix $\kappa=1/4$ and let $A$ vary.
Figure~\ref{fig:71a} shows $\phi_0$ as a function of $A$ and pairs of $V$ and $V_t$ for several choices of $A$, as indicated. We see that, as expected, there is a critical value for $A$, $A_{c}=1/\kappa=4$, at which the CdL range shrinks to zero ($\phi_0\to 0$), in accordance with the bound $\kappa<1/A$  derived above.
In figure~\ref{fig:VtL} we show, for the same choice of parameters,
the actions for decay via the HM transition or via the CdL bounce
(fixing $V_{\scriptscriptstyle +}=0.01$ for concreteness), confirming that the latter action is smaller. This is not guaranteed (see discussion in section~\ref{sec:HM}) and must be checked in each case. The rest of the examples we present do pass this test.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.45\textwidth]{phi0kappa.pdf}\hspace{0.2cm}
\includegraphics[width=0.45\textwidth]{VtVkappa.pdf}
\end{center}
\vspace{-0.5cm}
\caption{
For example \ref{7.1} with $A=2$, varying $\kappa$. Left plot: Limit, $\phi_0$, of the CdL interval. Right plot: For the fixed $V_t$,  several $V$'s in the CdL range for the indicated values of $\kappa$.
\label{fig:71b}
}
\end{figure}
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.45\textwidth]{phi0V0.pdf}\hspace{0.2cm}
\includegraphics[width=0.45\textwidth]{VtVV0.pdf}
\vspace{-0.5cm}
\end{center}
\caption{For example \ref{7.1} with $\kappa=1/4$, varying $A$. Left plot: Limit, $\phi_0$, of the CdL interval. Right plot: $V$ and $V_t$ pairs in the CdL range for the indicated values of $A$.
\label{fig:71a}
}
\end{figure}

When a potential $V$ is obtained from a postulated $V_t$, as done in this section, in principle $V$ is defined only in the CdL interval $({\phi_{0{\scriptscriptstyle +}},\phi_{0{\scriptscriptstyle -}}})$. Outside that interval one is free to
extend $V$ almost arbitrarily, although the continuity of $V$ and $V'$ at $\phi_{0\pm}$ are natural conditions to impose.
In addition, the location of the potential minima should be compatible with the decay considered. For instance, a $V_t$ with a maximum describes dS decay and therefore, the false vacuum
(in the region of $V$ outside the CdL interval) should have $V_{\scriptscriptstyle +}>0$.

In some cases, the function $V_t$ that has been used to construct $V$ is defined and well behaved outside the CdL interval and can be used to extend also $V$ in that field range. Figure~\ref{fig:VtL} illustrates this, for $A=3$ and $\kappa=1/4$, with $V$ and $V_t$ both simply extended beyond the CdL range. The extension of $V_t$, labeled $V_{tL}$ in the figure, can be simply ignored or if not, needs to be reinterpreted, as it breaks the crucial relation $V_t=V-\dot\phi^2/2$ which implies $V_t\leq V$. This extension is
similar to the Lorentzian continuation of Euclidean CdL geometries performed in \cite{exact0} to extend the exact solutions to describe the aftermath of bubble nucleation.
Exploring the precise meaning of this extension for $V_t$ would be interesting but goes beyond the goals of the present paper.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.485\textwidth]{SHMS.pdf}
\includegraphics[width=0.45\textwidth]{VtL.pdf}
\end{center}
\vspace{-0.5cm}
\caption{For example \ref{7.1}, right plot: Actions for decay via HM transition or via CdL bounce as a function of $A$, with $\kappa=1/4$. Left plot: Potential and tunneling potential, extended beyond the CdL range,  with $A=3$ and $\kappa=1/4$. The extension of $V_t$ is labeled $V_{tL}$ and shown by dashed lines.
\label{fig:VtL}
}
\end{figure}

\subsection{$\bma{V_t(\phi)=e^{-\phi^2}}$\label{7.2}}
For this $V_t$, integrating (\ref{EoMF}),
we get
\begin{equation}
F(\phi)\equiv \ \frac{\kappa}{2}(-\phi^2)^{\kappa/2}\left[\Gamma(-\kappa/2,-\phi^2)-\Gamma(-\kappa/2)\right] +C (\phi^2)^{\kappa/2} \ ,
\end{equation}
where $\Gamma(a,z)$ is the incomplete gamma function. Expanding $F(\phi)$ around $\phi_T=0$ one gets
\begin{equation}
F(\epsilon)=2+\frac{\kappa}{\kappa-2}\epsilon^2+{\cal O}(\epsilon^3)+ C (\epsilon^2)^{\kappa/2}\ ,
\end{equation}
from which we read $\kappa_c=2$, $C=0$, in accordance with the general discussion around (\ref{Fexp}).
The potential is then obtained by plugging $F$ above on (\ref{formalV}) and $\phi_0$ is obtained from $F(\phi_0)=0$.

Figure~\ref{fig:72} shows $\phi_0$ as a function of $\kappa$ and different $V$'s for the same $V_t$ for several choices of $\kappa$, as indicated. We see how the CdL range shrinks to zero ($\phi_0\to 0$) as $\kappa\to\kappa_c$. Above that critical value the transition occurs via the Hawking-Moss instanton.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.4\textwidth]{phi0kappa2.pdf}\hspace{0.2cm}
\includegraphics[width=0.4\textwidth]{VtVkappa2.pdf}
\vspace{-0.5cm}
\end{center}
\caption{For example \ref{7.2}, varying $\kappa$. Left plot: Limit, $\phi_0$, of the CdL interval. Right plot: For the fixed $V_t$,  several $V$'s in the CdL range for the indicated values of $\kappa$.
\label{fig:72}
}
\end{figure}

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.34\textwidth]{phi0kappa3.pdf}\hspace{0.2cm}
\includegraphics[width=0.4\textwidth]{VtVkappa3.pdf}
\vspace{-0.5cm}
\end{center}
\caption{For example \ref{7.3}, varying $\kappa$. Left plot: Limits, $\phi_{0\pm}$, of the CdL interval. Right plot: For the fixed $V_t$,  several $V$'s in the CdL range for the indicated values of $\kappa$.
\label{fig:73}
}
\end{figure}

\subsection{$\bma{V_t(\phi)=e^{-\phi}/\cosh^2\phi}$\label{7.3}}
This is one simple example  with $V_t$ and $V$ that are not symmetric under $\phi-\phi_T\to\phi_T-\phi$.
For these cases the critical equation $F(\phi)=0$ has two non symmetric zeros,  $\phi_{0\pm}$, that give the extremes of the CdL interval. For the example above we have\footnote{For the particular case $\kappa=3/4$, $F(\phi)$ can be expressed in terms of elementary functions, although it is not particularly simple.}
\begin{eqnarray}
F(\phi)&=&\frac{\kappa}{\sqrt{3}(9-2\kappa)(9-4\kappa^2)}\left\{\frac{1}{6\sqrt{z}}\left[9(111-30z-z^2)-144\kappa(1+z)+4\kappa^2(3+z)^2\right]
\right.\nonumber\\
&+&\left.
48(1-z)^{4\kappa/3}z^{-\kappa}(3-4\kappa)
\left[B(z;\kappa-1/2,-4\kappa/3)-C\right]\right\}\ ,
\end{eqnarray}
where $z\equiv 3\,e^{2\phi}$ and $B(z;a,b)$ is the incomplete beta function. Expanding $F(\phi)$ around
the maximum of $V_t$, at $\phi_T=-\log(3)/2$, we get
\begin{equation}
F(\phi_T+\epsilon)=\frac{4}{3\sqrt{3}}+\frac{2\kappa\epsilon^2}{\sqrt{3}(2\kappa-3)}+{\cal O}(\epsilon^3)+{\cal O}(\epsilon^{4\kappa/3})
\left[C-C_\kappa+{\cal O}(\epsilon)\right]\ ,
\end{equation}
with
\begin{equation}
C_\kappa=-\frac{\pi\csc(4\pi\kappa/3)\Gamma(\kappa-1/2)}{\Gamma(-1/2-\kappa/3)\Gamma(1+4\kappa/3)}\ .
\end{equation}
From this expansion we obtain $\kappa_c=3/2$ and $C=C_\kappa$.

The potential is obtained from (\ref{formalV}) and the extremes of the CdL interval come from solving $F(\phi_{0\pm})=0$. They
are shown in figure~\ref{fig:73}, left plot, with $\phi_{0+}=\phi_{0-}$ for $\kappa=3/2$. The right plot gives $V_t$ and $V$ for several values of $\kappa$ as indicated, showing again how for $\kappa\to\kappa_c=3/2$
the CdL interval disappears.

\subsection{$\bma{V_t(\phi)=2\left(\cos\theta+\cos\phi\right)/\sin^2\theta}$\label{7.4}}
As mentioned, we can also get simple examples by postulating a simple $F$, from which one gets $V_t$ and then $V$. For instance, if we take $F=a V_t+b$ and solve for $V_t$ we end up with the simplest example we have found. We get
\begin{equation}
V_t(\phi)=\frac{1}{4a}\left[-2b+(1+4a+b^2)\cos(\sqrt{2\kappa}\phi+C)+i (1-4a-b^2)\sin(\sqrt{2\kappa}\phi+C)\right]\ ,
\end{equation}
where $C$ is an integration constant.
Setting $\kappa=1/2$,  $C=0$ and $b=-\sqrt{1-4a}$ (to have a real $V_t$), we end up with
\begin{equation}
V_t(\phi)=\frac{2}{\sin^2\theta}\left(\cos\theta+\cos\phi\right)\ ,
\end{equation}
where we have introduced the angular parameter $\theta$, given by $4a=\sin^2\theta$, with $0\leq \theta\leq \pi/2$. From this we obtain the axion-like potential
\begin{equation}
V(\phi)=\frac{4}{3\sin^2\theta}\left(\cos\theta+2\cos\phi\right)\ ,
\end{equation}
and the CdL range is simply the interval $(-\theta,\theta)$, with $V(\pm\theta)=V_t(\pm\theta)=4\cos\theta/\sin^2\theta$ at the contact points of $V$ and $V_t$.
The CdL part of the action can be calculated exactly as
\begin{equation}
\Delta S_{CdL}=24\pi^2\sin\theta\,(\tan\theta-\theta)\ .
\end{equation}
The limit $\theta\to 0$ corresponds to the CdL instanton disappearance (which in this particular example only happens for $V_B=V(0)\to\infty$) with $\Delta S_{CdL}\to 0$. The limit $\theta\to\pi/2$ corresponds instead to a forbidden Minkowski to Minkowski  transition and has $\Delta S_{CdL}\to\infty$.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.4\textwidth]{phix.pdf}\hspace{0.2cm}
\includegraphics[width=0.4\textwidth]{rhox.pdf}
\end{center}
\caption{For example~\ref{7.4} and the indicated values of $\theta$. Left plot: Field profile $\phi(\xi)$ of the Euclidean bounce. Right plot: Profile of the Euclidean metric function $\rho(\xi)$.
\label{fig:phirho}
}
\end{figure}

It is instructive to derive the field and metric Euclidean profiles, $\phi(\xi)$ and $\rho(\xi)$ for this simple example. By integrating $d\phi/d\xi=-\sqrt{2(V-V_t)}$, see (\ref{dphi}), we get
\begin{equation}
\phi(\xi)=-2\ \mathrm{am}\left(\left.\frac{2\xi-\xi_e}{2\sqrt{6}\cos(\theta/2)}\right|\csc^2(\theta/2)\right)
\end{equation}
where $\mathrm{ am}(u|m)$ is the Jacobi amplitude function
and
\begin{equation}
\xi_e=\sqrt{6}\sin\theta\ K(\sin^2(\theta/2))\ ,
\end{equation}
with $K(m)$ the complete elliptic function of the first kind.  The metric function is obtained from $\rho=3\sqrt{2(V-V_t)}/D$, see (\ref{rho}), as
\begin{equation}
\rho(\xi)=\sqrt{3[\cos\phi(\xi)-\cos\theta]}\ .
\end{equation}
The finite interval over which the CdL instanton is defined is $\xi\in (0,\xi_e)$, with $\rho$ being zero at both extremes. Figure~\ref{fig:phirho} shows the profiles of both $\phi(\xi)$ and $\rho(\xi)$ for two choices of the $\theta$ parameter. This example nicely illustrates the complementarity between the Euclidean and $V_t$ formalisms showing how a complicated description in one case can become elementary in the other.

\subsection{$\bma{V_t(\phi)=A e^{\phi}-e^{a\phi}}$\label{7.5}}

This example, with $A>0$ and $a>1$, is interesting as
it can feature a dS vacuum decaying to AdS.
This $V_t$ peaks at
 $\phi_{T}=\log(A/a)/(a-1)$ and is
an asymmetric example with two different zeros $\phi_{0\pm}$.
The function $F$ is obtained as
\begin{equation}
F(\phi)=\frac{r}{q(a-1)}\left[\left(
\frac{a}{A}\right)^q\left(e^{-\phi}-\frac{A}{a}e^{-a\phi}\right)^rC+e^{-a\phi} {}_2F_1\left(1,p;1+q;\frac{A}{a}e^{-(a-1)\phi}\right)
\right]\ ,
\end{equation}
where $r=2\kappa/a$, $p=(a-2\kappa)/(a-1)$, $q=1+p/a$ while ${}_2F_1(a,b;c;z)$ is the hypergeometric function and $C$ is an integration constant.
The expansion of $F$ around $\phi_T$ reads
\begin{equation}
F(\phi_T+\epsilon)=\frac{1}{a-1}\left(\frac{a}{A}\right)^{a/(a-1)}
\left[1+\frac{a\kappa \epsilon^2}{2(\kappa-a)}+{\cal O}(\epsilon^3)
\right]+{\cal O}(\epsilon^{2\kappa/a})(C-C_\kappa)\ ,
\end{equation}
with
\begin{equation}
C_\kappa=\frac{\pi\csc(\pi r)}{B(p,1+r)}\ ,
\end{equation}
where $B(a,b)$ is the Euler beta function. We read off from this expansion $\kappa_c=a$ and $C=C_\kappa$, as usual.
As in all previous cases,
the potential is obtained from (\ref{formalV}) and the extremes of the CdL interval come from solving $F(\phi_{0\pm})=0$.

For the numerical examples we take $A=1/2$, $a=2$ and leave $\kappa$ free. Fig.~\ref{fig:75}, left plot, shows $\phi_{0{\scriptscriptstyle{\pm}}}$ as well as the value $\phi_B$ for the barrier top of $V$ as a function of $\kappa$. For $\kappa<2/3$, $V$ grows without reaching a maximum.  For $\kappa<3/2$, $V_t$ never reaches $V$ while for  $\kappa=3/2$, $V$ reaches $V_t$  only asymptotically with $\phi_{0{\scriptscriptstyle -}}\to\infty$, corresponding to a CdL field range of infinite extension. Finally, for $\kappa>\kappa_c=2$ there is no CdL instanton. We are therefore interested in principle in the interval $(3/2,2)$ for $\kappa$.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.4\textwidth]{PhidSAdS.pdf}
\includegraphics[width=0.4\textwidth]{dStoAdS2.pdf}
\end{center}
\vspace{-0.5cm}
\caption{For example \ref{7.5}, varying $\kappa$. Left plot: Limits, $\phi_{0\pm}$, of the CdL interval and $\phi_B$, location of the barrier maximum of $V$. Right plot: For the fixed $V_t$,  several $V$'s in the CdL range for the indicated values of $\kappa$.
\label{fig:75}
}
\end{figure}

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.4\textwidth]{actionratio.pdf}
\includegraphics[width=0.4\textwidth]{dSdk.pdf}
\end{center}
\vspace{-0.5cm}
\caption{For example \ref{7.5}, ratio of action for decay via Hawking-Moss instanton ($S_{HM}$) to action via Coleman-De Luccia bounce ($S$) as a function of $\kappa$ (left plot). For $\kappa>2$ only the HM transition is possible. Right plot: Slope  $dS/d\kappa$ for the action via the CdL bounce, showing that both signs are possible
\label{fig:actiondSAdS}}
\end{figure}

The tunneling potential, $V_t$, and several potentials, $V$, for different choices of $\kappa$ are plotted in figure~\ref{fig:75}, right plot. We confirm that, as $\kappa\to \kappa_c=2$ the CdL interval shrinks to zero. For $\kappa\stackrel{<}{{}_\sim} 1.81$  the transition is from dS to AdS
(the left plot also marks by a dashed line this special value).
 For $\kappa=3/2$ the intersection point $\phi_{0{\scriptscriptstyle -}}$ of $V$ and $V_t$ goes to $\infty$. This case is discussed below.

 Figure \ref{fig:actiondSAdS} shows the ratio between the  action for HM decay and the action for decay via the CdL instanton, as a function of $\kappa$. For concreteness we have assumed that the false vacuum
$\phi_{\scriptscriptstyle +}$ sits at a potential height $V_{\scriptscriptstyle +}=0.03$ (this just affects the overall value of the actions but not which one dominates).
The plot shows that decay occurs via a CdL
instanton below $\kappa=3/2$, while for $\kappa>3/2$ the HM instanton has lower action and drives the
decay. Notice also that the CdL part of the action remains finite also below
$\kappa=3/2$ even though the CdL range is infinite in that case (and moreover, $V_t$ never reaches $V$ for $\kappa<3/2$).  This does not seem to be physically accessible for decay, but the action thus obtained might be relevant to bound the true decay action in that range of $\kappa$ (similarly to what happens when a CdL bounce only exists at asymptotic infinity). The situation might be worth looking at in more detail, but goes beyond the goals of this paper. We can also examine  $dS[V_t]/d \kappa$, discussed in section 4.
Performing the integral (\ref{dSdk}) numerically we get the behaviour shown in Fig.~\ref{fig:actiondSAdS}, right plot, with both signs of  $dS[V_t]/d \kappa$ being possible although $dS/d\kappa>0$ occurs only in the suspicious case with $\kappa<3/2$.

The special case $\kappa=3/2$ (still with $A=1/2$ and $a=2$) is particularly interesting. For that critical value of $\kappa$, the potential simplifies dramatically to
\begin{equation}
V(\phi) = \frac49 e^{\phi}-\frac23 e^{2\phi}\ .
\end{equation}
Although the CdL field range  is infinite, $(\phi_{0{\scriptscriptstyle +}},\phi_{0{\scriptscriptstyle -}})=(-\log 6,\infty)$,   the CdL part of the action turns out to be finite, and can be computed exactly: $\Delta S_{CdL}=48\pi^2$. This leads to a total action $S=\Delta S_{HM}+\Delta S_{CdL}=(24\pi^2/\kappa^2)(1/V_{\scriptscriptstyle +}-1/V_{0{\scriptscriptstyle +}})+48\pi^2$. Remarkably, this is the same as the HM action $S_{HM}=(24\pi^2/\kappa^2)(1/V_{\scriptscriptstyle +}-1/V_{B})$. Therefore,  in this case  both instantons coexist and have the same action. Notice that this is very different from previous cases in which the HM case is reached as a limit in which the CdL instanton disappears. It is also interesting that, in the current case, one has  $-3V_B''=4\kappa V_B=4/9$, saturating the usual bound (\ref{mbound}) for the onset of HM.

In this simple example one can also obtain analytic expressions for the Euclidean bounce and metric profiles, by integrating (\ref{dphi}) and (\ref{rho}),  as
\begin{equation}
\phi(\xi)=\log\frac{\xi_e^2}{6(2\xi_e-\xi)\xi}\ ,\quad
\rho(\xi)=\frac{\xi(\xi_e-\xi)(2\xi_e-\xi)}{\xi_e^2}\ ,
\end{equation}
with $\xi_e=6\sqrt{6}$. The instanton is compact with the $\xi$ interval being $(0,\xi_e)$. In this case, $\xi_e<\xi_{dS,B}\equiv \pi\sqrt{3/(\kappa V_B)}$, so that the CdL bounce fits comfortably inside the $\phi_B$ horizon.

\section{Summary and conclusions \label{sec:conclusions}}

In this work we have resorted to the tunneling potential ($V_t$) formulation of vacuum decay in quantum field theory, including gravitational corrections \cite{Eg}, to obtain exactly solvable potentials in which to study such decays. The idea  is to find a simple
$V_t$ function (or the auxiliary function $F$ introduced in section~\ref{sec:exact}) whose differential equation of motion can be integrated to get $V$. The method can be used to study decays from any type of false vacua: Minkowski, anti-de Sitter or de Sitter, although we have focused on the latter, which is especially interesting and was lacking in \cite{Eg}. Section~\ref{sec:ex} contains a selection of several examples of such dS decays, including some strikingly simple ones, like the one in subsection \ref{7.4}. The method is quite powerful and many other examples can be found. Previous solvable examples discussed in the literature include some simple cases, like a conformally coupled scalar with a quartic potential with a particular value of the quartic coupling and an AdS false vacuum \cite{exact2} or more general methods based on postulating the Euclidean metric function $\rho(\xi)$, either by deforming the one of  Hawking-Moss instantons \cite{exact1}, or by imposing necessary constraints on the general form of $\rho(\xi)$ \cite{exact0}. Compared with this last more general method, the one we follow has some built-in advantages: for instance, the null-energy condition
$\dot\rho^2-\rho\ddot\rho-1\geq 0$,
imposed by \cite{exact0} (and identified there as the most challenging condition to find a physically meaningful $\rho$) is automatically satisfied by the $V_t$ formulation [as one can check by substituting in the condition above the relations in (\ref{rho})].

As we were particularly interested in finding examples of solvable dS
decays, we first examined how the $V_t$ formulation describes such decays. In particular we showed how these transitions are composed of two different pieces: a Hawking-Moss part (with $V_t=V$) and a Coleman-de Luccia part (with $V_t<V$), with their relative importance set by a balance between their competing needs to be minimized.
The illuminating discussion of \cite{BW} described a similar split  between a thermal excitation part (corresponding to our HM part) and a tunneling CdL part (corresponding to our $V_t<V$ part).
Our formalism also allows to derive in a straightforward way the simple relation between the actions for two-way transitions between dS vacua  and how, as the energy scale of the potential is raised, the dS decay is dominated by HM instantons. The method also allows
to understand the influence of the shape of the barrier top on this switch from CdL-dominated to HM-dominated decays.
For potentials with a quadratic barrier top we derived in a simple manner the critical value of $V''$ at the top of the barrier below which only the HM persists, showing also how the CdL instanton shrinks away and disappears at that critical value. For potentials with a flatter top the switch from the CdL-dominated decay to the HM-dominated one is qualitatively different, with the CdL instanton coexisting with the HM one although with higher action.

In addition, the general expression for the tunneling action integral allows to obtain several scaling properties of the action for dS decays: (1) Increasing all potential scales by an overall factor (thus increasing the height of the false vacuum but also that of the barrier and of the true vacuum by the same amount) lowers the action, making the potential more unstable; (2) Increasing gravitational effects (e.g. by increasing all mass scales towards $m_P$) lowers the action of dS to dS transitions but can have the opposite effect for dS to AdS transitions, depending on the potential shape and the relative balance of dS and AdS parts of the CdL range; (3)  As one would expect, wider barriers make the decay action
larger, making the vacuum more stable; (4) There is a limit to the effects in (2) and (3) above, given by the relations in (\ref{scalingrange}) which give lower and upper bounds on the action after the indicated rescalings.

redpajama_set_name: RedPajamaArXiv


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summary: "Base classes for creating simple 'nav-like' lists. Ideal for simple nav menus or for forming the basis of more complex navigation componenets."
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redpajama_set_name: RedPajamaGithub


The significant date of your choice can be engraved on this solid gold disk in our Maximalist Font which utilizes all of the given space with bold, deep engraving. 10mm in diameter, 18 kt gold. Sold as is to be worn directly on our earring fobs. If you would like a jumpring to be added so that it can be worn on a necklace or annex link, please indicate in the 2nd option for an additional fee.


It's been 8 years since I stopped driving a car. It was by choice that I let my car go. I got to the point where I was not willing to subscribe to the Petrol Industry. I do not regret this decision.
I had a turning point the summer of 2008. I had let my house go, a $5000 mortgage, not including property taxes and bills, and a $9000 overhead, which I proudly commuted to and from Los Angeles, to "own" as a Single Mom. I was keeping up with the "Joneses", the other set of parents, trying to outdo me, every step of my existence. It was nauseating at best, and I worked 90 hour weeks to pay for my life. I was exhausted, and had been diagnosed with cervical cancer and had collapsed on the job, which then got me "let go". Silly, isnt it? That we spend so much time and energy to pay to live.
So for that exit point, I decided to live, and rent a loft in Downtown LA, for a mere $2150 per month, a far cry from a $9000 overhead. The trouble was, I didnt have the wherewithal to maintain such a high-cost standard of living, even though I had clients and things were supposedly running smoothly. They weren't. When the Housing Market crashed, all the deposits in my account disappeared. I was at a decision-making point, where I had to decide if I was willing to continue the slave status, or step out. I was 100% Raw Vegan at the time, and had been discovering and realizing huge things about the world we live in. In 2007, I became a Raw Vegan, and cured myself from Cancer, without any assistance from the rapacious allopathic industry, aka medical doctors, hospitals, cancer treatments, chemo, etc. I took my life into my own hands and went on a 14-day master cleanse, and a silent retreat. During that cleanse, and silent retreat, I wrote 7 songs. When I came out of the silent retreat, I recorded a spoken word album. It was amazing how much energy I had and how creative I was. I was super inspired by life, and saw things as they were, right in front of me, at face value. I got excited about this new way of eating, and decided to become 100% Raw Vegan to assist with any health issues, and to experience a different way of living in the world. My veil was officially lifted during that time. Although I had been pretty psychic up until that point, and a "Spiritual Healer", this was a whole new level. The veil was never to go back down. I saw the demonic system set-up and saw how we were being manipulated on every level. At first, I tried to work with the system, on the private side, by becoming a secured party creditor, and then finally doing my freedom documents, that made me officially a sovereign. I created a ministry during that time as well. I decided that I answered to only one power, and that was the power of Source, otherwise predominantly referred to as "God". I dont see source as a Gender, however I do experience a more feminine caring presence in my world that supports and holds me infinitely in her arms.
From that point forward, when I left my house in 2008, I made a commitment to never turn back, to continue moving forward in the direction of freedom. I would never look back at how my life once was.
As I continued to stay present and listen, Spirit (Source) would reveal things to me, and I would follow the leads. In the summer of 2008, I was led to the Light School in Sedona, where I blasted open completely, saw myself as a Buddha with a Golden Orb around my head, just like you see in the pictures, and I activated my DNA on a whole new level. It took me 9 months to reground back into my physical body. Five years prior, in 2003, I had commanded the Activation of all of my DNA Strands, by myself, in a little meditation room, where I had been living. I had gotten the message it was time. Within a week of my activating my DNA the first time, I went under full-on attack. I ended up in an Ex-parte hearing with the Father of my child, and the beginning of a 5-year court battle ensued. Within that next few years, the DaVinci Code appeared, and it was clear that I had activated codes on a multi-dimensional level that were directly related to my soul, and applied to the collective. I was grateful for this, but it continued to be an incredibly hard road. Before this point, my life had been sorted with different levels of trauma, which I will speak to in my book, but I was able to manage, because I was under the illusion that I was in control of my life, from the perspective of "You create your reality". I had been attending Agape for several years at that point, a spiritual center in Los Angeles, where "The Secret" was first introduced, through the Science of Mind, and Religious Science teachings. I was studying to be an Ordained Minister of Religious Science. I didnt realize that there could potentially be outside influences, that would make my life seemingly hell. When the first activation happened, it started a harrowing journey that has continued to this day, and that I am extremely grateful for, every moment.
So, I ask you… What is it, about living in this society that really makes you want to show up every day for it? What false beliefs are still operating in your consciousness? I personally could not continue living in this societal illusion that was constructed for us, to render us slaves to a system we never chose for ourselves, as Organic Humans, and within which we are required to "pay to live". The cost of living in America is the most outrageous set-up I have seen to date. At this point, I have traveled to several countries within the past year, and have not witnessed the cost of living as high as in America, anywhere else. Cities gentrify and then it becomes impossible for most people to live comfortably, without being a slave to their job, families and the government. The idea that the government requires us to pay their bailouts should be clue enough that we are not in control. The thing is though, is that we are. We have the element of choice available to us in every moment. It never changes. It is always present, this element of choice. Its about looking at what our choices have been and how we choose to move forward.
We DO have a choice. It really is a choice that we have, to make decisions based on discernment and resonance. I have been living this way since 2008. So like I said, when 2009 came around, and the unraveling of my life had been in full swing for about 9 months, I decided to give up my last chain that was tying me to this society. I had been living out of my car during the summer of 2008, with my 6-year old son, having to find a place to sleep when he was with me, and sleeping in my car if I couldn't find someone to let me stay on their couch for the night. People would project their fears onto me, scream at me and say, "What are you going to do?!?!!" I would calmly respond and say, "Does it look like I am afraid?", and they would say, "Well no actually, you seem pretty calm". That's because I had one thing going for me, the action of Faith. Faith in my higher Source light energy that was providing for me, no matter what. Private Investigators had been hired to follow me around, by the other set of parents, so that was a thing as well. It was nervewracking at best, and I survived it. I am here, present and available, and I did not die. "What doesnt kill you makes you stronger", an old adage I had learned as a tike, has kept me alive, and I am so grateful for it, because I really know what the meaning of that is.
So today, with the world as unstable as it is, with no assurance that we are not going to be bombed any moment, it is important for us to remember what we are buying into exactly, and do our best to step out of the proverbial "matrix" and into the true natural reality of Gaia, who is fully supporting us in every moment. We can make choices for ourselves that support the collective, to support her, the earth, and all beings living on and in it. It is in fact a choice. We no longer need to be slaves to an outdated system that in fact, is designed to be broken.
My friend Ben Stewart, if you haven't heard of him, has put out some incredibly amazing documentaries about seeing the system for what it is. I highly recommend taking a look at his website, and watching some of his films to catch up on what's happening from his perspective. They are definitely thought provoking, even now, when I go back and watch them, I receive so much information that I forgot, or missed the first few times I saw them. They are that good. His website is http://talismanicidols.net.
At 20 years old, after going through what I now understand as a Spontaenous Kundalini Activation, and almost burning my mother's house down, my mother said to me, "Saralise, you need to write a book about your life." I said, "Umm…. Mom, I'm only 20 years old." She raised her voice, breaking through the walls of programming and penetrating straight to the core of my being, and said, "LISTEN TO ME! … YOU NEED….TO WRITE…. A BOOK…. ABOUT YOUR LIFE!!!!! AM I CLEAR?!?!" My response was nothing short of "CRYSTAL!"…….. Silence leaped through the ethers. She had made her point. Little did I understand the true repercussions of her words. To this day, she has come in very distinct moments to speak to me from the other side, she passed in 2004, and has only spoken a few times over the years, but that message rings in my mind, over and over, nagging me to complete this mountainous task. I dont think there will ever be an ending to my life. I just dont. It will continue, no matter what state I arrive at. There will always be The Saga of Saralise, and any other lifestream connected to this soul's embodiment will experience the nature of infinite life, never-ending, always eternal in its nature. But I digress.
Stepping out of the Matrix, filling your cup, can be an enormous decision and one that should be contemplated thoroughly. This does not mean in any way that you step out of life, or stop participating, quite the opposite. This means showing up, in the world, but not of it, in a whole new way, with a new set of eyes, that require you to claim your sovereign space within that said matrix. Not an easy task, by any means, because as you know, the construct of society requires us to "Pay to Live" and continues to encourage looking outside of ourselves for our sustenance. Do we need to go through such a tremendous experience? Do we need to completely leave all that we know? Only you can answer that for yourself. What I can say is that I did it. It doesnt mean that my life is easy, quite the opposite, but is it worth it?….. I wouldn't have it any other way.
Am I being responsible with my energy?
Do I really need all these items in my life, or are they just clutter taking up space in my life?
How am I spending my money, am I investing in my wellbeing, or just spending frivolously, and without care?
How many times a day am I eating? Am I consuming too much food/ too little? Am I eating properly?
Am I investing in relationships that nurture and sustain themselves, or am I being depleted by my connections with people?
Am I spending too much time NOT taking care of myself? And too much time focusing on others rather than my wellbeing?
Am I maintaining a peaceful state of being, or am I finding myself in a constant state of emotional crisis… consistently?
These are just a few examples, the list goes on and on. I came to a place where I was willing to ask myself any question that needed to be asked. I was willing to look at myself on all levels, and to see where I was denying life. There were a lot of places where I was really holding on and contracting, rather than just allowing my life to move. Life has a tendency to keep going, so you either get in the boat going downstream, or you miss the boat altogether, or you swim upstream, until you are so exhausted that life nudges you to pay attention. There is an easier, softer way, and that is to LISTEN TO LIFE.
No politician, or self-proclaimed "authority" will ever be able to take that away from us as organic humans. We are their battery. If we stop feeding the system that has been operating, it will eventually collapse, and all their sources of food with it. The powers that designed this system will no longer have a feeding mechanism and will be forced to either leave or change. It is already happening. My most recent escapades to Peru, are proof of that, which I will post in my Captain's Log.
The Majick Here is Palpable.
Here, in the Isle of Avalon, do not be surprised to run into a few Merlins, a few Ladies of the Lake, and Priestesses of the Goddess. It seems that almost everyone here is "speaking the language". There is not a moment that has gone by so far that I don't feel right at home. However, I have had moments lately of feeling like I am in a different timeline, and that I have jumped 80,000 light years into the center of the Milky Way galaxy, and am living on a parallel earth. Don't ask me why I think this, but I do. My scar on my right foot has disappeared into thin air, among other things. I haven't been able to get in touch with my son for almost a month, since I noticed the timeline shift. It is a bit worrisome. I am sure I will eventually get in touch with him, but for now, it has been disconcerting at best.
Since I have been in Europe and the UK, it's been full-on. I have gone through so many changes, and yet, I feel more at home here in this area of the world, than I ever did in the states. It is quite noticeable also the way I am not being as targeted as I was in the US. I have noticed a considerable difference. This could be from the timeline shift, or it could be that I am "out of reach" so to speak, and my focus has become more global rather than informing US citizens of the state of affairs. I will say however, I have continued to walk my talk, and have continued posting things that I feel are of massive importance to world affairs. For one, I can see, after being in another country, looking back, and hearing other people's opinions of the United States and its citizens, that we are one of the most disrespected countries in the world, for good reason. In most of Europe, if they heard my accent, they would immediately change their tune and begin to treat me differently, as if I was a "stupid American", it was quite hilarious. I decided to treat them with respect and energetically and verbally let them know that most Americans are not stupid whatsoever, but rather under mind control, and that I had a lot of compassion for what is happening there, even if I choose not to be there. At least for now.
Quite frankly, I do not see how I can return to the United States, in the state of affairs that it is in. I have absolutely no desire to be there, not one bit. My son lives there however, so if I need to come back, I will, but for now, no bueno. I am starting to feel like an ex-Pat, after only two months. Its like I got out of the "matrix" of the US and can feel that there is a whole world out here that sees things very differently than us, and for that I am deeply grateful. And Gaia is my home, after all.
As for Avalon, she has welcomed me home. It is a place I can rest, where I can just be me. I haven't had to try and figure out which language to speak, or how to communicate, how to find my way around, I can just be here, in this familiar place, and re-member all that I once knew, that I have taken for granted.
I have received so many downloads here, have activated the Albion codes within the land once more, and have decided that I could potentially live here. We shall see if that is what Spirit is calling forth. I have yet to visit a few key places on the map of the trajectory, and so we shall see how that plays itself out. There are many shifts and changes still to come.
For now, I have a few treats for you, some videos to watch for your activation and viewing pleasure!
If you feel to donate to the Global Pilgrimage, and can donate whatever you feel seems appropriate for the energy exchange of receiving these downloads, go ahead and donate via "family and friends" on paypal to the following email address: LegionOfLightGlobal@gmail.com.
Many Blissful Blessings your way!


Moving a house is one of the most difficult tasks in a person life. It can be more stressful when moving with kids and pets. Some people may move to a new house after a few years while others will only move once or twice in their lifetime. One may decide to move houses after they purchase a new home, get a job in a different location and change of environment. Moving to a new house can be stress-free and easy when you follow the right procedures. here are some tips from my brother, who works for a house removals Bristol company.
When moving, start packing early enough. You should give yourself a month to pack and arrange things in an orderly manner. Have a list of the things in your house and how you will start packing them. Packing early enough will help you not to leave important things behind when moving. When packing at the last minute, you will not remember where you kept some things.
You will be surprised at the amount of stuff that you have in the home that you do not use. When moving home, take the things that are important, and you can sell or give out the things that you do not need to your friends. Taking everything to your new house, especially large items like beds, will be expensive, and things might not fit into the moving van. Buying new large items isn't expensive. And having new stuff in your new home can't be a bad thing!
Moving homes can be tiresome. If you have kids in the house, you can ask them to help you out in packing the stuff in the right places. You should mark the moving boxes for them so that they can know which box to put certain things. You can also ask for assistance from your family members and friends who live near your current home.
If you have a busy schedule, you may not have the time to pack and plan on how you will move. Today, the number of moving companies has increased, and you should look for one. Moving companies will help you to pack, move you to your new house and unpack all your belongings. When selecting a moving company, ask them about the number of customers they have dealt with and their costs. Their prices are competitive hence no need to stress yourself over moving home.
After you have decided to move houses, ensure that you clear all your utility bills before moving. You should request for final bills, and after you have paid, you can request for change of address to your new location. If you were any contract, make sure that you cancel or wait for the contract to be over before moving.
When moving homes, ensure that you plan well before the moving day. The best time to move to a new house is during the weekend or when you are away from work. It will give you enough time to prepare and get used to your new environment.


text:
Home | Health
Which Late Night TV Host has Narcolepsy?
Commonly referred to as sleeping sickness, narcolepsy is a rare genetic disease affecting a person's sleep-wake cycles. According to Sleepfoundation.org, one in every 2,000 – 3,000 people suffers from this sleep disorder. Many people confuse it with insomnia because it is characterized by trouble sleeping but the condition primarily affects the brain's function regarding sleeping and waking cycles.
If you have narcolepsy, you may unwillingly fall asleep in the middle of an important task such as eating, driving, or even talking. The sleeping disorder is quite difficult to diagnose. You can have it for years and not know about it until the symptoms become more intense.
Jimmy Kimmel's Story with Narcolepsy
Through an essay written to Esquire in 2003, the late-night TV show Jimmy Kimmel claimed that he is mildly narcoleptic. Like many other patients suffering from the chronic disorder, Jimmy didn't know he was narcoleptic until he was 30. "I would sleep every afternoon for three hours and get tired for no apparent reason. I would sleep during meetings, driving, and watching TV," said Kimmel.
Jimmy claimed in the essay that he was once stopped by a traffic cop for dozing off on the wheel. His brain was as tired, which made him lack focus. By then, he would have had to down countless glasses of iced tea to get through the afternoon. He eventually went to the doctor when he noticed his body was constantly tired. The doctor did confirm his condition and gave him Provigil to help control it.
The essay entitled What It Feels like to Have Narcolepsy went on to reveal quite a lot about what the leading TV host had gone through with narcolepsy. Kimmel claimed to have fallen asleep once while being an MC on Ben Stein's show. Kimmel joked about the sleep disorder saying he'd rather have it than not have it. "When I hop on a plane to Vegas, I will sleep right before the plane takes off and only wake up after it has landed. I am ever that close to sleep."
Other Celebrities with Narcolepsy
Jimmy Kimmel isn't the only narcoleptic showbiz star. Here are a few other A-listers suffering from the disorder.
Justin Chambers, famous for playing the role of Alex Karev on the medical drama Grey's Anatomy, is also narcoleptic. He checked himself into UCLA's psychiatric ward in 2008 seeking help on managing the illness. He claims his mind would race while his body showed signs of extreme fatigue.
The vocally-endowed Maria Carey also admitted to sleeping a little longer than usual. "I have to get 15 hours of sleep before any performance to put my best foot forward. I also sleep in a room full of humidifiers to get rid of dry air in the room." She spoke to Interview Magazine in 2007. She eventually got help and is in a better position now.
The famous historical icon, British Prime Minister Winston Churchill wasn't apologetic about napping during the day. Churchill told the New York Times that napping during the day makes you accomplish more. "You get two days in one, or at the very least, one and a half."
Narcolepsy may be viewed as an unfortunate sleep disorder but Jimmy Kimmel, Winston Churchill, and other famous names have found a way to fit it into their lifestyles. Talk about making lemonade from lemons!
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