$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)\approx\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau}\Big{[}{\tilde{g}}_{1}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,e^{-m^{2}(1+R)\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\,|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|^{\nu}+{\tilde{g}}_{2}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,e^{m^{2}(R-1)\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\,|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|^{-\nu-1}\Big{]}~{},$
(B.33)

with

${\tilde{g}}_{1}=(2\zeta)^{\nu}g_{\Theta}~{},~{}~{}~{}{\tilde{g}}_{2}=\frac{\sqrt{\pi}\,g_{1}g_{2}}{\Gamma(-\nu)\zeta^{\nu+1}g_{\Theta}}~{},~{}~{}~{}g_{\Theta}=\left\\{\begin{array}[]{cc}g_{1}&{\rm
for~{}}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi>0\\\ g_{2}&{\rm
for~{}}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi<0\end{array}\right.~{}.$
(B.34)

For $m^{2}=3\xi/2$, the solution is

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau-\frac{m^{2}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\Big{[}g_{1}({\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\,e^{-\sqrt{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}-m^{2}}\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}+g_{2}({\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\,e^{\sqrt{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}-m^{2}}\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\Big{]}~{}.$
(B.35)

For $m^{2}<3\xi/2$, the parameter $R$ becomes imaginary and the solutions,
which are obtained by analytic continuation, exhibit an oscillatory behaviour.

####  B.2.1 ${\boldmath\delta}$-function Initial Condition

Consider absorbing boundary conditions set at infinite field values (with the
proviso explained at the beginning of sect. B.1.1)

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!\pm\infty,t)=0~{}~{}~{}~{}~{}{\rm(boundary~{}conditions)}$
(B.36)

and choose an initial condition with a $\delta$-function as in eq. (A.24). The
FPV solution for $m^{2}>3\xi/2$ is (see identities in sect. C.2)

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\frac{\zeta}{\sqrt{\pi}}\,\sum_{n=0}^{\infty}\frac{H_{n}(\zeta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0})H_{n}(\zeta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)}{2^{n}\,n!}\,e^{3H_{0}t+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}\tau-m^{2}(1+R)\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}+m^{2}(1-R)\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}^{2}}{2}}~{},$
(B.37)

where the eigenvalue spectrum is

${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}=m^{2}(1-R-2nR)~{},~{}~{}n\in\mathbb{N}_{0}~{}.$
(B.38)

The maximum eigenvalue is obtained for $n=0$ and it is such that

${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm
max}=m^{2}(1-R)~{}~{}\Rightarrow~{}~{}\frac{3\xi}{4}\leq{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}\leq\frac{3\xi}{2}~{}.$
(B.39)

The summation over spectral modes, which can be performed using eq. (C.38),
gives the same Gaussian function as in eq. (B.13) with

$\sigma^{2}=\frac{1-u^{2}}{m^{2}\,S}~{},~{}~{}~{}{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}=\frac{2R\,u\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}}{S}~{},~{}~{}~{}\chi=\ln\sqrt{\frac{2R}{S}}+\frac{R-1}{R}\ln
u+\frac{3\xi(1-u^{2}){\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}^{2}}{4S}~{},$
(B.40) $S=R+1+(R-1)u^{2}~{},~{}~{}~{}u=\exp(\\!-2m^{2}R\tau)~{}.$ (B.41)

As expected, the asymptotic behaviour at large times satisfies eq. (A.15).

In all $\xi$-gauges, the peak of the distribution
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ falls to the
bottom of the parabolic potential exponentially fast. The width starts by
growing as $\sigma\propto t^{1/2}$, but eventually freezes at the value

$\sigma^{2}_{\infty}=\frac{1}{m^{2}(R+1)}~{}.$ (B.42)

For $m^{2}<3\xi/2$, we are entering the quantum regime and the distribution
eventually climbs the potential. Therefore the approximation of using eq.
(B.36) is valid only up to a finite time cutoff and eventually breaks down.
The asymptotic behaviour can be computed by evaluating ${\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}$ from the boundary
condition
$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi=\pm{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E})=0$,
following the same procedure as in the case of the linear potential. Numerical
calculations or an analysis based on the discriminant condition discussed in
sect. 2.6 show that the asymptotic distribution has a symmetric double-peak
structure located in the vicinity of the endpoints.

###  B.3 Quadratic Potential (Negative Curvature)

Consider a potential quadratic in the field with negative curvature

$V(\phi)=V_{0}-\frac{M^{2}\phi^{2}}{2}~{},~{}~{}~{}v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=-\frac{m^{2}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}~{},$
(B.43)

with $M^{2}>0$ and $m^{2}$ is defined as in eq. (B.28). The most general
solution of the perturbative FPV is

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau+m^{2}(1-{\hat{R}})\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\Big{[}g_{1}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,H_{\hat{\nu}}({\hat{\zeta}}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)+g_{2}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,{H}_{{\hat{\nu}}}(-{\hat{\zeta}}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\Big{]}~{},$
(B.44)
${\hat{\zeta}}=\sqrt{{m^{2}{\hat{R}}}}~{},~{}~{}~{}{\hat{R}}=\sqrt{1+\frac{3\xi}{2m^{2}}}~{},~{}~{}~{}{\hat{\nu}}=\frac{-m^{2}(1+{\hat{R}})-{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}{2m^{2}{\hat{R}}}~{}.$ (B.45)

Sufficiently away from the minimum
($|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|\gg
1/{\hat{\zeta}}$), the solution can be approximated as

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)\approx\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau}\Big{[}{\tilde{g}}_{1}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,e^{m^{2}(1-{\hat{R}})\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\,|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|^{\hat{\nu}}+{\tilde{g}}_{2}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,e^{m^{2}(1+{\hat{R}})\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\,|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|^{-{\hat{\nu}}-1}\Big{]}~{},$
(B.46)

where ${\tilde{g}}_{1,2}({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})$ are given by eq. (B.34) after the replacement
$\nu,\zeta\to{\hat{\nu}},{\hat{\zeta}}$.

####  B.3.1 ${\boldmath\delta}$-function Initial Condition

Consider absorbing boundary conditions set at infinite field values (with the
proviso explained at the beginning of sect. B.1.1)

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!\pm\infty,t)=0~{}~{}~{}~{}~{}{\rm(boundary~{}conditions)}$
(B.47)

and choose an initial condition with a $\delta$-function as in eq. (A.24). The
FPV solution is

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\frac{{\hat{\zeta}}}{\sqrt{\pi}}\,\sum_{n=0}^{\infty}\frac{H_{n}({\hat{\zeta}}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0})H_{n}({\hat{\zeta}}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)}{2^{n}\,n!}\,e^{3H_{0}t+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}\tau+m^{2}(1-{\hat{R}})\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}-m^{2}(1+{\hat{R}})\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}^{2}}{2}}~{}.$
(B.48)

The eigenvalue spectrum is

${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}=-m^{2}({\hat{R}}+1+2n{\hat{R}})~{},~{}~{}n\in\mathbb{N}_{0}$
(B.49)

with a maximum eigenvalue

${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm
max}=-m^{2}({\hat{R}}+1)\approx\left\\{\begin{array}[]{lc}-2m^{2}&({\rm
C~{}regime~{}or}~{}\xi=0)\vspace{0.1cm}\\\ -\sqrt{\frac{3\xi m^{2}}{2}}&({\rm
QV~{}regime})\end{array}\right.~{}.$ (B.50)

The summation over spectral modes gives the same Gaussian function as in eq.
(B.13) with

$\sigma^{2}=\frac{{\hat{u}}^{2}-1}{m^{2}\,{\hat{S}}}~{},~{}~{}~{}{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}=\frac{2{\hat{R}}\,{\hat{u}}\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}}{{\hat{S}}}~{},~{}~{}~{}\chi=\ln\sqrt{\frac{2{\hat{R}}}{{\hat{S}}}}-\frac{{\hat{R}}+1}{{\hat{R}}}\ln{\hat{u}}+\frac{3\xi({\hat{u}^{2}}-1){\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}^{2}}{4{\hat{S}}}~{},$
(B.51)
${\hat{S}}={\hat{R}}+1+({\hat{R}}-1){\hat{u}}^{2}~{},~{}~{}~{}{\hat{u}}=\exp(2m^{2}{\hat{R}}\tau)~{}.$
(B.52)

As expected, the asymptotic behaviour at large times satisfies eq. (A.15).

In $e$-folding gauge ($\xi=0$), the peak of the distribution
(${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$) falls down the
hillside slope exponentially fast, unless it starts exactly at the top of the
hump (${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}=0$). The
width starts increasing as $\sigma\propto t^{1/2}$, but then grows
exponentially with time.

In all other gauges ($\xi\neq 0$),
$\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ starts by rolling
down the hill but then, once $\sigma$ reaches the value $(2/3\xi)^{1/2}$,
which is of Planckian size in physical units, it reverses its motion climbing
back the potential. The ‘idling time’ necessary for
$\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ to get back to the
initial point is

$\frac{t_{\rm
idle}}{t_{S}}=\frac{1}{2m^{2}{\hat{R}}}\ln\Big{(}\frac{{\hat{R}}+1}{{\hat{R}}-1}\Big{)}\approx\left\\{\begin{array}[]{c}\frac{1}{2m^{2}}\ln\frac{8{m}^{2}}{3\xi}\vspace{0.1cm}\\\
\frac{2}{3\xi}\end{array}\right.\begin{array}[]{c}({\rm
C~{}regime})\vspace{0.2cm}\\\ ({\rm QV~{}regime})\end{array}~{}.$ (B.53)

As mentioned before, $t_{\rm idle}$ is only a feature of the chosen initial
condition.

At later times, $\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$
continues to climb the hill exponentially fast, asymptotically reaching its
top. The time $\Delta t$ for
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ to travel from
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}$ to a field value
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}/\delta$ (with
$\delta\gg 1$ and
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}<0$), measured from
the end of the idling phase, is

$\frac{\Delta
t}{t_{S}}=\frac{\ln\delta}{2m^{2}{\hat{R}}}\approx\left\\{\begin{array}[]{c}\frac{\ln\delta}{2m^{2}}\vspace{0.1cm}\\\
\frac{\ln\delta}{\sqrt{6\xi m^{2}}}\end{array}\right.\begin{array}[]{c}({\rm
C~{}regime})\vspace{0.2cm}\\\ ({\rm QV~{}regime})\end{array}~{}.$ (B.54)

This shows that the typical time scale for the peak to explore the field range
is smaller than $t_{S}$ in the C regime and larger than $t_{S}$ in the QV
regime.

A cautionary note is warranted. It would appear from this analysis that even
in the C regime the field wishes to ultimately climb the potential. However,
this outcome is the result of the symmetry of the potential and boundary
conditions. The final stationary solution must be symmetric and there is no
special point on either slope, since they are infinitely long, hence it can
only rest atop the potential. This is not, however, due to a volume
contribution overcoming the classical rolling, as it is in the Q regimes.
Thus, in the C regime, were the potential or the boundary conditions not
exactly symmetric, the solution would not come to rest at the top. As a
result, it seems that the ‘climbing’ behaviour with a sub-$t_{S}$ time scale
is possible only as a result of symmetry, rather than dynamics.

The width of the distribution starts growing as $\sigma\propto t^{1/2}$, but
then freezes at the asymptotic value

$\sigma^{2}_{\infty}=\frac{1}{{m}^{2}({\hat{R}}-1)}\approx\left\\{\begin{array}[]{c}\frac{4}{3\xi}\vspace{0.2cm}\\\
\sqrt{\frac{2}{3\xi m^{2}}}\end{array}\right.\begin{array}[]{c}({\rm
C~{}regime})\vspace{0.2cm}\\\ ({\rm QV~{}regime})\end{array}~{}.$ (B.55)

The asymptotic width in the QV regime is always larger than in the C regime,
where it is independent of $m^{2}$.

###  B.4 Linear-Linear Potential

To illustrate the role of junction conditions, in this section we consider FPV
solutions for systems with different branches of linear potentials.

####  B.4.1 Pyramid Potential

Consider two linear potentials with different slopes matched together at a
critical point, where they form a cusp. With an appropriate choice of
coordinates we choose the critical point at the origin and, for simplicity, we
take equal and opposite slopes on the two sides of the critical point.
Therefore, the potential is

$v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=-\gamma|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|~{}.$
(B.56)

The junction condition in eq. (A.36) requires that, at the critical point,

$\frac{\Delta
p^{\prime}}{p(0)}=4\gamma~{}~{}~{}~{}~{}{\rm(junction~{}condition)}.$ (B.57)

We choose absorbing boundary conditions for a bounded field range

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!\pm{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E},t)=0~{}~{}~{}~{}~{}{\rm(boundary~{}conditions)},$
(B.58)

where ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\ll
S_{\\!{{\rm dS}}}/\gamma$ so that the perturbative approach is valid for all
field values.

We start by considering the case
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\to\infty$, which
is a valid approximation as long as the distribution is localised away from
the endpoints. Using the results obtained in sect. B.1 for the linear
potential, we find that the general FPV solution satisfying continuity and
boundary conditions at infinity is

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{n=1}^{\infty}\,e^{3H_{0}t+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}\tau+\gamma|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|}\,g_{n}\,{\rm
Ai}(Q_{n}+c|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|)~{},~{}~{}~{}Q_{n}\equiv\frac{\gamma^{2}+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}}{c^{2}}~{},~{}~{}~{}c\equiv(3\xi\gamma)^{\frac{1}{3}}~{},$
(B.59)

where $g_{n}$ are arbitrary constants. The eigenvalues ${\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{n}$ are determined by the
junction condition in eq. (B.57) and are given by the solutions of the
equation

$\frac{{\rm Ai}^{\,\prime}(Q_{n})}{{\rm Ai}(Q_{n})}=\frac{\gamma}{c}~{},$
(B.60)

with the maximum eigenvalue ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{\rm max}$ corresponding to $n=1$.

The distribution $P$ at asymptotically large times has two peaks, one at each
side of the critical point, located at the field values
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm}$ given by
the solutions of the equation

$\frac{{\rm
Ai}^{\,\prime}(Q_{1}+c|{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm}|)}{{\rm
Ai}(Q_{1}+c|{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm})|}=-\frac{\gamma}{c}~{}.$
(B.61)

The width of each peak, defined as
$\sigma^{2}=-P({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm})/P^{\prime\prime}({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm})$,
is

$\sigma^{2}=\frac{1}{\gamma^{2}-c^{2}(Q_{1}\pm
c{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm})}~{}.$
(B.62)

Equations (B.60)–(B.62) can be solved analytically with power expansions, in
the appropriate regimes depending on the size of $\gamma$.

When ${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm}$ are
in the vicinity of the endpoints, the approximation of taking
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\to\infty$ is no
longer valid and we need to extend eq. (B.59). Taking for simplicity
$c\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\\!\gg\\!1$,
the stationary solution corresponding to the maximum eigenvalue is

$p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm
max})=e^{\gamma|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|}\left[2\,{\rm
Ai}(x)-e^{-\frac{4}{3}x_{\\!E}^{3/2}}\,{\rm Bi}(x)\right]~{},~{}~{}~{}{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}=ac^{2}-\gamma^{2}~{},$
(B.63)
$x=a+c|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|~{},~{}~{}~{}x_{\\!E}=a+c{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}~{},$
(B.64)

where the parameter $a$, up to negligible corrections proportional to
$\exp(-\frac{4}{3}x_{\\!E}^{3/2})$, is the largest solution to the equation

$\frac{{\rm Ai}^{\,\prime}(a)}{{\rm
Ai}(a)}=\frac{\gamma}{c}~{}~{}~{}\Rightarrow~{}~{}~{}a=\left\\{\begin{array}[]{cc}a_{1}&({\rm
for~{}}\gamma\gg\sqrt{\xi})\\\ a_{1}^{\prime}&({\rm
for~{}}\gamma\ll\sqrt{\xi})\end{array}\right.~{}.$ (B.65)

Therefore, $a$ always lies in the range $-2.34\\!<\\!a\\!<\\!-1.02$.

#### Q2V regime

For $\gamma\ll\sqrt{\xi}$, the eigenvalues are

${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}=a_{n}^{\prime}(3\xi\gamma)^{\frac{2}{3}}+\frac{(3\xi)^{\frac{1}{3}}\gamma^{\frac{4}{3}}}{a_{n}^{\prime}}+{\mathcal{O}}(\gamma^{2})~{},$
(B.66)

with ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}$
corresponding to $n=1$. Working at leading order in the $\gamma/\\!\sqrt{\xi}$
expansion, the location and width of the two peaks of the asymptotic
distribution are

${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm}=\mp\frac{2\gamma^{\frac{1}{3}}}{a_{1}^{\prime}(3\xi)^{\frac{2}{3}}}~{},~{}~{}~{}~{}\sigma=\frac{1}{\sqrt{-a_{1}^{\prime}}\,(3\xi\gamma)^{\frac{1}{3}}}~{}.$
(B.67)

In the Q2V regime, the two peaks are much broader than their separation, since
$({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{+}-{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{-})/\sigma={\mathcal{O}}(\gamma^{2/3}/\xi^{1/3})\ll
1$, and therefore in practice they form a single peak centred at the critical
point. The shallower the potential, the broader the distribution becomes.

#### QV regime

For $\gamma\gg\sqrt{\xi}$, the eigenvalues are

${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}=-\gamma^{2}+a_{n}(3\xi\gamma)^{\frac{2}{3}}+3\xi+{\mathcal{O}}(\xi^{\frac{4}{3}}\gamma^{-\frac{2}{3}})~{},$
(B.68)

with ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}$
corresponding to $n=1$. Working at leading order in the $\sqrt{\xi}/\gamma$
expansion, the location and width of the two peaks of the asymptotic
distribution are

${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm}=\pm\frac{\gamma}{3\xi}~{},~{}~{}~{}~{}\sigma=\sqrt{\frac{2}{3\xi}}~{}.$
(B.69)

For the peaks to lie within the perturbative region defined in eq. (B.2) we
must require $\gamma\ll\sqrt{\xi S_{\\!{{\rm dS}}}}$ (which corresponds to
$\Gamma\ll\sqrt{\xi}H_{0}^{3}$). As $\gamma$ is decreased and the potential
gets shallower, the location of the peaks becomes closer to the critical
point. However, the two peaks are always well separated in units of width
since
$({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{+}-{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{-})/\sigma={\mathcal{O}}(\gamma/\\!\sqrt{\xi})\gg
1$.

#### C regime

As $\gamma$ grows larger, the peaks move further away from the critical point
and eventually the approximation
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\to\infty$ is
inconsistent with perturbativity, but the properties of the asymptotic
distribution can be derived from eq. (B.63). For
$\gamma/\\!\sqrt{\xi}\\!\gg\\!\sqrt{\xi}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\\!\gg\\!1$,
the asymptotic locations and width of the two peaks are

${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}_{\pm}=\pm\Big{(}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-\frac{1}{\gamma}\Big{)}~{},~{}~{}~{}~{}\sigma=\frac{1}{\gamma}~{},$
(B.70)

and the maximum eigenvalue is ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{\rm max}=-\gamma^{2}+a_{1}(3\xi\gamma)^{2/3}$.
The peaks settle as low as possible in the potential, compatibly with boundary
conditions.

Notice the difference between the outcomes in the C regime for the pyramid
potential and the negative-curvature quadratic potential studied in sect. B.3.
For the pyramid, the peak of the distribution asymptotically reaches the
maximum of the potential only in the Q2V regime, while it settles somewhere
along the slope but close to the top in the QV regime and slides down in the C
regime. Instead, for the quadratic potential with negative curvature, the peak
of the distribution asymptotically reaches the maximum in all regimes. The
difference is related to the fact that in the quadratic potential with
negative curvature the peak is forced to the top even in the C regime by the
symmetry of the boundary conditions, which are placed at infinity, and the
form of the potential whose vanishing gradient at the top requires, through
symmetry, a vanishing gradient for the solution also. This does not occur for
the pyramid, despite the symmetry, since the junction condition allows for a
non-vanishing gradient at the top of the slope, consistent with the symmetry
of the system.

####  B.4.2 Cliff Potential

Consider two potentials juxtaposed on the two sides of a critical point
$\phi_{c}$, chosen to be at the origin

$V(\phi)=\Theta(-\phi)\,V_{-}(\phi)+\Theta(\phi)\,V_{+}(\phi)~{}.$ (B.71)

Assume that $V_{+}$ is in the classical regime, while $V_{-}$ is in a quantum
regime, and impose absorbing or reflecting boundary conditions at some values
of the field well separated from the critical point. In a neighbourhood of the
critical point, we can always approximate $V_{\pm}$ as two linear potentials
with corresponding couplings

$\gamma_{\pm}=\frac{V^{\prime}_{\pm}(0)S_{\\!{{\rm
dS}}}}{6M_{P}H_{0}^{2}}~{}.$ (B.72)

Under these assumptions, the general study of the FPV in the classical regime
presented in sect. 2.9 shows that the solution proportional to ${\rm Bi}(x)$
is singled out just above the critical point, while the solution proportional
to ${\rm Ai}(x)$ dominates below. Therefore, in the vicinity of the critical
point, the FPV solution that satisfies boundary conditions and continuity
takes the form

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}\tau}\,g({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,\bigg{[}\Theta(-{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\,e^{-\gamma_{-}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,\frac{{\rm
Ai}(x_{-})}{{\rm
Ai}(Q_{-})}+\Theta({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\,e^{-\gamma_{+}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,\frac{{\rm
Bi}(x_{+})}{{\rm Bi}(Q_{+})}\bigg{]}~{},$ (B.73)
$x_{\pm}=Q_{\pm}-c_{\pm}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi~{},~{}~{}~{}Q_{\pm}\equiv\frac{\gamma_{\pm}^{2}+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}{c_{\pm}^{2}}~{},~{}~{}~{}c_{\pm}\equiv(3\xi\gamma_{\pm})^{\frac{1}{3}}~{},$
(B.74)

where $g({\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})$ are arbitrary
constants. The eigenvalues ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}$ are determined by the junction condition in eq.
(A.36), $\Delta p^{\prime}/p(0)=2(\gamma_{-}-\gamma_{+})$, which corresponds
to

$c_{+}\,\frac{{\rm Bi}^{\prime}(Q_{+})}{{\rm Bi}(Q_{+})}-c_{-}\,\frac{{\rm
Ai}^{\prime}(Q_{-})}{{\rm Ai}(Q_{-})}=\gamma_{+}-\gamma_{-}~{}.$ (B.75)

Since the system is in the C regime for positive
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$, the volume-weighted
distribution will preferentially populate low-potential field values and
therefore will be pushed towards the critical point. If the maximum eigenvalue
is positive, the distribution for negative
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ will favour high-
potential field values. As a result, the distribution will accumulate at the
transition between the two regimes to form a peak.

The condition to have a positive eigenvalue and keep the C regime above the
critical point up to a value
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{E}$ is

$\sqrt{\xi}\ll\xi{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{E}\ll\gamma_{+}\ll(\xi^{2}/\gamma_{-})^{\frac{1}{3}}~{}.$
(B.76)

Under this condition, we can expand eq. (B.75). Since $Q_{+}$ and $Q_{-}$ are
both large and positive when evaluated at the maximum eigenvalue, we can use
the asymptotic expansions for the Airy functions finding that, at leading
order, the maximum eigenvalue is

${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm
max}=\left(\frac{3\xi}{4\gamma_{+}}\right)^{2}-2\gamma_{+}\gamma_{-}~{}.$
(B.77)

As expected, the asymptotic distribution is monotonically increasing for
negative ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ and
decreasing for positive
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ with a cusp at the
critical point. The peak at
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}=0$ is asymmetric
and the spread on either side of the critical point is characterised by the
quantities

$\sigma_{-}\equiv\frac{P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!0_{-},t_{\infty})}{P^{\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!0_{-},t_{\infty})}=\frac{4\gamma_{+}}{3\xi}~{},~{}~{}~{}\sigma_{+}\equiv-\frac{P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!0_{+},t_{\infty})}{P^{\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!0_{+},t_{\infty})}=\frac{1}{2\gamma_{+}}~{}.$
(B.78)

The distribution is well localised in the negative
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ region because the
spread is much smaller than the full perturbative field range
($\sigma_{-}/\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\approx\gamma_{+}\gamma_{-}/(\xi
S_{\\!{{\rm dS}}})\ll 1$). It is also well localised in the positive
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ region because
$\sigma_{+}/{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\approx
1/(\gamma_{+}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{E})\ll 1$.
The spread of the distribution in the negative
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ region largely exceeds
the field interval in the positive
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ region where we can
trust the perturbative classical regime of the potential. However, this is
just an artefact of our linear approximation of the potential. By taking a
potential $V_{+}$ that satisfies the classical behaviour beyond the
perturbative domain, one can extend the validity of the model up to
arbitrarily large field values.

Because of the strong asymmetry in the distribution, the average field value
is not at the origin but at
$\langle{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\rangle\approx\sigma_{+}\\!-\\!\sigma_{-}$.
Since $\sigma_{-}\gg\sigma_{+}$, the average field value occurs for large and
negative ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$, although the
most probable value is
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi=0$. The probability of
finding the field in the negative region is also much larger than in the
positive region, since

$\frac{\int_{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi<0}d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,P}{\int_{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi>0}d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,P}\approx\frac{8\gamma_{+}^{2}}{3\xi}\gg
1~{}.$ (B.79)

In the cliff potential, the origin of the localisation is directly linked to
the junction condition at the critical point $\phi_{c}$. Indeed, if the
distribution to the left of $\phi_{c}$, called $P_{-}$, has a gradient at the
critical point satisfying

$0<\frac{P_{-}^{\prime}(\phi_{c},t)}{P_{-}(\phi_{c},t)}<\frac{24\pi^{2}M_{P}^{4}\,\Delta
V^{\prime}}{\hbar\,V^{2}(\phi_{c})}~{},$ (B.80)

then the junction condition implies that the solution $P_{+}$ on the right of
$\phi_{c}$ must take a value

$\frac{{P_{+}}^{\prime}(\phi_{c},t)}{P_{+}(\phi_{c},t)}<0~{}.$ (B.81)

Since the gradient of $P$ changes sign at $\phi_{c}$, a local peak must exist
at the critical point, at any time during which eq. (B.80) is satisfied. In
particular, if eq. (B.80) is satisfied at asymptotically large times, the peak
will persist indefinitely.

A remarkable feature of the cliff potential is that the localisation of the
field occurs at a point which is neither a minimum nor a maximum of the
potential. The localisation is entirely driven by the junction conditions
imposed by a discontinuity of the potential gradient. However, the cliff
potential has limited interest for critical phenomena, since it requires a
positive discontinuity $\Delta V^{\prime}>0$, while a first-order phase
transition usually leads to a negative gradient discontinuity.

###  B.5 EFT Notation

In the body of this paper we have used a compact parametrisation of the
potential which is particularly suitable for an EFT treatment. The potential
is expressed in terms of an overall coupling $g_{\epsilon}$ and the field
range $f$, according to

$V(\phi)=V_{0}+g_{\epsilon}^{2}f^{4}\omega(\varphi)~{},~{}~{}~{}~{}\varphi\equiv\frac{\phi}{f}~{},$
(B.82)

where $\omega(\varphi)$ is a generic function such that $\omega(0)=0$ and
$|\omega(\varphi)|\leq 1$. The perturbative domain corresponds to
$g_{\epsilon}^{2}f^{4}\ll V_{0}$.

The parameters that characterise the behaviour of the FPV solutions are

$\alpha\equiv\frac{3\hbar
H_{0}^{4}}{4\pi^{2}g_{\epsilon}^{2}f^{4}}~{},~{}~{}~{}~{}\beta\equiv\frac{3\xi
f^{2}}{2M_{P}^{2}}~{},~{}~{}~{}~{}t_{R}\equiv\frac{3H_{0}}{g_{\epsilon}^{2}f^{2}}=\frac{\alpha\beta
S_{\\!{{\rm dS}}}}{3\xi H_{0}}~{}.$ (B.83)

A further advantage of this parametrisation is that the gauge variable $\xi$
is completely reabsorbed into the definition of $\beta$.

The parameters $\alpha$ (which measures the strength of the diffusion term in
units of the drift term) and $\beta$ (which measures the strength of the
volume term in units of the drift term) characterise the following regimes,
valid for $\beta>1$.

$\begin{array}[]{lc}{\mbox{\bf Classical (C) :}}&~{}~{}~{}\frac{1}{S_{\\!{{\rm
dS}}}}\ll\alpha\ll\frac{1}{\beta}\\\\[8.0pt] {\mbox{\bf Quantum+Volume (QV)
:}}&~{}~{}~{}\frac{1}{\beta}\ll\alpha\ll\frac{1}{\sqrt{\beta}}\\\\[8.0pt]
{\mbox{\bf Quantum{\boldmath${}^{2}$}+Volume (Q{\boldmath${}^{2}$}V)
:}}&~{}~{}~{}\frac{1}{\sqrt{\beta}}\ll\alpha\ll\beta\\\\[8.0pt] {\mbox{\bf
Quantum (Q) :}}&~{}~{}~{}\alpha\gg\beta\end{array}$

For convenience, we summarise in table 1 how to convert the notations used in
this appendix to those of the EFT approach.

|  | Appendix | EFT
---|---|---|---
Scalar field | (dimensionful) | $\phi={{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,M_{P}$ | $\phi=\varphi\,f$
| (dimensionless) | ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ | $\sqrt{\frac{2\beta}{3\xi}}~{}\varphi$
Potential | (dimensionful) | $V(\phi)=V_{0}\left[1+\frac{2v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)}{S_{\\!{{\rm dS}}}}\right]$ | $V(\phi)=V_{0}+g_{\epsilon}^{2}f^{4}\omega(\varphi)$
| (dimensionless) | $v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)$ | $\frac{\omega(\varphi)}{\alpha}$
Eigenvalues | (dimensionful) | ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}/t_{S}$ | $\lambda/t_{R}$
| (dimensionless) | ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}$ | $\frac{3\xi}{\alpha\beta}\,\lambda$
Linear | Appendix: $v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=\gamma{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ | $\gamma$ | $\sqrt{\frac{3\xi}{2\alpha^{2}\beta}}$
potential | EFT: $\omega(\varphi)=\varphi$ |  |
Quadratic | Appendix: $v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=\frac{m^{2}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}$ | $m^{2}$ | $\frac{3\xi}{2\alpha\beta}$
potential | EFT: $\omega(\varphi)=\frac{\varphi^{2}}{2}$ |  |
Table 1: The conversion between the notations used in this appendix and those
of the EFT approach. Quantities in the same line of the table are equal, but
expressed in the corresponding convention.

##  Appendix C Useful Identities of Some Special Functions

###  C.1 Airy’s Equation

Consider the equation

${p}^{\prime\prime}+2A{p}^{\prime}+(B+C{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi){p}=0~{},$
(C.1)

where $A,B,C$ are generic constants. First, take the case $C\neq 0$. With the
definitions

${p}=e^{-A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,\Phi~{},~{}~{}~{}x=\frac{k}{c^{2}}-c\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi~{},~{}~{}~{}k=A^{2}-B~{},~{}~{}~{}c=C^{1/3}~{},$
(C.2)

eq. (C.1) turns into

$\frac{d^{2}\Phi}{dx^{2}}-x\,\Phi=0~{},$ (C.3)

which is known as Airy’s differential equation [137].

Two linearly-independent solutions of eq. (C.3) are given by the Airy
functions ${\rm Ai}(x)$ and ${\rm Bi}(x)$. Their power expansions for small
argument are

${\rm Ai}(x)=f(x)-\frac{g(x)}{\sqrt{3}}~{},~{}~{}~{}{\rm
Bi}(x)=\sqrt{3}\,f(x)+g(x)~{},$ (C.4)
$f(x)=a_{0}\left[1+\frac{x^{3}}{6}+{\mathcal{O}}(x^{6})\right]~{},~{}~{}~{}g(x)=\frac{1}{2\pi
a_{0}}\left[x+\frac{x^{4}}{12}+{\mathcal{O}}(x^{7})\right]~{},$ (C.5)

with $a_{0}\equiv{\rm Ai}(0)=3^{-2/3}/\Gamma(2/3)=0.355$.

Their asymptotic behaviours are

${\rm Ai}(x)~{}\stackrel{{\scriptstyle
x\to+\infty}}{{\longrightarrow}}~{}\frac{\exp\big{(}-\frac{2}{3}\,x^{\frac{3}{2}}\big{)}}{2\sqrt{\pi}\,x^{\frac{1}{4}}}\Big{[}1-\frac{5x^{-\frac{3}{2}}}{48}+{\mathcal{O}}(x^{-3})\Big{]}~{},$
(C.6) ${\rm Bi}(x)~{}\stackrel{{\scriptstyle
x\to+\infty}}{{\longrightarrow}}~{}\frac{\exp\big{(}\frac{2}{3}\,x^{\frac{3}{2}}\big{)}}{\sqrt{\pi}\,x^{\frac{1}{4}}}\Big{[}1+\frac{5x^{-\frac{3}{2}}}{48}+{\mathcal{O}}(x^{-3})\Big{]}~{},$
(C.7) ${\rm Ai}(x)~{}\stackrel{{\scriptstyle
x\to-\infty}}{{\longrightarrow}}~{}\frac{1}{\sqrt{\pi}\,|x|^{\frac{1}{4}}}\Big{[}\cos\Big{(}\frac{2}{3}\,|x|^{\frac{3}{2}}-\frac{\pi}{4}\Big{)}+\frac{5}{48|x|^{\frac{3}{2}}}\sin\Big{(}\frac{2}{3}\,|x|^{\frac{3}{2}}-\frac{\pi}{4}\Big{)}\Big{]}$
(C.8) ${\rm Bi}(x)~{}\stackrel{{\scriptstyle
x\to-\infty}}{{\longrightarrow}}~{}\frac{1}{\sqrt{\pi}\,|x|^{\frac{1}{4}}}\Big{[}-\sin\Big{(}\frac{2}{3}\,|x|^{\frac{3}{2}}-\frac{\pi}{4}\Big{)}+\frac{5}{48|x|^{\frac{3}{2}}}\cos\Big{(}\frac{2}{3}\,|x|^{\frac{3}{2}}-\frac{\pi}{4}\Big{)}\Big{]}$
(C.9) $\frac{{\rm Ai}^{\,\prime}(x)}{{\rm Ai}(x)}~{}\stackrel{{\scriptstyle
x\to+\infty}}{{\longrightarrow}}~{}-\sqrt{x}-\frac{1}{4x}+\frac{5x^{-\frac{5}{2}}}{32}+{\mathcal{O}}(x^{-4})~{},$
(C.10) $\frac{{\rm Bi}^{\,\prime}(x)}{{\rm Bi}(x)}~{}\stackrel{{\scriptstyle
x\to+\infty}}{{\longrightarrow}}~{}\sqrt{x}-\frac{1}{4x}-\frac{5x^{-\frac{5}{2}}}{32}+{\mathcal{O}}(x^{-4})$
(C.11)

and their Wronskian is

$W[{\rm Ai},{\rm Bi}]={\rm Ai}(x)\,{\rm Bi}^{\,\prime}(x)-{\rm
Ai}^{\,\prime}(x)\,{\rm Bi}(x)=\frac{1}{\pi}~{}.$ (C.12)

The Airy functions and their derivatives can be expressed in terms of the
modified Bessel functions

${\rm
Ai}(x)=\frac{\sqrt{x}}{3}\left[I_{-\frac{1}{3}}(t)-I_{\frac{1}{3}}(t)\right]~{},~{}~{}~{}{\rm
Bi}(x)=\sqrt{\frac{x}{3}}\left[I_{-\frac{1}{3}}(t)+I_{\frac{1}{3}}(t)\right]~{},$
(C.13) ${\rm
Ai}^{\,\prime}(x)=-\frac{x}{3}\left[I_{-\frac{2}{3}}(t)-I_{\frac{2}{3}}(t)\right]~{},~{}~{}~{}{\rm
Bi}^{\,\prime}(x)=\frac{x}{\sqrt{3}}\left[I_{-\frac{2}{3}}(t)+I_{\frac{2}{3}}(t)\right]~{},$
(C.14)

where $t=(2/3)x^{3/2}$.

The zeroes of the Airy functions and their derivatives

${\rm Ai}(a_{n})=0~{},~{}~{}~{}{\rm Bi}(b_{n})=0~{},~{}~{}~{}{\rm
Ai}^{\,\prime}(a_{n}^{\prime})=0~{},~{}~{}~{}{\rm
Bi}^{\,\prime}(b_{n}^{\prime})=0$ (C.15)

are negative numbers approximately given by

$n$ | 1 | 2 | 3 | …
---|---|---|---|---
$a_{n}$ | $-2.34$ | $-4.09$ | $-5.52$ | …
$b_{n}$ | $-1.17$ | $-3.27$ | $-4.83$ | …
$a_{n}^{\prime}$ | $-1.02$ | $-3.25$ | $-4.82$ | …
$b_{n}^{\prime}$ | $-2.29$ | $-4.07$ | $-5.51$ | …

The Airy functions of the first kind provide an orthonormal basis

$\delta(x-y)=\int_{-\infty}^{+\infty}dk\,{\rm Ai}(k+x)\,{\rm Ai}(k+y)~{}.$
(C.16)

They also satisfy the identity

$\int_{-\infty}^{+\infty}dk\,e^{kt}\,{\rm Ai}(k+x){\rm
Ai}(k+y)=\frac{1}{2\sqrt{\pi
t}}\exp\left[\frac{t^{3}}{12}-\frac{(x-y)^{2}}{4t}-\frac{(x+y)t}{2}\right]~{}.$
(C.17)

In summary, the most general solution of eq. (C.1) for $C\neq 0$ is

$p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=g_{1}\,e^{-A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,{\rm
Ai}(x)+g_{2}\,e^{-A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,{\rm
Bi}(x)~{},$ (C.18)

where $x$ is defined in eq. (C.2) and $g_{1,2}$ are two arbitrary constants.

For $C=0$, the solution is

$p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=g_{1}\,e^{-(A+\sqrt{A^{2}-B}){\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}+g_{2}\,e^{-(A-\sqrt{A^{2}-B}){\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}~{}.$
(C.19)

###  C.2 Hermite’s Equation

Consider Hermite’s differential equation

${p}^{\prime\prime}+2A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi{p}^{\prime}+(B+C{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}){p}=0~{},$
(C.20)

where $A,B,C$ are generic constants. First, consider the case $A^{2}>C$. With
the definitions

${p}=e^{-\frac{A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\,\Phi~{},~{}~{}~{}x=\zeta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi~{},~{}~{}~{}\zeta=(A^{2}-C)^{1/4}~{},~{}~{}~{}\nu=\frac{B-A}{2\,\zeta^{2}}-\frac{1}{2}~{},$
(C.21)

eq. (C.20) turns into

$\frac{d^{2}\Phi}{dx^{2}}+(2\nu+1-x^{2})\Phi=0~{},$ (C.22)

which is known as Weber’s differential equation.

Two linearly-independent (for non-integer $\nu$) solutions of eq. (C.22)
are777The function $\Phi_{1}(\nu,x)$ is related to other special functions
$\frac{1}{\sqrt{2^{\nu}}\,c_{\nu}}\Phi_{1}(\nu,x)=D_{\nu}(\sqrt{2}\,x)=U(-\nu-\frac{1}{2},\sqrt{2}\,x)=\sqrt{\frac{2^{\nu}}{x}}\,W_{\frac{\nu}{2}+\frac{1}{4},-\frac{1}{4}}(x^{2})~{},$
where $D_{\nu}(z)$ is the Whittaker function defined as
ParabolicCylinderD[nu,z] in Mathematica, $U(a,z)$ is the parabolic cylinder
function defined in ref. [138], and $W_{k,m}(z)$ is Whittaker’s confluent
hypergeometric function defined as WhittakerW[k,m,z] in Mathematica.

$\Phi_{1}(\nu,x)=c_{\nu}\,e^{-\frac{x^{2}}{2}}\,H_{\nu}(x)~{},~{}~{}~{}\Phi_{2}(\nu,x)=c_{\nu}\,e^{-\frac{x^{2}}{2}}\,H_{\nu}(-x)~{},$
(C.23)
$c_{\nu}=\left[2^{\nu}\sqrt{\pi}\,\Gamma(\nu+1)\right]^{-\frac{1}{2}}~{},$
(C.24)
$H_{\nu}(x)=2^{\nu}\sqrt{\pi}\left[\frac{{}_{1}F_{1}\big{(}-\frac{\nu}{2},\frac{1}{2},x^{2}\big{)}}{\Gamma\big{(}\frac{1-\nu}{2}\big{)}}-2x\,\frac{{}_{1}F_{1}\big{(}\frac{1-\nu}{2},\frac{3}{2},x^{2}\big{)}}{\Gamma\big{(}-\frac{\nu}{2}\big{)}}\right]~{}.$
(C.25)

Here ${{}_{1}F_{1}}(a,b,z)$ is Kummer’s confluent hypergeometric function of
the first kind, $\Gamma(z)$ is Euler’s gamma function and $H_{\nu}(x)$ is the
Hermite function defined in Mathematica as HermiteH[nu,x].

The function $H_{\nu}(x)$ satisfies the relations

$H_{\nu+1}(x)=2xH_{\nu}(x)-2\nu H_{\nu-1}(x)~{},$ (C.26)
$H_{\nu}(ix)=\frac{2^{\nu}\,\Gamma(\nu+1)}{\sqrt{\pi}}\,e^{-x^{2}}\left[e^{-\frac{i\pi\nu}{2}}H_{-\nu-1}(x)+e^{\frac{i\pi\nu}{2}}H_{-\nu-1}(-x)\right]~{},$
(C.27)
$H_{\nu}(x){H}_{\nu+1}(-x)+H_{\nu+1}(x){H}_{\nu}(-x)=2\,e^{x^{2}}b_{\nu}{\bar{b}}_{\nu}~{},~{}~{}~{}~{}~{}~{}b_{\nu}=2^{\nu}~{},~{}~{}~{}{\bar{b}}_{\nu}=\frac{\sqrt{\pi}}{\Gamma(-\nu)}~{}.$
(C.28)

The asymptotic behaviours are

$H_{\nu}(x)~{}~{}\stackrel{{\scriptstyle
x\to+\infty}}{{\longrightarrow}}~{}~{}b_{\nu}\,x^{\nu}~{},~{}~{}~{}~{}H_{\nu}(x)~{}\stackrel{{\scriptstyle
x\to-\infty}}{{\longrightarrow}}~{}{\bar{b}}_{\nu}\,e^{x^{2}}|x|^{-\nu-1}~{},$
(C.29)
$H_{\nu}(x)~{}~{}\stackrel{{\scriptstyle\nu\to-\infty}}{{\longrightarrow}}~{}~{}b_{\nu}{\bar{b}}_{\frac{\nu-1}{2}}\,\,e^{\frac{x^{2}}{2}-|x|\sqrt{-2\nu-1}}~{},$
(C.30) $\left.\begin{array}[]{l}\Phi_{1}(\nu,x)~{}~{}\stackrel{{\scriptstyle
x\to+\infty}}{{\longrightarrow}}\\\
\Phi_{2}(\nu,x)~{}~{}\stackrel{{\scriptstyle
x\to-\infty}}{{\longrightarrow}}\end{array}\right\\}=c_{\nu}b_{\nu}\,e^{-\frac{x^{2}}{2}}|x|^{\nu}~{}~{}~{}~{}\left.\begin{array}[]{l}\Phi_{1}(\nu,x)~{}~{}\stackrel{{\scriptstyle
x\to-\infty}}{{\longrightarrow}}\\\
\Phi_{2}(\nu,x)~{}~{}\stackrel{{\scriptstyle
x\to+\infty}}{{\longrightarrow}}\end{array}\right\\}=c_{\nu}{\bar{b}}_{\nu}\,e^{\frac{x^{2}}{2}}|x|^{-\nu-1}~{}.$
(C.31)

The power expansion for small $x$ is

$H_{\nu}(x)=b_{\nu}{\bar{b}}_{\frac{\nu-1}{2}}\left[1-\nu
x^{2}-{\scriptstyle\frac{\nu(2-\nu)}{6}}x^{4}\right]-b_{\nu+1}{\bar{b}}_{\frac{\nu}{2}}\left[x+{\scriptstyle\frac{1-\nu}{3}}x^{3}+{\scriptstyle\frac{(1-\nu)(3-\nu)}{30}}x^{5}\right]+{\mathcal{O}}(x^{6})~{}.$
(C.32)

The power expansion for small $\nu$ is

$H_{\nu}(x)~{}~{}\stackrel{{\scriptstyle\nu\to
0}}{{\longrightarrow}}~{}~{}\left\\{\begin{array}[]{lc}1+\nu(\psi+\sqrt{\pi}\,x)&({\rm
for~{}}x\to 0)\\\ 1+\nu\log 2x&({\rm for~{}}x\to+\infty)\\\
\frac{\nu\sqrt{\pi}\,e^{x^{2}}}{|x|}[-1+\nu(\gamma+\log|x|)]&({\rm
for~{}}x\to-\infty)\end{array}\right.~{},$ (C.33)

where $\psi=\log 2+\psi^{(0)}(1/2)/2=-0.29$, $\psi^{(n)}(x)$ is the polygamma
function, and $\gamma=0.58$ is the Euler-Mascheroni constant.

The derivative of the function $H_{\nu}(x)$ is

$\frac{dH_{\nu}(x)}{dx}=2\nu H_{\nu-1}(x)~{}.$ (C.34)

The Wronskian of the two solutions is

$W[\Phi_{2},\Phi_{1}]=\Phi_{2}(\nu,x)\frac{d\Phi_{1}(\nu,x)}{dx}-\frac{d\Phi_{2}(\nu,x)}{dx}\Phi_{1}(\nu,x)=\frac{2\,\sin\pi\nu}{\pi}~{},$
(C.35)

which shows that $\Phi_{1,2}(\nu,x)$ are no longer linearly independent for
integer $\nu$. In this case, an independent solution is found by choosing a
linear combination of the hypergeometric functions different from the one
defined by eq. (C.25).

For non-negative integer $\nu=n$, $H_{n}(x)$ are the Hermite polynomials and
the functions $\Phi_{1}(n,x)$ form an orthonormal basis

$\int_{-\infty}^{+\infty}dx\,\Phi_{1}(n,x)\,\Phi_{1}(m,x)=\delta_{nm}~{}.$
(C.36)

They also satisfy the properties

$\sum_{n=0}^{\infty}\Phi_{1}(n,x)\Phi_{1}(n,y)=\delta(x-y)~{},$ (C.37)
$\sum_{n=0}^{\infty}t^{n}\,\Phi_{1}(n,x)\Phi_{1}(n,y)=\frac{1}{\sqrt{\pi(1-t^{2})}}\exp\left[\frac{4txy-(1+t^{2})(x^{2}+y^{2})}{2(1-t^{2})}\right]~{}~{}(0<t<1).$
(C.38)

In summary, the most general solution of eq. (C.20) for $A^{2}>C$ is

$p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=e^{-\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}(A+\zeta^{2})}\left[g_{1}\,H_{\nu}(\zeta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)+g_{2}\,{H}_{\nu}(-\zeta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\right]~{},$
(C.39)

where $\zeta$ and $\nu$ are defined in eq. (C.21) and $g_{1,2}$ are two
arbitrary constants.

For $A^{2}=C$, the solution is

$p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=g_{1}\,e^{-\frac{A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}-\sqrt{A-B}\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}+g_{2}\,e^{-\frac{A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}+\sqrt{A-B}\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}~{}.$
(C.40)

For $A^{2}<C$, we define

${p}=e^{-\frac{A{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\,{\bar{\Phi}}~{},~{}~{}~{}{\bar{x}}={\bar{\zeta}}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi~{},~{}~{}~{}{\bar{\zeta}}=(C-A^{2})^{1/4}~{},~{}~{}~{}{\bar{\nu}}=\frac{B-A}{2\,{\bar{\zeta}}^{2}}-\frac{1}{2}~{}$
(C.41)

and eq. (C.20) turns into

$\frac{d^{2}{\bar{\Phi}}}{d{\bar{x}}^{2}}+(2{\bar{\nu}}+1+{\bar{x}}^{2}){\bar{\Phi}}=0~{},$
(C.42)

which is Weber’s differential equation of the second kind. The solutions of
this equation are obtained from eq. (C.23) with the replacements
$\Phi\to{\bar{\Phi}}$, $\zeta\to\sqrt{i}\,{\bar{\zeta}}$ and
$2\nu+1\to-i(2{\bar{\nu}}+1)$.

The solutions to eq. (C.42) have an oscillatory behaviour. For instance, in
the limit of fixed ${\bar{\nu}}$ and positive and large ${\bar{x}}$, two
linearly-independent solutions have the asymptotic behaviour (for
${\bar{x}}\to+\infty$)

${\bar{\Phi}}_{1}({\bar{\nu}},{\bar{x}})=\frac{1}{\sqrt{{\bar{x}}}}\left(s_{1}\cos{\bar{X}}-s_{2}\sin{\bar{X}}\right)~{},~{}~{}~{}{\bar{\Phi}}_{2}({\bar{\nu}},{\bar{x}})=\frac{1}{\sqrt{{\bar{x}}}}\left(s_{1}\sin{\bar{X}}+s_{2}\cos{\bar{X}}\right)~{},$
(C.43)
${\bar{X}}=\frac{{\bar{x}}^{2}+(2{\bar{\nu}}+1)\ln{\bar{x}}+\omega}{2}~{},~{}~{}~{}\omega=\frac{\pi}{2}+(2{\bar{\nu}}+1)\ln\sqrt{2}+\arg\Gamma\left[\frac{1-i(2{\bar{\nu}}+1)}{2}\right]~{},$
(C.44)
$s_{1}=1-\frac{2{\bar{\nu}}+1}{4{\bar{x}}^{2}}+{\mathcal{O}}({\bar{x}}^{-4})~{},~{}~{}~{}s_{2}=\frac{2{\bar{\nu}}({\bar{\nu}}+1)-1}{8{\bar{x}}^{2}}+{\mathcal{O}}({\bar{x}}^{-4})~{}.$
(C.45)

In the limit of large and negative ${\bar{\nu}}$, the solutions
${\bar{\Phi}}_{1,2}$ can be expressed in terms of Airy functions.

## References

  * [1] P. Bak, C. Tang and K. Wiesenfeld, _Self-organized criticality: An Explanation of 1/f noise_ , Phys. Rev. Lett. 59 (1987) 381.
  * [2] P. Ehrenfest, _Phasenumwandlungen im üblichen und erweiterten Sinn, classifiziert nach dem entsprechenden Singularitäten des thermodynamischen Potentiales_ , Verhandlingen der Koninklijke Akademie van Wetenschappen (Amsterdam) 36 (1933) 153 ; Communications from the Physical Laboratory of the University of Leiden, Supplement No. 75b.
  * [3] J. Khoury and O. Parrikar, _Search Optimization, Funnel Topography, and Dynamical Criticality on the String Landscape_ , JCAP 12 (2019) 014 [arXiv:1907.07693].
  * [4] J. Khoury, _Accessibility Measure for Eternal Inflation: Dynamical Criticality and Higgs Metastability_ , JCAP 06 (2021) 009 [arXiv:1912.06706].
  * [5] G. Kartvelishvili, J. Khoury and A. Sharma, _The Self-Organized Critical Multiverse_ , JCAP 02 (2021) 028 [arXiv:2003.12594].
  * [6] A. Vilenkin, _The Birth of Inflationary Universes_ , Phys. Rev. D 27 (1983) 2848.
  * [7] A. A. Starobinsky, _Multicomponent de Sitter (Inflationary) Stages and the Generation of Perturbations_ , JETP Lett. 42 (1985) 152.
  * [8] A. A. Starobinsky, _Stochastic de Sitter (Inflationary) Stage in the Early Universe_ , Lect. Notes Phys. 246 (1986) 107.
  * [9] S.-J. Rey, _Dynamics of Inflationary Phase Transition_ , Nucl. Phys. B 284 (1987) 706.
  * [10] A. S. Goncharov, A. D. Linde and V. F. Mukhanov, _The Global Structure of the Inflationary Universe_ , Int. J. Mod. Phys. A 2 (1987) 561.
  * [11] M. Sasaki, Y. Nambu and K.-i. Nakao, _Classical Behavior of a Scalar Field in the Inflationary Universe_ , Nucl. Phys. B 308 (1988) 868.
  * [12] K.-i. Nakao, Y. Nambu and M. Sasaki, _Stochastic Dynamics of New Inflation_ , Prog. Theor. Phys. 80 (1988) 1041.
  * [13] H. E. Kandrup, _Stochastic Inflation as a Time Dependent Random Walk_ , Phys. Rev. D 39 (1989) 2245.
  * [14] Y. Nambu and M. Sasaki, _Stochastic Stage of an Inflationary Universe Model_ , Phys. Lett. B 205 (1988) 441.
  * [15] Y. Nambu, _Stochastic Dynamics of an Inflationary Model and Initial Distribution of Universes_ , Prog. Theor. Phys. 81 (1989) 1037.
  * [16] M. Mijic, _Random Walk After the Big Bang_ , Phys. Rev. D 42 (1990) 2469.
  * [17] D. S. Salopek and J. R. Bond, _Stochastic inflation and nonlinear gravity_ , Phys. Rev. D 43 (1991) 1005.
  * [18] A. D. Linde and A. Mezhlumian, _Stationary universe_ , Phys. Lett. B 307 (1993) 25 [gr-qc/9304015].
  * [19] A. D. Linde, D. A. Linde and A. Mezhlumian, _From the Big Bang theory to the theory of a stationary universe_ , Phys. Rev. D 49 (1994) 1783 [gr-qc/9306035].
  * [20] A. A. Starobinsky and J. Yokoyama, _Equilibrium state of a selfinteracting scalar field in the De Sitter background_ , Phys. Rev. D 50 (1994) 6357 [astro-ph/9407016].
  * [21] S. Winitzki and A. Vilenkin, _Uncertainties of predictions in models of eternal inflation_ , Phys. Rev. D 53 (1996) 4298 [gr-qc/9510054].
  * [22] A. Vilenkin, _On the factor ordering problem in stochastic inflation_ , Phys. Rev. D 59 (1999) 123506 [gr-qc/9902007].
  * [23] Y. Nambu and M. Sasaki, _Stochastic Approach to Chaotic Inflation and the Distribution of Universes_ , Phys. Lett. B 219 (1989) 240.
  * [24] S. Winitzki, _Eternal inflation_ , World Scientific, Singapore (2008) .
  * [25] A. H. Guth, _Eternal inflation and its implications_ , J. Phys. A 40 (2007) 6811 [hep-th/0702178].
  * [26] A. D. Linde, _Eternal Chaotic Inflation_ , Mod. Phys. Lett. A 1 (1986) 81.
  * [27] A. D. Linde, _Eternally Existing Selfreproducing Chaotic Inflationary Universe_ , Phys. Lett. B 175 (1986) 395.
  * [28] B. Freivogel, _Making predictions in the multiverse_ , Class. Quant. Grav. 28 (2011) 204007 [arXiv:1105.0244].
  * [29] M. Aryal and A. Vilenkin, _The Fractal Dimension of Inflationary Universe_ , Phys. Lett. B 199 (1987) 351.
  * [30] J. Garcia-Bellido and A. D. Linde, _Stationarity of inflation and predictions of quantum cosmology_ , Phys. Rev. D 51 (1995) 429 [hep-th/9408023].
  * [31] S. Winitzki, _The Eternal fractal in the universe_ , Phys. Rev. D 65 (2002) 083506 [gr-qc/0111048].
  * [32] A. Borde and A. Vilenkin, _Eternal inflation and the initial singularity_ , Phys. Rev. Lett. 72 (1994) 3305 [gr-qc/9312022].
  * [33] A. Vilenkin, _Making predictions in eternally inflating universe_ , Phys. Rev. D 52 (1995) 3365 [gr-qc/9505031].
  * [34] A. Borde, A. H. Guth and A. Vilenkin, _Inflationary space-times are incompletein past directions_ , Phys. Rev. Lett. 90 (2003) 151301 [gr-qc/0110012].
  * [35] A. Mithani and A. Vilenkin, _Did the universe have a beginning?_ , arXiv:1204.4658.
  * [36] L. Susskind, _Was There a Beginning?_ , arXiv:1204.5385.
  * [37] L. Susskind, _Is Eternal Inflation Past-Eternal? And What if It Is?_ , arXiv:1205.0589.
  * [38] A. Linde and M. Noorbala, _Measure Problem for Eternal and Non-Eternal Inflation_ , JCAP 09 (2010) 008 [arXiv:1006.2170].
  * [39] A. De Simone, A. H. Guth, M. P. Salem and A. Vilenkin, _Predicting the cosmological constant with the scale-factor cutoff measure_ , Phys. Rev. D 78 (2008) 063520 [arXiv:0805.2173].
  * [40] R. Bousso, B. Freivogel and I.-S. Yang, _Properties of the scale factor measure_ , Phys. Rev. D 79 (2009) 063513 [arXiv:0808.3770].
  * [41] A. De Simone, A. H. Guth, A. D. Linde, M. Noorbala, M. P. Salem and A. Vilenkin, _Boltzmann brains and the scale-factor cutoff measure of the multiverse_ , Phys. Rev. D 82 (2010) 063520 [arXiv:0808.3778].
  * [42] R. Bousso, _New Scale Factor Measure_ , Phys. Rev. D 86 (2012) 023532 [arXiv:1205.6634].
  * [43] A. Vilenkin and M. Yamada, _Four-volume cutoff measure of the multiverse_ , Phys. Rev. D 101 (2020) 043520 [arXiv:1912.02187].
  * [44] K. D. Olum, P. Upadhyay and A. Vilenkin, _Black holes and uptunneling suppress Boltzmann brains_ , Phys. Rev. D 104 (2021) 023528 [arXiv:2105.00457].
  * [45] N. Arkani-Hamed, S. Dubovsky, A. Nicolis, E. Trincherini and G. Villadoro, _A Measure of de Sitter entropy and eternal inflation_ , JHEP 05 (2007) 055 [arXiv:0704.1814].
  * [46] P. Creminelli, S. Dubovsky, A. Nicolis, L. Senatore and M. Zaldarriaga, _The Phase Transition to Slow-roll Eternal Inflation_ , JHEP 09 (2008) 036 [arXiv:0802.1067].
  * [47] S. Dubovsky, L. Senatore and G. Villadoro, _The Volume of the Universe after Inflation and de Sitter Entropy_ , JHEP 04 (2009) 118 [arXiv:0812.2246].
  * [48] S. Dubovsky, L. Senatore and G. Villadoro, _Universality of the Volume Bound in Slow-Roll Eternal Inflation_ , JHEP 05 (2012) 035 [arXiv:1111.1725].
  * [49] G. Dvali and C. Gomez, _Quantum Compositeness of Gravity: Black Holes, AdS and Inflation_ , JCAP 01 (2014) 023 [arXiv:1312.4795].
  * [50] G. Dvali, C. Gomez and S. Zell, _Quantum Break-Time of de Sitter_ , JCAP 06 (2017) 028 [arXiv:1701.08776].
  * [51] M. Lewandowski and A. Perko, _Leading slow roll corrections to the volume of the universe and the entropy bound_ , JHEP 12 (2014) 060 [arXiv:1309.6705].
  * [52] M. Baumgart and R. Sundrum, _De Sitter Diagrammar and the Resummation of Time_ , JHEP 07 (2020) 119 [arXiv:1912.09502].
  * [53] T. Cohen and D. Green, _Soft de Sitter Effective Theory_ , JHEP 12 (2020) 041 [arXiv:2007.03693].
  * [54] V. Gorbenko and L. Senatore, _$\lambda\phi^{4}$ in dS_, arXiv:1911.00022.
  * [55] E. Palti, _The Swampland: Introduction and Review_ , Fortsch. Phys. 67 (2019) 1900037 [arXiv:1903.06239].
  * [56] H. Matsui and F. Takahashi, _Eternal Inflation and Swampland Conjectures_ , Phys. Rev. D 99 (2019) 023533 [arXiv:1807.11938].
  * [57] G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, _De Sitter Space and the Swampland_ , arXiv:1806.08362.
  * [58] H. Ooguri and C. Vafa, _On the Geometry of the String Landscape and the Swampland_ , Nucl. Phys. B 766 (2007) 21 [hep-th/0605264].
  * [59] D. Klaewer and E. Palti, _Super-Planckian Spatial Field Variations and Quantum Gravity_ , JHEP 01 (2017) 088 [arXiv:1610.00010].
  * [60] K. Dimopoulos, _Steep Eternal Inflation and the Swampland_ , Phys. Rev. D 98 (2018) 123516 [arXiv:1810.03438].
  * [61] W. H. Kinney, _Eternal Inflation and the Refined Swampland Conjecture_ , Phys. Rev. Lett. 122 (2019) 081302 [arXiv:1811.11698].
  * [62] S. Brahma and S. Shandera, _Stochastic eternal inflation is in the swampland_ , JHEP 11 (2019) 016 [arXiv:1904.10979].
  * [63] T. Rudelius, _Conditions for (No) Eternal Inflation_ , JCAP 08 (2019) 009 [arXiv:1905.05198].
  * [64] Z. Wang, R. Brandenberger and L. Heisenberg, _Eternal Inflation, Entropy Bounds and the Swampland_ , Eur. Phys. J. C 80 (2020) 864 [arXiv:1907.08943].
  * [65] J. J. Blanco-Pillado, H. Deng and A. Vilenkin, _Eternal Inflation in Swampy Landscapes_ , JCAP 05 (2020) 014 [arXiv:1909.00068].
  * [66] M. Tegmark, _What does inflation really predict?_ , JCAP 04 (2005) 001 [astro-ph/0410281].
  * [67] A. D. Linde, D. A. Linde and A. Mezhlumian, _Do we live in the center of the world?_ , Phys. Lett. B 345 (1995) 203 [hep-th/9411111].
  * [68] A. D. Linde, D. A. Linde and A. Mezhlumian, _Nonperturbative amplifications of inhomogeneities in a selfreproducing universe_ , Phys. Rev. D 54 (1996) 2504 [gr-qc/9601005].
  * [69] A. D. Linde, _Towards a gauge invariant volume-weighted probability measure for eternal inflation_ , JCAP 06 (2007) 017 [arXiv:0705.1160].
  * [70] A. D. Linde, V. Vanchurin and S. Winitzki, _Stationary Measure in the Multiverse_ , JCAP 01 (2009) 031 [arXiv:0812.0005].
  * [71] G. W. Gibbons and S. W. Hawking, _Cosmological Event Horizons, Thermodynamics, and Particle Creation_ , Phys. Rev. D 15 (1977) 2738.
  * [72] D. L. Bennett and H. B. Nielsen, _Predictions for nonAbelian fine structure constants from multicriticality_ , Int. J. Mod. Phys. A 9 (1994) 5155 [hep-ph/9311321].
  * [73] L. Smolin, _Cosmology as a problem in critical phenomena_ , Lect. Notes Phys. 461 (1996) 184 [gr-qc/9505022].
  * [74] M. Born and H. W. Peng, _Statistical Mechanics of Fields and the ‘Apeiron’_ , Nature 153 (1944) 164.
  * [75] E. Silverstein and A. Westphal, _Monodromy in the CMB: Gravity Waves and String Inflation_ , Phys. Rev. D 78 (2008) 106003 [arXiv:0803.3085].
  * [76] K. Choi and S. H. Im, _Realizing the relaxion from multiple axions and its UV completion with high scale supersymmetry_ , JHEP 01 (2016) 149 [arXiv:1511.00132].
  * [77] D. E. Kaplan and R. Rattazzi, _Large field excursions and approximate discrete symmetries from a clockwork axion_ , Phys. Rev. D 93 (2016) 085007 [arXiv:1511.01827].
  * [78] G. F. Giudice and M. McCullough, _A Clockwork Theory_ , JHEP 02 (2017) 036 [arXiv:1610.07962].
  * [79] ATLAS Collaboration, G. Aad et al., _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214].
  * [80] CMS Collaboration, S. Chatrchyan et al., _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_ , Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235].
  * [81] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori et al., _Higgs mass and vacuum stability in the Standard Model at NNLO_ , JHEP 08 (2012) 098 [arXiv:1205.6497].
  * [82] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio et al., _Investigating the near-criticality of the Higgs boson_ , JHEP 12 (2013) 089 [arXiv:1307.3536].
  * [83] A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner and O. L. Veretin, _Stability of the Electroweak Vacuum: Gauge Independence and Advanced Precision_ , Phys. Rev. Lett. 115 (2015) 201802 [arXiv:1507.08833].
  * [84] A. Andreassen, W. Frost and M. D. Schwartz, _Scale Invariant Instantons and the Complete Lifetime of the Standard Model_ , Phys. Rev. D 97 (2018) 056006 [arXiv:1707.08124].
  * [85] S. W. Hawking and I. G. Moss, _Supercooled Phase Transitions in the Very Early Universe_ , Phys. Lett. B 110 (1982) 35.
  * [86] Particle Data Group Collaboration, P. A. Zyla et al., _Review of Particle Physics_ , PTEP 2020 (2020) 083C01.
  * [87] ATLAS, CDF, CMS and D0 Collaborations, _First combination of Tevatron and LHC measurements of the top-quark mass_ , arXiv:1403.4427.
  * [88] G. F. Giudice, A. Notari, M. Raidal, A. Riotto and A. Strumia, _Towards a complete theory of thermal leptogenesis in the SM and MSSM_ , Nucl. Phys. B 685 (2004) 89 [hep-ph/0310123].
  * [89] G. F. Giudice and R. Rattazzi, _Living Dangerously with Low-Energy Supersymmetry_ , Nucl. Phys. B 757 (2006) 19 [hep-ph/0606105].
  * [90] G. F. Giudice, _Naturally Speaking: The Naturalness Criterion and Physics at the LHC_ , arXiv:0801.2562.
  * [91] G. Dvali and A. Vilenkin, _Cosmic attractors and gauge hierarchy_ , Phys. Rev. D 70 (2004) 063501 [hep-th/0304043].
  * [92] G. Dvali, _Large hierarchies from attractor vacua_ , Phys. Rev. D 74 (2006) 025018 [hep-th/0410286].
  * [93] P. W. Graham, D. E. Kaplan and S. Rajendran, _Cosmological Relaxation of the Electroweak Scale_ , Phys. Rev. Lett. 115 (2015) 221801 [arXiv:1504.07551].
  * [94] N. Arkani-Hamed, T. Cohen, R. T. D’Agnolo, A. Hook, H. D. Kim and D. Pinner, _Solving the Hierarchy Problem at Reheating with a Large Number of Degrees of Freedom_ , Phys. Rev. Lett. 117 (2016) 251801 [arXiv:1607.06821].
  * [95] A. Arvanitaki, S. Dimopoulos, V. Gorbenko, J. Huang and K. Van Tilburg, _A small weak scale from a small cosmological constant_ , JHEP 05 (2017) 071 [arXiv:1609.06320].
  * [96] A. Herraez and L. E. Ibanez, _An Axion-induced SM/MSSM Higgs Landscape and the Weak Gravity Conjecture_ , JHEP 02 (2017) 109 [arXiv:1610.08836].
  * [97] M. Geller, Y. Hochberg and E. Kuflik, _Inflating to the Weak Scale_ , Phys. Rev. Lett. 122 (2019) 191802 [arXiv:1809.07338].
  * [98] C. Cheung and P. Saraswat, _Mass Hierarchy and Vacuum Energy_ , arXiv:1811.12390.
  * [99] G. F. Giudice, A. Kehagias and A. Riotto, _The Selfish Higgs_ , JHEP 10 (2019) 199 [arXiv:1907.05370].
  * [100] N. Kaloper and A. Westphal, _A Goldilocks Higgs_ , Phys. Lett. B 808 (2020) 135616 [arXiv:1907.05837].
  * [101] G. Dvali, _Cosmological Relaxation of Higgs Mass Before and After LHC and Naturalness_ , arXiv:1908.05984.
  * [102] A. Strumia and D. Teresi, _Relaxing the Higgs mass and its vacuum energy by living at the top of the potential_ , Phys. Rev. D 101 (2020) 115002 [arXiv:2002.02463].
  * [103] C. Csáki, R. T. D’Agnolo, M. Geller and A. Ismail, _Crunching Dilaton, Hidden Naturalness_ , Phys. Rev. Lett. 126 (2021) 091801 [arXiv:2007.14396].
  * [104] N. Arkani-Hamed, R. T. D’Agnolo and H. D. Kim, _The Weak Scale as a Trigger_ , arXiv:2012.04652.
  * [105] D. Wyler and L. Wolfenstein, _Massless Neutrinos in Left-Right Symmetric Models_ , Nucl. Phys. B 218 (1983) 205.
  * [106] R. N. Mohapatra and J. W. F. Valle, _Neutrino Mass and Baryon Number Nonconservation in Superstring Models_ , Phys. Rev. D 34 (1986) 1642.
  * [107] E. Ma, _Lepton Number Nonconservation in $E(6)$ Superstring Models_, Phys. Lett. B 191 (1987) 287.
  * [108] CMS Collaboration, A. M. Sirunyan et al., _Search for vector-like leptons in multilepton final states in proton-proton collisions at $\sqrt{s}$ = 13 TeV_, Phys. Rev. D 100 (2019) 052003 [arXiv:1905.10853].
  * [109] ATLAS Collaboration, G. Aad et al., _Search for heavy lepton resonances decaying to a $Z$ boson and a lepton in $pp$ collisions at $\sqrt{s}=8$ TeV with the ATLAS detector_, JHEP 09 (2015) 108 [arXiv:1506.01291].
  * [110] S. Bißmann, G. Hiller, C. Hormigos-Feliu and D. F. Litim, _Multi-lepton signatures of vector-like leptons with flavor_ , Eur. Phys. J. C 81 (2021) 101 [arXiv:2011.12964].
  * [111] S. Gopalakrishna and A. Velusamy, _Higgs vacuum stability with vectorlike fermions_ , Phys. Rev. D 99 (2019) 115020 [arXiv:1812.11303].
  * [112] G. Isidori, G. Ridolfi and A. Strumia, _On the metastability of the standard model vacuum_ , Nucl. Phys. B 609 (2001) 387 [hep-ph/0104016].
  * [113] Y. B. Zeldovich, _Cosmological Constant and Elementary Particles_ , JETP Lett. 6 (1967) 316.
  * [114] Y. B. Zel’dovich, A. Krasinski and Y. B. Zeldovich, _The Cosmological constant and the theory of elementary particles_ , Sov. Phys. Usp. 11 (1968) 381.
  * [115] S. Weinberg, _The Cosmological Constant Problem_ , Rev. Mod. Phys. 61 (1989) 1.
  * [116] S. Nobbenhuis, _Categorizing different approaches to the cosmological constant problem_ , Found. Phys. 36 (2006) 613 [gr-qc/0411093].
  * [117] J. Polchinski, _The Cosmological Constant and the String Landscape_ , in _23rd Solvay Conference in Physics: The Quantum Structure of Space and Time_ , 3, 2006, hep-th/0603249.
  * [118] R. Bousso, _TASI Lectures on the Cosmological Constant_ , Gen. Rel. Grav. 40 (2008) 607 [arXiv:0708.4231].
  * [119] J. Martin, _Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)_ , Comptes Rendus Physique 13 (2012) 566 [arXiv:1205.3365].
  * [120] C. P. Burgess, _The Cosmological Constant Problem: Why it’s hard to get Dark Energy from Micro-physics_ , in _100e Ecole d’Ete de Physique: Post-Planck Cosmology_ , 9, 2013, arXiv:1309.4133, DOI.
  * [121] A. Padilla, _Lectures on the Cosmological Constant Problem_ , arXiv:1502.05296.
  * [122] S. Weinberg, _Anthropic Bound on the Cosmological Constant_ , Phys. Rev. Lett. 59 (1987) 2607.
  * [123] P. W. Graham, D. E. Kaplan and S. Rajendran, _Relaxation of the Cosmological Constant_ , Phys. Rev. D 100 (2019) 015048 [arXiv:1902.06793].
  * [124] I. M. Bloch, C. Csáki, M. Geller and T. Volansky, _Crunching away the cosmological constant problem: dynamical selection of a small $\Lambda$_, JHEP 12 (2020) 191 [arXiv:1912.08840].
  * [125] P. Svrcek and E. Witten, _Axions In String Theory_ , JHEP 06 (2006) 051 [hep-th/0605206].
  * [126] M. Kawasaki, K. Kohri and N. Sugiyama, _Cosmological constraints on late time entropy production_ , Phys. Rev. Lett. 82 (1999) 4168 [astro-ph/9811437].
  * [127] M. Kawasaki, K. Kohri and N. Sugiyama, _MeV scale reheating temperature and thermalization of neutrino background_ , Phys. Rev. D 62 (2000) 023506 [astro-ph/0002127].
  * [128] S. Hannestad, _What is the lowest possible reheating temperature?_ , Phys. Rev. D 70 (2004) 043506 [astro-ph/0403291].
  * [129] K. Ichikawa, M. Kawasaki and F. Takahashi, _Constraint on the Effective Number of Neutrino Species from the WMAP and SDSS LRG Power Spectra_ , JCAP 05 (2007) 007 [astro-ph/0611784].
  * [130] P. F. de Salas, M. Lattanzi, G. Mangano, G. Miele, S. Pastor and O. Pisanti, _Bounds on very low reheating scenarios after Planck_ , Phys. Rev. D 92 (2015) 123534 [arXiv:1511.00672].
  * [131] Planck Collaboration, N. Aghanim et al., _Planck 2018 results. VI. Cosmological parameters_ , Astron. Astrophys. 641 (2020) A6 [arXiv:1807.06209].
  * [132] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog et al., _High precision determination of the gluon fusion Higgs boson cross-section at the LHC_ , JHEP 05 (2016) 058 [arXiv:1602.00695].
  * [133] T. Aoyama et al., _The anomalous magnetic moment of the muon in the Standard Model_ , Phys. Rept. 887 (2020) 1 [arXiv:2006.04822].
  * [134] S. Borsanyi et al., _Leading hadronic contribution to the muon magnetic moment from lattice QCD_ , Nature 593 (2021) 51 [arXiv:2002.12347].
  * [135] C. Vafa, _The String landscape and the swampland_ , hep-th/0509212.
  * [136] H. Risken, _The Fokker-Planck Equation_ , Springer Verlag, Berlin (1989) .
  * [137] O. Vallée and M. Soares, _Airy Functions and Applications to Physics_ , Imperial College Press, London (2004) .
  * [138] M. Abramowitz and I. Stegun, _Handbook of Mathematical Functions_ , National Bureau of Standards, Washington (1964) .