will be taken to be positive. In eq. (4.3) we have not shown explicitly a
constant vacuum energy, eventually tuned to reproduce today’s cosmological
constant.

Figure 6: A sketch of the scalar potential $V(\varphi,\langle h\rangle)$ with
the two branches corresponding to the IR ($\langle h\rangle=v$) and UV
($\langle h\rangle\sim M/g_{*}$) Higgs phases. The two phases coexist for
$0<\varphi<\varphi_{+}$. During inflation, the supercooled IR phase terminates
at $\varphi_{\star}$ because of Hubble-induced transitions into the UV phase.

The form of the potential in eq. (4.3) is sketched in fig. 6. The IR branch is
degenerate with the UV branch at $\varphi=0$ (the point corresponding to the
RG trajectory such that $\lambda(M/g_{*})=0$) and terminates at
$\varphi=\varphi_{+}$ (corresponding to the RG trajectory such that
$\lambda(v)=0$). Between these two points, the IR branch is in a supercooled
phase characterised by $\lambda(\Lambda_{I})=0$ with $\Lambda_{I}$ varying
from $M/g_{*}$ to $v$, as $\varphi$ varies from $0$ to $\varphi_{+}$. In fig.
6, the UV branch terminates at $\varphi=0$ since we make the plausible
assumption that the UV vacuum disappears immediately beyond the critical
point, although the precise location of the endpoint depends on the features
of the UV completion. The potential in eq. (4.3) is pyramid-like, as the
example studied in sect. 3.1, with microscopic features at the cusp analogous
to those discussed in sect. 3.2.

###  4.2 SOL Predictions

To solve the FPV corresponding to the potential in eq. (4.3) we need to
determine the boundary conditions that define the matching between the two
phases. We will restrict our considerations to the case in which the Higgs
fluctuations around the UV vacuum are damped, so that the Higgs cannot have
large excursions away from its UV minimum. Since the typical Higgs mass at the
UV vacuum is $M$, this condition implies $M>{3H_{0}}/{2}$. However, even when
this condition holds, the potential is relatively shallow for low Higgs
fields. Large Higgs fluctuations around the IR minimum are expected during
inflation, and these can bring $h$ to explore high field values, possibly
getting trapped into the deeper UV minimum. This means that, during the
inflationary era, the system cannot access the full supercooled branch that
extends up to $\varphi_{+}$ because the Higgs field prematurely drops from the
IR to the UV phase for $\varphi>\varphi_{\star}$. Consequently, we must impose
the boundary condition that, on the IR branch, the FPV distribution $P$
vanishes at $\varphi_{\star}$.

To estimate the value of $\varphi_{\star}$ we must calculate the condition for
Hubble fluctuations to drive efficiently the Higgs from the IR to the UV
vacuum. Quantum tunnelling is inefficient because the bubbles of true vacuum
do not percolate as they are swamped by the fast expansion of the surrounding
space. On the other hand, the Higgs field can overcome the potential barrier
by means of de-Sitter ‘thermal’ effects proportional to the Gibbons-Hawking
temperature $T_{H}=H_{0}/2\pi$ [71]. Quantitatively, the condition for vacuum
transition corresponds to the requirement that the rate for processes mediated
by Hawking-Moss instantons [85] is unsuppressed, namely

$\Delta V_{\rm max}<\frac{3\hbar\,H_{0}^{4}}{8\pi^{2}}~{},$ (4.5)

where $\Delta V_{\rm max}$ is the potential barrier that the Higgs field has
to climb to reach the UV minimum.

Expressing the running coupling constants in terms of SM physical observables,
we find that, in the proximity of the SM values for the strong coupling
constant and for the Higgs and top-quark masses, the value of $H_{0}$ that
saturates eq. (4.5) is

$\log_{10}\frac{H_{0}}{\rm GeV}=9.8+0.7\Big{(}\frac{m_{h}}{\rm
GeV}-125.10\Big{)}-1.0\Big{(}\frac{m_{t}}{\rm
GeV}-173.34\Big{)}+0.3\frac{\alpha_{s}(m_{Z})-0.1184}{0.0007}~{}.$ (4.6)

This result corresponds to a full NNLO calculation of the barrier of the
effective potential in the SM as performed in ref. [82]. Since $\Delta V_{\rm
max}$ is the difference between two extrema of the potential, the result is
scheme and gauge independent. Note that the result is independent of $M$ and
it is therefore robust against unknown features of the UV completion. Equation
(4.6) gives, for any given value of $H_{0}$, the SM parameters that correspond
to the RG trajectory identified by $\varphi_{\star}$ and such that the Higgs
quartic at the UV scale is equal to $\lambda_{\star}\equiv-
g_{*}^{2}\varphi_{\star}$.

As shown in sect. 2.8, the solution of the FPV equation for a linear potential
in the Q2V regime with absorbing boundary conditions at $\varphi_{\star}$ is
peaked, at asymptotically large times, in the interval between
$\varphi_{\star}-\Delta\varphi$ and $\varphi_{\star}$ where

$\Delta\varphi\approx\left(\frac{\hbar\,H_{0}^{4}M_{P}^{2}}{4\pi^{2}\kappa_{{\rm
IR}}M^{4}f^{2}}\right)^{1/3}~{}.$ (4.7)

In other words, the UV value of the Higgs quartic is determined to be equal to
$\lambda_{\star}$ with an uncertainty $\Delta\lambda\approx
g_{*}^{2}\Delta\varphi$, which is completely negligible since the bound on the
dark-energy equation of state in eq. (3.29) implies

$\Delta\lambda<\left(\frac{H_{0}}{10^{10}~{}{\rm
GeV}}\right)^{\frac{4}{3}}\left(\frac{10^{12}~{}{\rm
GeV}}{M}\right)^{4}10^{-66}~{}.$ (4.8)

However, this estimate omits the effects of Hubble-induced Higgs fluctuations,
which are unavoidable and must be accounted for to effectively capture the SOL
prediction. A complete treatment would require the computation of the two-
field FPV solution $P(\varphi,h)$, from which the maximum likelihood and
statistical uncertainty on $\lambda$ could be extracted, subject to the
boundary conditions imposed. For the sake of brevity we may instead estimate
the scale of the effect by assuming that $h^{2}$ is a random variable with RMS
fluctuations of $\mathcal{O}(H_{0}^{2})$. In this case $\varphi$ will see a
smeared background value of $h^{2}$, rather than a value pinned to a local
minimum. Consequently, this will smear the apparent endpoint $\varphi_{+}$ by
an amount proportional to $\Delta{\varphi_{+}}\sim H_{0}^{2}/M^{2}$. Hence the
final uncertainty in the prediction of the Higgs self-coupling is

$\Delta\lambda\approx\frac{g_{*}^{2}H_{0}^{2}}{M^{2}}~{},$ (4.9)

which is still parametrically small but, in the quantum regime we are
considering, is much larger than what was estimated from eq. (4.7).

In conclusion, SOL predicts that the SM parameters lie in the proximity of the
critical point where two Higgs phases coexist and the ordinary Higgs vacuum is
at the verge of collapsing into a new high-field configuration. The proper
combination of SM parameters is uniquely determined by the Hubble rate
$H_{0}$, which identifies an RG trajectory for the Higgs quartic that crosses
zero at a field value of order $H_{0}$. It is also expected that, at the end
of inflation, the Higgs field lies in the IR phase. Although near-criticality
of the SM is a robust consequence of SOL and the related statistical
uncertainty is small, the precise prediction for the SM parameters has
theoretical uncertainties related to the determination of $\varphi_{\star}$.
Here we have calculated $\varphi_{\star}$ on the basis of the Hawking-Moss
rate but the criterion we have used has an intrinsic order-one uncertainty on
the determination of $H_{0}$, which is associated with the detailed features
of the transition between IR and UV vacua.

Figure 7: The SOL prediction as a function of the Higgs and top-quark masses,
for given values of $H_{0}$ in GeV (black lines), alongside the absolute
stability (green) and instability (red) regions, as calculated in ref. [82].
Also shown is the 95% CL ellipse corresponding to the SM values. The dashed
blue lines show the Higgs field value (in GeV) at which the effective
potential vanishes and therefore, for $M$ equal to the same value, the IR and
UV vacua are exactly degenerate. Figure 8: The region of $H_{0}$ (the Hubble
rate during inflation) and $M$ (the energy cutoff of the Higgs sector) in
which the SOL prediction is consistent with the SM values of the Higgs and
top-quark masses at the 1-$\sigma$ to 3-$\sigma$ levels. Also shown are the
corresponding values of the reheating temperature $T_{\rm RH}$. In the shaded
regions, the apeiron energy density exceeds the inflationary background
($\rho_{\phi}>\rho_{\rm inf}$) or quantum fluctuations destabilise the UV
phase ($H_{0}>2M/3$).

In fig. 7 we show the SOL prediction as a function of the Higgs and top-quark
masses, for certain values of $H_{0}$. We also show the lines corresponding to
the critical point at which the IR and UV vacua are degenerate. This is
calculated by imposing that the SM effective potential at NNLO, as calculated
in ref. [82], vanishes at $h=M/g_{*}$. These critical lines are uniquely
identified by the value of $M$ but, unlike the lines corresponding to
$\varphi_{\star}$, contain a residual scheme and gauge dependence because the
point $h=M/g_{*}$ does not correspond to an extremum of the SM potential and
are also sensitive to the unknown features of the UV completion.

In fig. 8 we show the region of $M$ and $H_{0}$ where the SOL prediction is
consistent with measurements of the Higgs and top-quark masses, allowing for
experimental uncertainties and taking $m_{h}=125.10\pm 0.14$ GeV [86] and
$m_{t}=173.34\pm 0.76_{\rm exp}\pm 0.30_{\rm th}$ GeV [87], where we have
added a theoretical error of 300 MeV to account for non-perturbative effects
in the top mass. The $M$–$H_{0}$ space is limited by the previously-mentioned
requirement that de-Sitter fluctuations do not affect the Higgs in its UV
minimum and by the condition that the inflaton energy density dominates over
the typical $\varphi$ contribution. These two conditions, shown in fig. 8,
correspond to

$\frac{M^{2}}{\sqrt{3}\,g_{*}M_{P}}<H_{0}<\frac{2M}{3}~{}.$ (4.10)

The condition of inflaton domination is not strictly a physical requirement,
but it allows us to simplify the calculation by treating the apeiron energy
density as a background perturbation.

The predicted values of $H_{0}$ are safely below the limit from non-
observation of primordial tensor modes but also lead to values of $r$ which
are beyond the reach of next-generation CMB experiments. Also shown in fig. 8
are the corresponding values of the reheating temperature after inflation,
given by eq. (3.30). Since the inflaton-domination condition automatically
implies $T_{\rm RH}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}M$, the final fate
of the Higgs field is insensitive to whether $h$ finds itself on the IR or UV
branch at the end of inflation. Once reheating starts, temperature
fluctuations dominate the evolution and the Higgs field will be adiabatically
driven towards the symmetric configuration as the Universe cools down. It is
also interesting that, in the full region shown in fig. 8, the reheating
temperature is compatible with the lower bound $T_{\rm RH}>2\times 10^{9}$ GeV
coming from the requirement that the right-handed neutrino explains the cosmic
baryon asymmetry through leptogenesis [88].

##  5 Higgs Naturalness from SOL

The SM Higgs bilinear parameter can be interpreted as near-critical with
respect to the electroweak phase transition because it happens to be right at
the edge of the separation between broken and unbroken phase [89]. Generically
it could take any value between $\pm M^{2}$, where $M$ is the cutoff of the
theory, associated with the scale at which the Higgs mass becomes calculable
in terms of more fundamental parameters. The near-criticality is particularly
baffling, as it bluntly violates EFT logic, leading to the well-known Higgs
naturalness problem. More than a decade ago, it was suggested that self-
organised criticality could provide an explanation of the peculiar value of
the Higgs mass [90], although no concrete realisation was put forward. By now,
several ideas have been proposed to explain the Higgs mass in terms of the
cosmological evolution [91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102,
103, 104].

The EW phase transition between the symmetric and symmetry-breaking vacua is
not immediately suited for SOL, as its quantum nature is second-order.
However, SOL may address Higgs naturalness by considering the running self-
coupling because the underlying quantum phase transition between IR and UV
phases, following from Higgs vacuum metastability, is first-order and
generates the required discontinuity between the two phases. Furthermore, the
metastability scale is naturally exponentially separated from the scale of UV
completion since it arises through the renormalisation group evolution of a
marginal parameter.

###  5.1 Phase Diagram

Consider a Higgs mass term which is scanned by the apeiron with a potential
given by

$V(\varphi,h)=\frac{M^{4}}{g_{*}^{2}}\,\omega(\varphi)-\frac{\varphi
M^{2}h^{2}}{2}+\frac{\lambda(h)\,h^{4}}{4}~{}.$ (5.1)

For simplicity, let us focus on the case in which $\lambda_{{\rm
UV}}\equiv\lambda(M/g_{*})$ is negative and $\varphi$-independent, while
$\beta_{\lambda}$ is negative at all scales so that $\lambda(h)$ vanishes at
an intermediate scale $\Lambda_{I}$. The Higgs vacuum can be in three
different phases and the corresponding apeiron potential, expanded in powers
of $\varphi$, is

$\\!\frac{V(\varphi,\langle
h\rangle)}{M^{4}}=\left\\{\\!\\!\begin{array}[]{rll}\kappa_{{\rm
EW}}\varphi+\kappa_{2}\varphi^{2}+\dots&{\rm
for}~{}\varphi\\!<\\!0&\\!\\!${{(}unbroken EW:}$~{}\langle
h\rangle\\!=\\!0)\vspace{0.2cm}\\\ \kappa_{{\rm EW}}\varphi+\kappa_{{\rm
IR}}\varphi^{2}+\dots&{\rm
for}~{}0\\!<\\!\varphi\\!<\\!\varphi_{+}&\\!\\!${{(}IR~{}phase:}$~{}\langle
h\rangle\\!=\\!v)\vspace{0.2cm}\\\ -\kappa_{0}+\kappa_{{\rm
UV}}\varphi+\kappa_{2}\varphi^{2}+\dots&{\rm
for~{}any}~{}\varphi&\\!\\!${{(}UV~{}phase:}$~{}\langle h\rangle\\!=\\!c_{{\rm
UV}}M)\end{array}\right.$ (5.2) $\kappa_{{\rm
EW}}=\frac{\omega^{\prime}(0)}{g_{*}^{2}}~{},~{}~{}~{}\kappa_{2}=\frac{\omega^{\prime\prime}(0)}{2g_{*}^{2}}~{},~{}~{}~{}\kappa_{{\rm
IR}}=\kappa_{2}-\Delta\kappa~{},~{}~{}~{}\kappa_{0}=\frac{-\lambda_{{\rm
UV}}c_{{\rm UV}}^{4}}{4}~{},~{}~{}~{}\kappa_{{\rm UV}}=\kappa_{{\rm
EW}}-\frac{c_{{\rm UV}}^{2}}{2}~{}.$ (5.3)

First consider the region $\varphi\ll(g_{*}\Lambda_{I}/M)^{2}$ and
$h\ll\Lambda_{I}$. A phase transition occurs at $\varphi=0$, which corresponds
to the critical point for EW symmetry breaking. The IR phase is characterised
by a Higgs vacuum $\langle h\rangle=v$, a Higgs mass $m_{h}$ and an additional
contribution to the apeiron potential $\Delta\kappa$ with

$v^{2}=\frac{\varphi M^{2}}{\lambda(v)}~{},~{}~{}~{}~{}m_{h}^{2}=2\varphi
M^{2}~{},~{}~{}~{}~{}\Delta\kappa=-\frac{1}{4\lambda(v)}~{}.$ (5.4)

The contribution $\Delta\kappa$ gives a discontinuity in
$\partial^{2}V/\partial\varphi^{2}$, characteristic of a second-order phase
transition. This can create a local maximum in the apeiron potential at
$\varphi=2\lambda\omega^{\prime}(\varphi)/g_{*}^{2}$. However, unless one
assumes $|\omega^{\prime}(0)|\ll 1$, contrary to the EFT power counting, the
maximum is at a generic field point on the scalar potential, not exponentially
close to the critical point $\varphi=0$. Thus the only gain in naturalness
that could be found is by assuming a non-generic potential, with the result of
pushing the SM UV cutoff only one or possibly two loop factors above the weak
scale, as noted in [98, 102]. Indeed, the model in ref. [102] ultimately
relies on anthropic selection. Alternatively, one could appeal to a periodic
potential from a new axion-like particle in order to create an additional
backreaction at the critical point, as proposed in ref. [97]. Because of these
limitations, the phase transition between broken and unbroken EW is not suited
for SOL and we will follow a different route.

The IR phase is separated from the UV phase, characterised by a high-field
Higgs vacuum $\langle h\rangle=c_{{\rm UV}}M$, by a barrier located at $h_{\rm
max}$ and of height $V_{\rm max}$ with

$h_{\rm max}^{2}=\frac{\Lambda_{I}^{2}}{\sqrt{e}}~{},~{}~{}~{}~{}V_{\rm
max}={\displaystyle\frac{-\beta_{I}\Lambda_{I}^{4}}{8e}}~{}.$ (5.5)

Here we have approximated the running coupling $\lambda$, in the neighbourhood
of the instability scale $\Lambda_{I}$, as

$\lambda(h)\approx\beta_{I}\ln\frac{h^{2}}{\Lambda_{I}^{2}}~{},~{}~{}~{}\beta_{I}\equiv\beta_{\lambda}(\Lambda_{I})~{}.$
(5.6)

When $\varphi$ approaches $(g_{*}\Lambda_{I}/M)^{2}$ and the mass parameter in
the Higgs potential becomes comparable to $g_{*}\Lambda_{I}$, the Higgs vacuum
structure in the IR phase changes into

$\displaystyle v^{2}=\frac{x_{1}\Lambda_{I}^{2}}{\sqrt{e}}~{},$
$\displaystyle~{}~{}~{}~{}{\displaystyle m_{h}^{2}=2\varphi
M^{2}+\frac{2\beta_{I}x_{1}\Lambda_{I}^{2}}{\sqrt{e}}}~{},$ (5.7)
$\displaystyle h_{\rm max}^{2}=\frac{x_{2}\Lambda_{I}^{2}}{\sqrt{e}}~{},$
$\displaystyle~{}~{}~{}~{}{\displaystyle V_{\rm
max}=\frac{-\beta_{I}x_{2}^{2}\Lambda_{I}^{4}}{8e}-\frac{x_{2}\varphi
M^{2}\Lambda_{I}^{2}}{4\sqrt{e}}}~{},~{}$ (5.8)

where $x_{1,2}$ (with $x_{1}<x_{2}$) are the two solutions of the equation

$x\ln x=\frac{\sqrt{e}\varphi M^{2}}{\beta_{I}\Lambda_{I}^{2}}~{}.$ (5.9)

In this case, we find $\Delta\kappa={\mathcal{O}}(1/\beta_{I})$. The IR phase
terminates at the value $\varphi_{+}$

$\varphi_{+}=\frac{-\beta_{I}\,e^{-\frac{3}{2}}\Lambda_{I}^{2}}{M^{2}}~{},$
(5.10)

where $x_{1}=x_{2}=1/e$. Beyond $\varphi_{+}$, the potential is a
monotonically decreasing function of $h$ and the IR vacuum no longer exists.

Figure 9: A sketch of the scalar potential $V(\varphi,\langle h\rangle)$ with
the three branches corresponding to the EW, IR and UV Higgs phases. For
$\varphi<\varphi_{+}$, two Higgs phases coexist.

The coefficients $\kappa$ in eq. (5.3) are of order one and we assume
$\kappa_{{\rm EW}}>0$ and $\kappa_{0}>|\kappa_{{\rm UV}}|,|\kappa_{2}|$. A
sketch of the apeiron potential in the three different phases is shown in fig.
9.

###  5.2 SOL Prediction

Assuming the apeiron to be in the Q2V regime, from the results presented in
sect. 2.8 and in the absence of Hubble-induced Higgs fluctuations, we would
conclude that the asymptotic FPV distribution is localised close to
$\varphi_{+}$, leading to the following prediction for the Higgs vacuum and
mass666More precisely, the FPV distribution is peaked at
${\bar{\varphi}}=\varphi_{+}-\delta$ with a width $\sigma=\delta$, where
$\delta\equiv\Big{(}\frac{\alpha}{2\beta}\Big{)}^{\frac{1}{3}}=\Big{(}\frac{\sqrt{\hbar}\,g_{*}H_{0}^{2}M_{P}}{2\pi
M^{2}f}\Big{)}^{\frac{2}{3}}~{}.$ Hence, for small $\delta$, the SOL
predictions and their corresponding statistical uncertainties are
$v=e^{-\frac{3}{4}}\Lambda_{I}-\Big{(}\frac{\delta}{-2\beta_{I}}\Big{)}^{\frac{1}{2}}M~{},~{}~{}~{}\frac{\Delta
v}{v}=\Big{(}\frac{e^{\frac{3}{2}}\delta}{-2\beta_{I}}\Big{)}^{\frac{1}{2}}\frac{M}{\Lambda_{I}}~{},~{}~{}~{}m_{h}=\Big{(}\frac{-8\beta_{I}\delta}{e^{\frac{3}{2}}}\Big{)}^{\frac{1}{4}}\sqrt{\Lambda_{I}M}~{},~{}~{}~{}\frac{\Delta
m_{h}}{m_{h}}=1~{}.$ However, the bound in eq. (3.29) is so strong that in
practice we can set $\delta=0$.

$v=e^{-\frac{3}{4}}\,\Lambda_{I}~{},~{}~{}~{}~{}m_{h}=0~{}.$ (5.11)

However, as discussed in sect. 3.6, the Hubble-induced Higgs fluctuations must
be accounted for. They will smear the endpoint by an amount proportional to
$H_{0}^{2}/M^{2}$ and hence the final prediction for $v^{2}$ at the end of
inflation will be subject to an additional uncertainty of
$\mathcal{O}(H_{0}^{2}/\lambda)$. As long as
$H_{0}^{2}\ll\lambda\Lambda_{I}^{2}$ the quantitative change in the prediction
for $v$ is negligible, although the change in the statistical uncertainty is
significant with respect to the fixed-Higgs assumption, leading to

$\Delta v\approx\frac{H_{0}^{2}}{\lambda v}~{}.$ (5.12)

The vanishing of $m_{h}$ in eq. (5.11) is signalling that the theory is at the
verge of an instability, with the vacuum about to disappear. The most
remarkable aspect of the result in eq. (5.11) is that SOL leads to a natural
hierarchy between $v$ and the cutoff energy $M$ as a consequence of the
dimensional transmutation which provides the exponentially small ratio
$v/M\sim\exp(-\lambda_{{\rm UV}}/2\beta_{\lambda})$. The relevant near-
criticality is not between the unbroken and broken EW vacua, which are not
separated by any barrier, but between two EW breaking vacua, one in the IR and
one in the UV. SOL drives the underlying parameters to a critical situation in
which the coexistence of two vacua is about to break down. For an appropriate
and generic function $\omega$ and coefficient $c_{{\rm UV}}$, the IR vacuum is
preferred. Just like in the case of the near-criticality of the Higgs quartic
discussed in sect. 4, the smoking gun of SOL is that SM parameters (in this
case the mass term in the Higgs potential) are dynamically driven to a point
where the vacuum of the theory is at a critical stage, near to collapse.

###  5.3 Reconciling the Prediction with the SM Higgs

The SOL prediction for the Higgs parameters in eq. (5.11) is obviously
untenable, since the instability scale in the SM is $\Lambda_{I}\sim
10^{10}$–$10^{12}$ GeV. A way to bring the prediction closer to reality is to
introduce new matter that makes $\beta_{\lambda}$ more negative, leading to an
earlier instability.

As simple illustrative examples consider two kinds of new vector-like fermions
with mass $M_{VL}$: a weak doublet $\chi$ and a SM singlet $\psi$. We study
two possible Yukawa interactions with the Higgs

$(a)~{}~{}\mathcal{L}=-y_{VL}\bar{\psi}\chi
H_{h}+\text{h.c.}~{},~{}~{}~{}~{}(b)~{}~{}\mathcal{L}=-y_{VL}\bar{\psi}LH_{h}+\text{h.c.}~{},$
(5.13)

where $L$ and $H_{h}$ are the lepton and Higgs doublets respectively. Case
$(b)$ corresponds to what is known as inverse seesaw [105, 106, 107], although
this model does not generate any neutrino mass. Direct searches for vector-
like fermions have led to mass limits of about 800 GeV for $\chi$ and 300 GeV
for $\psi$ [108, 109, 110]. Figure 10 shows the two-loop running of the Higgs
quartic coupling for two choices of masses and couplings for the two cases:
$(a)$ $M_{VL}=1.5$ TeV, $y_{VL}=1.5$, and $(b)$ $M_{VL}=400$ GeV, $y_{VL}=2$.
We see that the instability scale can be lowered to $\Lambda_{I}\sim 2$ TeV in
case $(a)$, or 500 GeV in case $(b)$, and that the Higgs potential barrier
separating the IR and UV vacua is reduced accordingly, as the apeiron scans
the Higgs mass.

Figure 10:  Two-loop RGE evolution of the Higgs quartic $\lambda$ (left) and
the Higgs potential for varying mass $\mu$ in the Higgs bilinear term (right)
with the addition of vector-like (VL) doublet and singlet leptons (solid
lines) or including only a VL singlet lepton (dashed line).

Figure 11:  Potential (left) with UV vacuum restored by a $|H_{h}|^{6}/M^{2}$
dimension-6 operator at a cut-off $M=700$ TeV and the corresponding bounce
solution (right) between the green and red points, calculated by solving
numerically the Euclidean equation of motion for the Higgs field $h$ as a
function of the radial coordinate $x$, as described, e.g., in ref. [111]. The
bounce action for this solution is $S_{B}=435$, corresponding to an
exponentially suppressed probability for vacuum decay, $P\sim e^{404-S_{B}}$.

The strongest obstruction to lowering $\Lambda_{I}$ comes from the constraint
that quantum tunnelling does not destabilise the Higgs vacuum today. This sets
a lower bound on the Higgs quartic
$\lambda\stackrel{{\scriptstyle>}}{{{}_{\sim}}}-0.06$ [112], which is quickly
violated in the multi-TeV scale for the running corresponding to the parameter
choices shown in fig. 10. Nonetheless, the bound can be considerably relaxed
by adding a dimension-6 operator $|H_{h}|^{6}/M^{2}$, which allows for a
sufficiently long-lived Universe even for a cutoff $M$ as large as about
$10^{6}$ GeV, see fig. 11. Or, even more effectively, the bound can be
completely evaded if new physics is added above $\Lambda_{I}$ such that
$\beta_{\lambda}$ is sufficiently small to keep the Higgs quartic in the range
$-0.06<\lambda<0$ for energies between $\Lambda_{I}$ and $M$. This situation,
which may appear artificial, could have a justification in setups where the
coupling has a quasi-fixed point in the UV and a quasi-conformal running down
to the instability scale.

In conclusion, it is conceivable that new matter in the TeV domain can lower
the SM instability scale, bringing the SOL prediction closer to the real
world. However, even in the presence of these new particles, the theory is
left with an unsatisfactory little-hierarchy problem, since the Higgs mass can
be at best in the hundreds of GeV or TeV range. This little-hierarchy problem
has a degree of severity comparable to that encountered today in low-energy
supersymmetry and is particularly reminiscent of what happens in composite
Higgs models, where the weak scale naturally wants to be of the order of the
Goldstone scale $f$, with the scale separation $v/f$ requiring some additional
dynamics. Similarly, some new ingredients have to be added to our theoretical
setup to cure the little hierarchy, and this comes at the price of a certain
degree of cancellation among parameters.

One can envisage several approaches to shift the SOL prediction in eq. (5.11)
to values slightly below the critical point, from inflaton couplings,
curvature couplings, or even trapping in a periodic potential after inflation.
Perhaps, the conceptually simplest solution is to appeal to a certain degree
of numerical coincidence (which does not imply any technical fine-tuning)
between the Hubble rate during inflation $H_{0}$ and the instability scale
$\Lambda_{I}$. In this case, as apparent from eqs. (5.11) and (5.12), the
width of the distribution can cover the breadth of the weak scale and a little
hierarchy is accommodated. Note that this does mean that we can make a sharp
prediction of the Higgs vacuum expectation value, but only that the result can
be compatible with observation, accepting a parameter coincidence at the level
of $\sim v^{2}/\Lambda_{I}^{2}$.

To summarise, the natural SOL prediction for the weak scale is
$v\sim\Lambda_{I}$, where $\Lambda_{I}$ is the instability scale in a theory
where the SM is augmented with new matter. Since $\Lambda_{I}$ is generated by
dimensional transmutation, the weak scale may be exponentially smaller than
the cutoff of the theory, thus offering the setup for a natural explanation of
the lightness of the Higgs. Our existence requires that $\Lambda_{I}$ is
separated from the weak scale by a small hierarchy, which is the SOL
incarnation of the little hierarchy problem. This discrepancy may be
accommodated by a coincidence between (finite) dS-induced Higgs fluctuations
and the instability scale.

##  6 A Small Cosmological Constant from SOL

The smallness of the cosmological constant is the greatest fine-tuning puzzle
in nature (see e.g. [113, 114, 115, 116, 117, 118, 119, 120, 121]) and
attempts to explain it have rarely been successful. Arguably the most
successful explanation is Weinberg’s anthropic approach [122] which requires a
vast landscape of dS and AdS vacua in order to render an explanation of the
smallness of the cosmological constant in our own multiverse allotment.
Recently, alternative cosmological approaches have been investigated [95, 123,
124]. In this section of the paper we present an alternative approach, which
exploits a landscape of cosmological constant values, but is based on a purely
dynamical selection process governed by SOL and does not appeal to any
anthropic criteria.

###  6.1 Theoretical Setup

We assume that the fundamental theory, in which the Standard Model is embedded
as a low-energy limit, is supersymmetric with some of its parameters being
scanned by the apeiron field. The theory possesses at least two different
vacua, called $v$ and $h$, each supporting a different quantum phase. Vacuum
$h$ preserves supersymmetry and an R-symmetry. In this phase of the theory,
the cosmological constant is zero because all auxiliary fields vanish (by
supersymmetry) and so does the superpotential (by R-symmetry). Obviously such
a configuration does not resemble our universe, and we will refer to it as the
‘hidden’ phase of the theory. We live instead on vacuum $v$, the ‘visible’
phase where both supersymmetry and R-symmetry are broken.

On vacuum $h$, the apeiron enjoys a shift symmetry that controls its
potential, making it at most exponentially shallow. This is not the case on
vacuum $v$, where the apeiron energy density changes as the field scans the
parameters of the theory. The different apeiron behaviour in the two phases
can be justified by the fact that a pseudo-Goldstone potential is very
sensitive to the vacuum structure of the QFT from which it emerges. To
illustrate this, consider a QFT not dissimilar from the SM in which the Higgs
potential has, due to RG effects, two different vacua. In the IR vacuum the
Higgs vev could be small, or even vanishing, yet in the UV vacuum it is large.
The pion mass depends on the Higgs vev, thus if the Higgs vev is
parametrically different in the two vacua so, too, is the pion mass. This
simple example serves as some justification of our underlying assumption, but
it does not imply that only a Higgs-like sector would have this required
vacuum structure. Moreover, we assume that the non-perturbative corrections to
the apeiron potential on vacuum $h$, as arising from worldsheet, brane,
gravitational or additional gauge instantons are small enough, potentially
leveraging the additional protection from supersymmetry, as to be a negligible
correction (see [125] for a discussion of the typical instanton action for
these contributions, in support of our assumption).

Next we add a second sector of the theory, which is responsible for driving an
inflationary process. We assume that this inflationary sector is sequestered,
in the sense that it is not coupled directly to other fields in the Kähler
potential or superpotential and therefore its interactions with the rest of
the theory are purely (super)gravitational. As a result, the inflationary
sector preserves the apeiron shift symmetry. In the ground state, which is
currently occupied in our Universe, the inflationary sector is both
supersymmetric and R-symmetric and therefore gives a vanishing contribution to
the vacuum energy today. However, during inflation, it is in an excited state
that breaks supersymmetry and possibly R-symmetry, uplifting the theory to dS
by an energy density $V_{I}$.

Figure 12: A sketch of the scalar potential $V(\phi)$ with the two branches
corresponding to phase $h$ (with the microscopic theory residing in the
‘hidden’ vacuum) and phase $v$ (the ‘visible’ vacuum occupied by our
Universe). The two branches are degenerate in energy at $\phi=\phi_{c}$ and
transitions are only allowed at $\phi=\phi_{T}$, where vacuum $h$ decays into
$v$.

The apeiron potential in phases $v$ and $h$, as based on our assumptions, is
sketched in fig. 12. With an appropriate choice of coordinates, we define
$\phi_{c}=0$ to be the field point where the two phases are degenerate in
energy.

Phases $v$ and $h$ do not live in isolation, but phase transitions between
them are possible. Transitions from vacuum $h$ to vacuum $v$ can occur for any
$\phi<0$, although the tunnelling rate is likely to be slow since the height
of the potential barrier separating the two vacua is characterised by some
high-energy scale of the fundamental theory. Moreover, for $\phi<\phi_{\rm
AdS}$, the tunnelling from vacuum $h$ to $v$ corresponds to a dS$\to$AdS
transition. Therefore, any bubble created in this transition will rapidly
crunch and any random-walk trajectory that ventures into the region
$\phi<\phi_{\rm AdS}$ will not contribute to the final census. Note that it
does not matter if the tunnelling is slow, since we are interested in the
asymptotic behaviour of the system after extraordinary long times: any field
configuration that belongs to phase $h$ and random walks in the region
$\phi<\phi_{\rm AdS}$ will eventually tunnel to vacuum $v$ and collapse.
Analogously, any field configuration in phase $v$ that explores the region
$\phi<\phi_{\rm AdS}$ will rapidly crunch and cannot contribute to the volume-
weighted field distribution in steady state. This is conveniently captured by
an absorbing boundary condition $P_{v}(\phi_{\rm AdS})=0$. The situation in
phase $h$ is different because of tunnelling $h\to v$. For modelling in a
simple way the phase transition, we assume that tunnelling becomes relevant
whenever $\phi<\phi_{T}$, where $\phi_{T}$ is somewhere in the range
$\phi_{\rm AdS}<\phi_{T}<0$ and we impose the condition $P_{h}(\phi_{T})=0$.

Transitions $v\to h$ are also possible in principle, but we assume that across
the entire field range the lifetime to tunnel to vacuum $h$ is so slow as to
be irrelevant, since the field in phase $v$ is more likely to slide into AdS
during its exploration of the potential than to tunnel to vacuum $h$. Note
that this cannot happen in phase $h$, which has no AdS region, justifying our
modelling assumption of having $h\to v$ transitions but not $v\to h$.

Finally we must define the conditions at the upper end of the field range,
$\phi=f$, where the theory meets its UV cutoff. In phase $h$ we impose an
absorbing boundary condition $P_{h}(f)=0$. For phase $v$, we impose a sourcing
boundary condition $P_{v}^{\prime}(f)=-k_{v}/M_{P}$, where $k_{v}$ is a
positive constant that parametrises a field flux injected from the UV theory.
This could be the result of tunnelling from an additional UV vacuum. The role
of this boundary condition is to preserve the trajectory of the distribution
close to the diffusionless solution $p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ (see
sect. 2.9) as the theory approaches the UV. See sect. 3.3 for a brief
discussion about the absence of fine-tuning for this class of boundary
conditions.

###  6.2 SOL Dynamics

Having defined the apeiron potential in the two phases and the corresponding
boundary conditions, we can now solve the FPV. The analogy with the waterfall
potential studied in sect. 3.3 is manifest and indeed the solution of the FPV
can be derived following the same procedure. We define the spectral
decomposition as in eq. (2.22).

In phase $h$ we find that the unique mode that satisfies positivity and the
boundary conditions is, up to an overall normalisation,

$p_{h}(\phi)=\sin\bigg{[}\sqrt{\frac{6(1-\lambda_{\scriptscriptstyle
H})}{\hbar}}\,\frac{2\pi(\phi-\phi_{T})}{H_{I}}\bigg{]}~{},~{}~{}~{}~{}\lambda_{\scriptscriptstyle
H}=1-\frac{\hbar\,H_{I}^{2}}{24(f-\phi_{T})^{2}}~{}.$ (6.1)

In phase $v$ we consider a monotonically increasing potential which is
degenerate in energy with the theory in phase $h$ at the field value $\phi=0$
and which scans the full range of cosmological constant values. Since
naturalness requires the cosmological constant to be as large as $M^{4}$,
where $M$ is the energy cutoff, we impose

$V_{v}(\phi_{\rm
AdS})=0~{},~{}~{}~{}V_{v}(0)=V_{I}~{},~{}~{}~{}V_{v}(f)=\frac{M^{4}}{g_{*}^{2}}~{}.$
(6.2)

We require that the potential satisfies slow roll in the full field range and
positivity at least for $\phi\stackrel{{\scriptstyle>}}{{{}_{\sim}}}0$. This
implies

$\frac{V_{v}^{3/2}}{M_{P}^{3}}<V_{v}^{\prime}<\frac{V_{v}}{M_{P}}~{}~{}~{}~{}({\rm
for}~{}\phi\stackrel{{\scriptstyle>}}{{{}_{\sim}}}0)~{}.$ (6.3)

If boundary conditions are such that the diffusionless solution
$p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ dominates over $p_{G}$ in a neighbourhood
of $\phi=0$, then the distribution will present a peak such that

$V_{v}({\bar{\phi}})=V_{I}\,\lambda_{\scriptscriptstyle
H}^{2/\xi}~{},~{}~{}~{}\sigma=\sqrt{\frac{2}{3\xi}}\,M_{P}~{},$ (6.4)

up to subleading corrections in the slow-roll parameters. As expected from eq.
(2.25), in the classical slow-roll regime, the potential height at the
location of the peak is determined by the eigenvalue.

Once the systems in the two phases are kept in ‘thermal equilibrium’ by the
phase transition occurring at $\phi_{T}$, their steady-state distributions
must have the same global expansion rate, which is measured by the eigenvalue
$\lambda_{\scriptscriptstyle H}$. Replacing in eq. (6.4) the maximum
eigenvalue determined in phase $h$, see eq. (6.1), we find that the location
of the peak is such that

$V_{v}({\bar{\phi}})=V_{I}\Big{(}1-\frac{\hbar\,H_{I}^{2}}{12\xi
f^{2}}\Big{)}~{},~{}~{}~{}\frac{\bar{\phi}}{H_{I}}=-\frac{\hbar}{4\xi}\frac{M_{P}^{2}}{f^{2}}\frac{H_{I}^{3}}{V_{v}^{\prime}(0)}~{},~{}~{}~{}\sigma=\sqrt{\frac{2}{3\xi}}\,M_{P}~{},$
(6.5)

taking $f\gg\phi_{T},H_{I}$. The first equation shows that, for
$f\gg\xi^{-1/2}H_{I}$, the peak in phase $v$ occurs at a field point such that
the potential energy matches the vacuum energy in phase $h$. The second and
third equations show that, in the classical regime
($V_{v}^{\prime}(0)>H_{I}^{3}$) and for $f>\xi^{-1/4}(H_{I}M_{P})^{1/2}$, the
displacement of the peak from the critical point at $\phi=0$ is smaller than
the width. Both conditions are satisfied in our theory. This result is
remarkable, as $\phi=0$ is not a special point when phase $v$ is considered in
isolation and there is no communication between the two phases around the
critical point.

###  6.3 SOL Prediction

The SOL mechanism ensures that, during inflation, the vacuum occupied by our
Universe is almost exactly degenerate in energy with the supersymmetric and
R-symmetric vacuum which has vanishing cosmological constant. Once inflation
is over, the energy of both vacua will be reduced by the same amount $V_{I}$
and therefore the degeneracy will be preserved. After inflation, our Universe
will undergo a reheating process, with various subsequent thermal phase
transitions (e.g. Higgs or quark-hadron transitions) which will alter its
vacuum energy. However, once it reaches a temperature equal to about $H_{I}$,
our Universe will return to a state with a vacuum energy almost identical to
the one it occupied during inflation, aside from the uplifting contribution
from $V_{I}$. Since this contribution is identical in the two phases, we
conclude that the cosmological constant in our Universe today remembers the
SOL condition of being degenerate with the supersymmetric and R-symmetric
vacuum, as long as $H_{I}$ is smaller than $10^{-3}$ eV. This condition is
necessary for any residual difference between the vacuum energies of the dS
state and the present state of the Universe to be smaller than the observed
value of the cosmological constant. As we will see in the following, this
condition is satisfied in our theory.

The prediction for the cosmological constant in phase $v$ today is obtained by
subtracting the inflationary contribution $V_{I}$ from the vacuum energy
$V_{v}(\phi)$ for a field value in the range
$|\phi-{\bar{\phi}}|<n_{\sigma}\sigma$, where $n_{\sigma}$ is the number of
standard deviations that we are willing to tolerate in the statistical
distribution. Since, as shown before, $\sigma>|{\bar{\phi}}|$, the SOL
prediction is dominated by the statistical uncertainty and is given by

$|\Lambda_{\rm
CC}^{4}|<\frac{V_{v}^{\prime}(0)M_{P}}{c_{\xi}}~{},~{}~{}~{}c_{\xi}=\sqrt{\frac{3\xi}{2n_{\sigma}^{2}}}~{}.$
(6.6)

The parameter $c_{\xi}$ is a measure of the intrinsic statistical nature of
the SOL prediction and combines the probabilistic uncertainty ($n_{\sigma}$)
with the uncertainty associated with the way we define the statistical
ensemble over which we average the patches in the multiverse ($\xi$). In the
following, we will assume that the bound in eq. (6.6) is saturated and we will
use it as an equality.

Combining eq. (6.6) with the CbQ condition in eq. (6.3), we find upper bounds
on the Hubble rate during inflation and on the reheating temperature after
inflation

$H_{I}<c_{\xi}^{1/3}(\Lambda_{\rm CC}^{4}/M_{P})^{1/3}\approx
c_{\xi}^{1/3}\,2\times 10^{-13}~{}{\rm eV}~{},~{}~{}~{}T_{\rm
RH}<c_{\xi}^{1/6}(\Lambda_{\rm CC}^{2}M_{P})^{1/3}\approx
c_{\xi}^{1/6}\,25~{}{\rm MeV}~{},$ (6.7)

where we have taken $g_{R}=10.75$ as the number of degrees of freedom in the
thermal bath. The first bound in (6.7) shows that, as previously advertised,
the Hubble rate amply satisfies the requirement $H_{I}<\Lambda_{\rm
CC}^{1/4}$. It also shows the remarkable fact that the field exchange between
phase $v$ and $h$ occurs at a point $\phi_{T}$ in which the vacuum energy
density is about 40 orders of magnitude different from the one corresponding
to the actual cosmological constant of our Universe. For an observer who lives
in phase $v$, like us, the field value where the cosmological constant vanish
is truly unexceptional.

The second bound in (6.7) shows that the reheating temperature is relatively
low, but sufficiently high to allow for ordinary nucleosynthesis to proceed
successfully. Indeed, nucleosynthesis only requires that $T_{\rm RH}$ be
larger than about 4 MeV [126, 127, 128, 129, 130]. This leads only to the very
weak constraint $c_{\xi}>10^{-5}$.

The timescale for the system to reach the steady-state configuration is
determined by the evolution in phase $h$, where the potential is in a quantum
regime. The timescale can be estimated by recalling that, due to the field
random walk, the spread of the distribution evolves as
$\sigma^{2}=H^{3}t/(2\pi)^{2}$. Since the asymptotic distribution in eq. (6.1)
has a spread $\langle\phi^{2}\rangle\approx f^{2}/\pi^{2}$, the number of
$e$-folds necessary to reach the steady state is about $N\approx
f^{2}/H_{I}^{2}$. Since $f$ is super-Planckian, $N$ exceeds the bound in eq.
(2.4) and inflation must be eternal. The same result can be obtained by
estimating the characteristic evolution timescale as $\Delta t\approx
1/(3H_{I}\Delta\lambda_{\scriptscriptstyle H})$, where
$\Delta\lambda_{\scriptscriptstyle H}$ is the typical splitting between
eigenvalues. In phase $h$, we find that two contiguous eigenvalues differ by
${\lambda_{\scriptscriptstyle H}}_{n}-{\lambda_{\scriptscriptstyle
H}}_{n+1}=(2n+1)\hbar H_{I}^{2}/(24f^{2})$, leading to the same estimate on
the number of required $e$-folds, $N\approx f^{2}/H_{I}^{2}$.

After inflation is over in our patch, the random-walk process terminates and
the apeiron will simply roll down its potential by an amount

$\delta\phi=-\frac{V_{v}^{\prime}(0)}{3H_{\rm now}^{2}}=-c_{\xi}M_{P}~{}.$
(6.8)

As long as $c_{\xi}<1/\sqrt{n_{\sigma}}$, the shift is within the statistical
uncertainty ($|\delta\phi|<\sigma$) and therefore does not affect the SOL
prediction.

Although post-inflationary dynamics does not pose any significant constraint
on the theory, it offers the possibility of an observational test of the
mechanism since it modifies the dark energy EoS, see eq. (3.26),

$w=-1+\left(\frac{V_{v}^{\prime}(0)}{3H_{\rm now}\Lambda_{\rm
CC}^{2}}\right)^{2}=-1+\frac{c_{\xi}^{2}}{3}~{}.$ (6.9)

Modifications from $w\\!=\\!-1$ (as predicted by an unchanging cosmological
constant) are expected from SOL to be positive and possibly sizeable. However,
the prediction is blurred by the statistical nature of SOL, as encoded in the
parameter $c_{\xi}$, and the effect could be unobservably small without
conflict with nucleosynthesis. Nevertheless, the prediction is robust in the
sense that it does not depend on model-dependent features such as the
functional form of the apeiron potential. Present fits on cosmological
parameters give [131] $w=-1.028\pm 0.031$ and therefore $w+1<0.02$ at 95% CL,
although these results depend on prior modelling assumptions. Measurements of
dark energy parameters are expected to become much more precise with future
weak lensing data, galaxy surveys, CMB and large-scale structure observations.

The results presented so far do not depend on the explicit form of
$V_{v}(\phi)$ but only on its local properties around the critical point
$\phi=0$. However, global information about the potential is necessary to
determine the maximum cutoff energy $M$ up to which the classical slow-rolling
conditions in eq. (6.3) are satisfied. A simple linear potential can reproduce
the observed value of the cosmological constant only assuming an energy cutoff
around some tens of MeV at most. This is already a remarkable result since SOL
could justify a hierarchy $M^{4}/\Lambda_{\rm CC}^{4}$ of 40 orders of
magnitude, although it falls short of explaining the remaining 80 orders of
magnitudes that are needed to have $M$ near the Planck scale. However, it is
not difficult to think of potentials beyond linear that can extend the
validity of classical slow roll up to very large values of $M$.

A simple example is a potential in phase $v$ which is exponential in $\phi$.
When parameters are chosen to satisfy eq. (6.2) and to reproduce the observed
value of the cosmological constant as in eq. (6.6), the exponential potential
takes the form

$V_{v}(\phi)=\frac{c_{\xi}\Lambda_{\rm
CC}^{4}}{\kappa}\left(e^{\frac{\kappa\phi}{M_{P}}}-1\right)+V_{I}~{},~{}~{}~{}\frac{f}{M_{P}}=\ln\left(\frac{\kappa
M^{4}}{g_{*}^{2}c_{\xi}\Lambda_{\rm CC}^{4}}+1\right)^{1/\kappa}~{},$ (6.10)

where $\kappa$ is a free constant. With this potential, the apeiron scans a
range of vacuum energies encompassing any positive value of the cosmological
constant in the present Universe, up to the cutoff $M$. The conditions for
classical slow-roll in eq. (6.3) are satisfied for positive $\phi$, if we
require

$\frac{M^{2}}{g_{*}M_{P}^{2}}<\kappa<1~{}.$ (6.11)

For $M\approx M_{P}$ the scale $f$ is moderately super-Planckian, with
$f/M_{P}\approx 280$. If we reduce the cutoff, the range of $f$ expands: for
instance, for $M=10^{16}$ GeV, we find
$10^{2}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}f/M_{P}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}10^{7}$.

The SOL mechanism, leading to field localisation at a point in phase $v$
nearly degenerate with phase $h$, works for the exponential potential in full
analogy with the case of the linear potential that was studied in sect. 3.3. A
difference is that the exponential potential is in the classical regime above
the peak at $\phi=0$, but not below. However, this does not change the nature
of the solution since the existence of the peak only requires classical slow
roll for $\phi>0$.

In principle, with knowledge of the potential in the full field range, we
could calculate the relative probability for a Universe to live in phase $v$
or $h$. However, the result depends on arbitrary boundary conditions at the
upper endpoint and therefore a definite prediction requires knowledge of the
UV completion.

##  7 Summary and Conclusions

Seemingly we understand perturbations about our vacuum state, at least at
currently accessible wavelengths, very well. For example, we can predict Higgs
boson production rates at hadron colliders with N3LO accuracy [132] or compute
the muon anomalous magnetic moment with precision at the level of a few
$10^{-7}$ [133, 134]. Yet, the structure of the vacuum itself still hides many
mysteries. We have, however, had some hints about the answers that may lurk
within the vacuum. For instance, the structure of the physical vacuum may be
intimately related to the structure of physics in the deep UV, whatever the
true nature of quantum gravity may be [135]. At more modest wavelengths, the
measurement of IR parameters, such as the Higgs and top quark masses, have
revealed that our current working hypothesis for describing perturbations
about the vacuum (i.e. the Standard Model) itself has a nontrivial vacuum
structure with a deeper energy-density vacuum lying far from our current
state. Clues such as these have led to a modern-day incarnation akin to the
days of the æther: when asking UV-motivated questions we increasingly seek
answers from the vacuum. In this context, we have proposed a new concept that
links the properties of the vacuum to physical observables: self-organised
localisation. SOL is a phenomenon that can drive physical parameters, measured
at low energy, towards values determined by the global vacuum structure of the
theory, even when this structure is inaccessible to direct low-energy
observations.

The first basic ingredient of SOL is the hypothesis that one, or more,
parameters of the microscopic theory (which can be identified with the SM or
any of its field-theoretical extensions) are promoted to dynamical variables
dependent on one, or more, scalar fields $\phi$. Generically referring to
these parameters as $\mu$, the hypothesis is $\mu\to\mu(\phi)$, where the
function $\mu(\phi)$ spans the full natural range of the parameter as $\phi$
varies in its domain. During inflation, the light scalar fields $\phi$ (where
by ‘light’ we mean relative to the expansion rate) are subject to large
stochastic fluctuations and effectively scan the physical parameters $\mu$,
exploring the global vacuum structure of the theory, even reaching locales far
from any local minima.

The second ingredient is the hypothesis that the microscopic theory has a non-
trivial vacuum structure as a function of $\mu$ with two, or more, quantum
phases possibly coexisting for the same value of $\mu$ and terminating at some
critical value $\mu_{c}$. Note that we refer to coexistence of phases as to
the situation in which the theory has different local minima for the same
$\mu$, but we do not necessarily require energy degeneracy among minima.
Therefore, some of the vacua will be metastable, but this does not preclude
them from being populated during inflation, even for timescales much longer
than the tunnelling time, because the volume expansion prevents nucleated
bubbles of true vacuum to percolate and take over the entire space.

With these two ingredients and some appropriate hypotheses on the $\phi$
potential, we have observed the emergence of a rather universal phenomenon
during inflation: points of criticality, where two phases are on the verge of
ceasing to coexist, can act as global attractors for the scalar fields that
govern the phase transition, with the volume-weighted field distribution
typically becoming highly localised at these points. We have referred to this
phenomenon as SOL, since the scalar field localises itself around a value
$\bar{\phi}$, intimately related to the critical point. As a result, the
physical parameter $\mu$ is dynamically selected to take the value
$\mu(\bar{\phi})$ across the entire Universe, even if that value does not
correspond to any enhanced symmetry and can appear to violate ordinary EFT
reasoning. Due to SOL’s probabilistic nature, the prediction
$\mu=\mu(\bar{\phi})$ is affected by an intrinsic uncertainty related to the
width of the $\phi$ distribution around the localisation point, although in
practice this does not significantly limit the predictive power.

The SOL condition for criticality can be realised in several different ways.
The most straightforward example is the one in which varying $\mu(\phi)$
across a critical value $\mu_{c}$ induces a first-order phase transition in
the microscopic theory. The variation of the background field $\phi$ is
analogous to changing the magnetic field around a ferromagnet. As $\phi$ is
varied, it may cross a point at which the two vacua of the microscopic theory
are degenerate. In the magnet analogy, this would be at $|B|=0$. Continuing
further, the vacuum degeneracy will be broken such that one now resides in a
metastable higher-energy vacuum until it reaches the point where decay to the
lower vacuum state is inevitable. This would be the critical magnetic field
strength for a ferromagnet, beyond which all spins are guaranteed to flip to
the lower-energy state. If the region of coexistence of the two vacua is small
compared to the total field range and if the phase transition is sufficiently
strong, the critical point corresponds to a local maximum of the vacuum energy
density and then, in Landau’s terminology, one will observe a first-order
discontinuity of the gradient of the potential. However, resolving this cusp
through the underlying microscopic physics exposes overlapping branches of
metastable vacua, and the highest energy-density point will correspond to the
field value at which the excited vacuum terminates. Ultimately, this field
value determines the localisation point $\bar{\phi}$ and therefore links the
prediction $\mu=\mu(\bar{\phi})$ to the critical behaviour of the microscopic
theory.

The case we have just illustrated is not, however, the only way for SOL to
make use of criticality. We have studied other cases in which criticality
could be a feature of a phase different from the one occupied by our Universe,
thus remaining invisible to parameter variations in the local minimum, or it
could emerge as a condition for degeneracy of the vacuum energy of different
phases. Future studies may reveal other contexts in which SOL could relate
field localisation to critical phenomena.

A crucial aspect for any of the quantitative results presented in this paper
is the existence of steady-state solutions emerging from the stochastic
equation that governs the time evolution of the volume-weighted field
distribution. Much as thermodynamic systems reach thermodynamic equilibrium in
a laboratory, these steady-state distributions give a fairly robust
description of the properties of the system, independently of any initial
condition of the Universe.

The question of how these steady-state distributions are reached dynamically
is important to determine the timescale for the system to attain equilibrium.
We have identified three relevant regimes of the $\phi$ potential, which are
qualitatively distinct in their properties. When the potential is sufficiently
steep, and the system is in what we call the C (Classical) regime, the field
distribution preferentially evolves towards configurations with lower energy
density, although the ultimate behaviour depends on the form of the boundary
conditions. In a second regime, termed QV (Quantum+Volume), quantum effects
combined with volume expansion effects conspire to give large fluctuations to
classical evolution and the volume-weighted distribution of the scalar field
preferentially evolves towards configurations with high energy density. In the
case of the multi-valued potentials characteristic of critical phenomena, this
leads to a distribution with a peak of a specified width and position near the
top of the metastable branch just before it terminates. In a third parameter
regime, termed Q2V, the second-order quantum effects combined with volume
effects give an even stronger preference for high values of the potential
energy, leading to a final volume-weighted distribution sharply peaked at the
critical point that marks the end of coexistence between the two phases.

The QV and Q2V regimes show a natural attraction towards the critical points,
whenever the phase transition is sufficiently strong to create a local maximum
of the $\phi$ potential. However, in QV and Q2V regimes, the characteristic
timescale for reaching the steady state is so long that the dynamics needs to
be supported by eternal inflation. Although it is possible to devise examples
of SOL with shorter relaxation timescales, eternal inflation was an ingredient
of all applications considered in this paper. By choosing parameters in the C
regime, there could be physical applications where SOL occurs during non-
eternal inflation, although additional field-theory ingredients such as post-
inflationary mechanisms would be necessary to prevent the field from rolling
too fast since reheating. On the contrary, eternal inflation appears
relatively generic in inflationary models [25] and, despite the onset of the
measure problem and the associated loss of calculability, seems well
motivated. Thus, to a large extent, eternal inflation appears to be a natural
setting for SOL.

Our exploration shows that SOL is a rich and versatile phenomenon, which can
take different forms and adapt to a variety of applications in cosmology.
Indeed, SOL may have profound physical implications for our understanding of
the vacuum. For example, if the background of a light scalar field, here
called apeiron, controls the values of Standard Model parameters, then we
could expect those parameters to take values close to critical points at which
a metastable vacuum is about to collapse. SOL implies a prevalence, in the
multiverse, of vacua on the cusp between stability and metastability with
respect to parameter variations. In this paper we have explored the relevance
of SOL to three key questions surrounding the Standard Model vacuum.

#### 1\. Near-criticality of the Higgs self-coupling

SOL is ideally suited to explain the observation that the measured values of
the Higgs and top masses place the SM in a peculiarly balanced metastable
state. As the apeiron scans the SM parameters, the theory develops two Higgs
vacua. One is the ordinary low-field vacuum, which persists as long the Higgs
self-coupling, renormalised at low energies, is positive. The other is a high-
field vacuum, which starts to exist as soon as the Higgs self-coupling,
renormalised at high energies, becomes negative. In an intermediate region of
parameters, the two vacua coexist and SOL predicts that this is the most
likely outcome after a long inflationary period, in perfect agreement with
observations. The mechanism is detailed in sect. 4.

#### 2\. Higgs naturalness

Since the electroweak quantum phase transition occurs smoothly as the Higgs
mass is varied, the problem of Higgs naturalness seems less suited for SOL.
However, the SM vacuum structure offers an intriguing possibility for
exploiting vacuum coexistence. As the Higgs mass-squared parameter is scanned
by the apeiron and its value becomes more negative, the vacuum moves to ever
greater field values. However, once it crosses a critical point, determined
from the renormalisation group evolution to be the scale at which the
effective quartic coupling passes through zero, the Higgs vacuum becomes
metastable with respect to a deeper high-field vacuum and eventually vanishes
entirely. SOL predicts that the apeiron localises during inflation in
correspondence with the critical value for the Higgs mass. The mechanism
dynamically creates an exponential hierarchy between the UV cutoff energy and
the Higgs mass. This is due to dimensional transmutation which generates a
natural separation between the cutoff and the instability scale where the
quartic coupling vanishes.

Although, on the back of dimensional transmutation, SOL creates a natural mass
hierarchy, the result cannot be directly applied to the SM. New matter must be
added to the SM in order to bring the instability scale closer to the
electroweak scale. Moreover, the mechanism suffers from a little hierarchy
problem which can be accommodated only at the price of a coincidence between
the Hubble rate and the instability scale, or by adding further dynamical
ingredients to the theory. This mechanism is discussed in sect. 5.

#### 3\. The cosmological constant

The vanishing of the cosmological constant does not appear to correspond to a
phase transition and cannot be immediately linked to a question of vacuum
coexistence. However, if the Standard Model is embedded within an overarching
framework, the situation may be propitious for SOL. Consider a supergravity
theory containing multiple moduli fields, of which the present vacuum, with
the Standard Model low-energy field content, is just one of multiple
possibilities. If this theory also contains a supersymmetric and R-symmetric
vacuum, then that vacuum will have vanishing energy density by dint of the
symmetries. In sect. 6 we explored how SOL could transfer information from the
hidden supersymmetric phase into our Universe.

Due to the interplay between the two phases, the volume-weighted distribution
is peaked at precisely the degeneracy point between the vacua, despite that
this is far from where the tunnelling occurs. Physically, the steady states in
the two phases must inflate at the same rate, hence the system is dynamically
attracted towards the point where the vacuum energies are degenerate. The
result is that the inflating multiverse tunes physical parameters towards
vacuum degeneracy between two different phases which, locally, are entirely
unaware of one another. Thanks to the interplay with supersymmetry and
R-symmetry, SOL has self-tuned the cosmological constant to be hierarchically
smaller than what one expects generically in our present vacuum, where
supersymmetry and R-symmetry are badly broken.

All these examples show that SOL is a sufficiently generic phenomenon that one
can imagine to find other physical applications, beyond those considered here.
Critical points are common in particle-physics theories and SOL can provide a
dynamical explanation for a cosmological selection of near-critical behaviour.
Moreover, the example of the cosmological constant studied in sect. 6 exhibits
a feature qualitatively different from those of the Higgs examples in sects. 4
and 5. The key ingredient is the presence of two distinct phases of the
theory. SOL singles out a specific value of a fundamental parameter in one
phase, although that parameter is not special nor corresponds to any enhanced
symmetry within that phase. The secret of the mechanism is that a value of a
parameter which looks completely generic in one phase, may be special in the
other phase. By relating parameters in the two phases, SOL makes a physical
prediction which may appear inexplicable to observers who are confined to live
in one vacuum and remain unaware of the existence of another phase of the
theory, which is hidden to them.

It is interesting to draw an analogy between the steady-state configurations
of the FPV solutions and thermal equilibrium in a gas. Microscopic properties
of atoms in a gas (such as individual energies) change incessantly but, in
thermal equilibrium, some macroscopic thermodynamical properties (such as
temperature) remain constant and characterise the behaviour of the global
system. Similarly, the Langevin trajectories describing the field value in a
single patch of the Universe keep on evolving incessantly but, when steady-
state is reached, the FPV solution identifies the field distribution that
characterise the global behaviour in the multiverse. For atoms in a gas, it
would be impossible to derive their microscopic properties from fundamental
principles by studying a single atom in isolation: any individual energy is as
good as any other. Only when we consider the atom as part of a statistical
ensemble, we can make probabilistic predictions on microscopic quantities
based on macroscopic equilibrium properties. Similarly, attempts to derive SM
parameters from fundamental principles may be futile: any parameter value is
as good as any other. Only when we consider our Universe as part of a
statistical ensemble, we may be able to make probabilistic predictions on SM
parameters across the multiverse.

The analogy with a gas in thermal equilibrium takes a more concrete form when
we think of the background Hubble rate as the Gibbons-Hawking temperature in
de Sitter space. Particularly revealing is the SOL application to the
cosmological constant, which shows how the physical result is linked to the
macroscopic behaviour of the system. A single Langevin trajectory (or, in
other words, a single universe) does not see anything special in the critical
field value where the two phases are degenerate, just like a single atom in a
gas does not see anything special in any particular energy value. And yet,
just like, when two boxes of gas are put in thermal contact, atoms arrange
themselves statistically to produce a single temperature of the system, so
field trajectories arrange themselves to reach a steady-state configuration in
which the expansion rate is roughly the same in both phases. Since the
expansion rate is proportional to the energy density, the energy density in
the visible phase becomes nearly equal to the one in the hidden supersymmetric
phase, even if the visible sector does not know anything about supersymmetry.
According to SOL, the secret of the smallness of the cosmological constant in
our Universe lies in the macroscopic statistical properties of the multiverse.

In our approach, the multiverse is not merely a setup suitable to anthropic
selection. Instead, we propose to interpret the multiverse as a quantum
statistical system, in which critical phenomena may play an essential role in
the selection process that determines physical parameters.

#### Open questions

A generic feature of SOL is that its probabilistic predictions refer to the
end of inflation and not to the present epoch. Therefore, they can be
modified, or even erased, by the subsequent thermal evolution of the Universe.
This can be a virtue or a curse. In some cases, it can be beneficial because
it allows for a certain flexibility of the SOL predictions. For instance,
particle masses can be subjected to ‘AdS thermal’ corrections proportional to
the Hubble rate that evaporate at the end of inflation, thus affecting the
original SOL prediction. In most cases, however, the requirement that the
scalar field has not evolved significantly between reheating and the present
day provides a strong constraint on the theory parameters, forcing the scalar
field potential to be extraordinarily shallow and the dynamics to enter a
regime of eternal inflation (see sect. 3.5).

Besides these aspects related to model building, a more important concern is
that the viability of SOL still faces enormous challenges in the context of
quantum cosmology. Some of the open issues have been commented upon in sect.
2.2 and we summarise them here.

One issue is linked to our use of the volume-weighted Fokker-Planck equation,
whose formulation requires a choice of space-time foliation, and therefore of
gauge. The formation of peaks in the vicinity of critical points (which is a
central element for SOL) is a gauge-independent result. However, the widths of
the distributions (i.e. the degree of localisation) are not. Their gauge
dependence is not lethal for the success of SOL, unless one considers the
somehow contrived case of $e$-folding gauge ($\xi=0$), which corresponds to a
time slicing in which every patch undergoes identical expansion.

Another open issue is related to our hypothesis that inflation is eternal.
This brings in the ‘measure problem’, which is endemic to any probabilistic
interpretation based on eternal inflation. Since eternal inflation is
characterised by having an infinite reheating surface, defining probabilities
for observables at the end of inflation is a task plagued with infinities.
Although various proposals for dealing with this problem have been put forward
in the literature (see e.g. refs. [24, 28] for reviews), the ‘measure problem’
remains a serious concern. This raises questions about quantitative results
concerning, in particular, the predicted width of the field distribution,
which is not only a gauge-dependent quantity, but also measure-dependent. On
the other hand, we see that the qualitative behaviour of SOL, which is the
central result of this work, is gauge-independent and seems to follow
automatically in a volume-based measure. Thus we expect that if the measure
problem were to be understood better, the qualitative physics of SOL should
remain, although quantitative aspects may change.

When in the QV or Q2V regimes, the completion of the dynamical process that
leads to SOL requires a number of inflationary $e$-folds much larger than
$M_{P}^{2}/H_{0}^{2}$. This is not in conflict with the bound based on the
maximum entropy available in de Sitter space [45, 46, 47, 48], which applies
only to non-eternal inflation. However, it raises concerns [49, 50] about the
validity of the semi-classical approach that we have followed here. As
discussed in sect. 2.2, it is not clear how problematic this issue is for the
generic predictions of SOL.

Finally, SOL requires super-Planckian field displacements and likely violates
both the de Sitter Conjecture and the Distance Conjecture [58, 59], placing
most of its applications firmly in the swampland. If one feels the need, the
theory could possibly be rescued from the swampland by employing mechanisms
that make these super-Planckian displacements only a low-energy delusion [75,
76, 77, 78].

#### Experimental tests

In spite of these profound conceptual difficulties in the interpretation of
predictions for physical observables, we believe that SOL offers a new route
towards answering fundamental questions concerning the physical vacuum. A
particularly appealing aspect of SOL is that, despite being a phenomenon
apparently confined to an abstract reality, as it takes place in a multiverse
populated by eternal inflation and whose expanse is largely beyond our causal
contact, it has instead a clear smoking gun amenable to experimental scrutiny.

The smoking gun of SOL is the prediction of coexistence of different vacua,
with the Universe we inhabit being in a metastable state at the edge of
collapse with respect to variations of some of the fundamental physical
parameter. The experimental test does not rely on waiting for the Universe to
collapse. Instead, it relies on making precise experimental measurements of
fundamental parameters and performing theoretical extrapolations to study the
vacuum structure as those parameters are altered. The case of near-criticality
of the Higgs self-coupling is a perfect example. The precise experimental
determination of the Higgs mass, top mass, and gauge coupling constants
allowed a theoretical extrapolation that has revealed a striking peculiarity
of the SM vacuum structure. That peculiarity could be interpreted as a hint
for SOL. Similarly, the approach followed in sect. 5 to address Higgs
naturalness requires the existence of new particles. If these particles were
discovered and their couplings with the Higgs precisely measured, one could
make a theoretical extrapolation to higher energies. Finding evidence for
near-criticality of the vacuum with respect to variations of the Higgs mass-
squared parameter would provide a clue in favour of SOL.

In some cases, it may be practically challenging to infer the existence of a
different phase from experimental measurements and theoretical extrapolation.
This is particularly true in cases where the visible phase (i.e. the one
occupied by our Universe) does not exhibit any sign of criticality, which is a
feature only of a hidden phase, unaccessible to us. A good example is the SOL
explanation of the cosmological constant proposed in sect. 6, where it is hard
to imagine how one could infer from measurements the existence of a hidden
supersymmetric phase with nearly the same vacuum energy as our Universe. In
these cases, one has to devise alternative observational tests, which depend
on the particular SOL implementation. For the theory presented in sect. 6, an
experimental signal is provided by the time-dependence of the cosmological
constant due to the residual motion of the apeiron. This effect leads to a
modification of the dark-energy equation of state which could be within reach
of the next generation of experiments in observational cosmology.

Decades of studying symmetries in quantum field theories in Minkowski space
have yielded enormous progress in fundamental physics, yet stubbornly refuse
to offer empirically viable and theoretically appealing answers to many
fundamental questions concerning the microscopic world beyond our present
reach. New vistas and fresh opportunities arise when we look beyond this
paradigm towards the interplay between cosmological evolution and the
structure of the vacuum. In this work we have shown that SOL may select low-
energy parameters to be naturally close to critical points as a result of the
multi-valued vacuum structure. We are accustomed to believing that points of
enhanced symmetry are preferred because they are stable under quantum
corrections. On the contrary, SOL’s assertion is that inflation is likely to
deliver a Universe which is at the edge of collapse with respect to some
parameter variation.

## Acknowledgments

We thank Tim Cohen, David Curtin, Gia Dvali, Michael Geller, Daniel Green,
Yonit Hochberg, Eric Kuflik, Marko Simonović, and Daniele Teresi for useful
conversations and particularly Andrei Linde for detailed comments on the first
version of this paper. T.Y. is supported by a Branco Weiss Society in Science
Fellowship and partially by the UK STFC via the grant ST/P000681/1.

## Appendix

##  Appendix A The Volume-Weighted Fokker-Planck Equation

###  A.1 Structure of the Equation

The volume-weighted Fokker-Planck equation (FPV) (for a review and a list of
references, see [24]) describes the time evolution of the distribution
$P(\phi,t)$, defined as the probability for a scalar field to take the value
$\phi$ within a spatial patch, weighted by the corresponding volume-expansion
factor in an inflating universe with Hubble rate $H$. It is the result of the
average over a large number of random-walk trajectories of the classical field
$\phi$ governed by a Langevin equation with Gaussian random noise. Expressed
in terms of the proper time $t$, the FPV is

$\frac{\partial P(\phi,t)}{\partial
t}=\frac{\partial}{\partial\phi}\left[\frac{\hbar
H^{a}}{8\pi^{2}}\frac{\partial}{\partial\phi}\Big{(}H^{3-a}P(\phi,t)\Big{)}+\frac{dV}{d\phi}\frac{P(\phi,t)}{3H}\right]+3HP(\phi,t)~{}.$
(A.1)

The first term in the right-hand side is called the diffusion term in
statistical mechanics and quantum term in cosmology, as it describes the
effect of the random quantum fluctuations of the scalar field in de Sitter
space. The power of $\hbar$ is required according to the dimensional analysis
of ref. [78], since $H$ has units of mass and $\phi$ units of scale. The
second term is named drift term in statistical mechanics and classical term in
cosmology, as it describes the field’s tendency to roll down the classical
potential $V=3H^{2}M_{P}^{2}$. The exponent $a$ parametrises the ambiguity in
translating the Langevin equation into the FPV, with $a=0$ known as Ito
ordering and $a=3/2$ as Stratonovich ordering. In the following, we will
choose $a=0$, but any choice is perfectly equivalent since the difference
amounts to corrections ${\mathcal{O}}(H^{2}/\pi^{2}M_{P}^{2})$ to the drift
term, which are negligible in the regime where quantum-gravity effects can be
ignored ($V\ll M_{P}^{4}$). The third term, which will be called volume term,
is what distinguishes FPV from the ordinary Fokker-Planck equation (FP) (for a
review, see [136]) and it describes the effect of the expansion of physical
space in de Sitter. Because of this term, FPV does not have a conserved
current and its solutions do not retain their normalisations when integrated
over $\phi$, unlike the ordinary FP. Hence, $P(\phi,t)$ does not describe a
probability density, but the volume-weighted distribution of $\phi$
configurations at time $t$.

As we are interested in the $\phi$ dynamics on a de Sitter background, we
split the scalar potential into a constant background value $V_{0}$ and a
field dependent part defining

$V(\phi)=V_{0}\left[1+\frac{2v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)}{S_{\\!{{\rm
dS}}}}\right]~{},~{}~{}~{}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\equiv\frac{\phi}{M_{P}}~{},~{}~{}~{}h({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\equiv\frac{H(\phi)}{H_{0}}=\sqrt{1+\frac{2v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)}{S_{\\!{{\rm
dS}}}}}~{},$ (A.2)
$H_{0}^{2}\equiv\frac{V_{0}}{3M_{P}^{2}}~{},~{}~{}~{}t_{S}\equiv\frac{S_{\\!{{\rm
dS}}}}{H_{0}}~{},~{}~{}~{}S_{\\!{{\rm
dS}}}\equiv\frac{8\pi^{2}M_{P}^{2}}{\hbar\,H_{0}^{2}}~{},~{}~{}~{}\tau\equiv\frac{t}{t_{S}}~{},$
(A.3)

where $M_{P}$ is the reduced Planck mass and $S_{\\!{{\rm dS}}}$ is the
maximum entropy of de Sitter space. We note that $S_{\\!{{\rm dS}}}^{-1}=\hbar
V_{0}/(24\pi^{2}M_{P}^{4})$, showing that this quantity measures the size of
quantum-gravity loops and thus our analysis can be valid only in the regime
$S_{\\!{{\rm dS}}}\gg 1$. The parameter $t_{S}$ is the time scale associated
with de Sitter entropy. With the above definitions, the FPV becomes

$\frac{\partial P}{\partial\tau}=\left[(h^{3}P)^{\prime}+2S_{\\!{{\rm
dS}}}h^{\prime}P\right]^{\prime}+3S_{\\!{{\rm dS}}}hP~{},$ (A.4)

where primes denote derivatives with respect to
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$. As already mentioned,
it is always possible to pull $h^{3}$ outside the derivatives in the diffusion
term since the difference only gives small corrections to the drift and volume
terms suppressed by ${\mathcal{O}}(H^{2}/\pi^{2}M_{P}^{2})$.

###  A.2 Time Reparametrisation

To study the effect of different time parametrisations, let us consider the
family of time coordinates $t_{\xi}$ such that

$\frac{dt_{\xi}}{dt}=\Big{(}\frac{H}{H_{0}}\Big{)}^{1-\xi}~{},$ (A.5)

where $\xi$, taken to vary between 0 and 1, will be called the gauge
parameter. After the time-coordinate transformation in eq. (A.5) and the
redefinition $P\to(H/H_{0})^{\xi-1}P$, the FPV in eq. (A.1) becomes

$\frac{\partial P(\phi,t_{\xi})}{H_{0}^{1-\xi}\partial
t_{\xi}}=\frac{\partial}{\partial\phi}\left[\frac{\hbar
H^{a}}{8\pi^{2}}\frac{\partial}{\partial\phi}\Big{(}H^{2+\xi-a}P(\phi,t_{\xi})\Big{)}+\frac{dV}{d\phi}\frac{P(\phi,t_{\xi})}{3H^{2-\xi}}\right]+3H^{\xi}P(\phi,t_{\xi})~{},$
(A.6)

where we have used the property that $H$ depends on time only through $\phi$.
Expressed in a general $\xi$-gauge, eq. (A.4) becomes

$\frac{\partial
P}{\partial\tau_{\xi}}=\left[(h^{\xi+2}P)^{\prime}+2S_{\\!{{\rm
dS}}}h^{\xi-1}h^{\prime}P\right]^{\prime}+3S_{\\!{{\rm dS}}}h^{\xi}P~{},$
(A.7)

where $\tau_{\xi}=t_{\xi}/t_{S}$.

The definition in eq. (A.5) allows us to interpolate continuously between
proper-time gauge ($\xi=1$) and the case $\xi=0$, which is often referred to
as $e$-folding gauge. The time coordinate $t_{\xi=0}$ measures the number of
$e$-foldings accrued by each patch for any given value of $\phi$. In
$e$-folding gauge, the coefficient of the volume term becomes field-
independent, and the FPV solution is given simply by the solution of the
corresponding FP equation times the field-independent factor
$\exp(3H_{0}t_{\xi=0})$. This is because $e$-folding gauge corresponds to a
time slicing in which every patch undergoes an identical expansion.

For simplicity, in the following we will drop the index $\xi$ in the time
coordinate and simply call it $t$. It is implicit that the time coordinate
refers to the corresponding choice of $\xi$.

###  A.3 Spectral Representation

We express the FPV solution as a spectral representation by defining

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau}\,p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})~{},$ (A.8)

where ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}$ and $p$ are the
eigenvalues and eigenfunctions determined by the FPV differential equation,
which can be expressed as

$\left({\mathcal{L}}-{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}\right)\,p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=0~{},~{}~{}~{}{\mathcal{L}}=\frac{\partial}{\partial{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}h^{\xi+2}U^{2}\frac{\partial}{\partial{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}U^{-2}+3S_{\\!{{\rm
dS}}}(h^{\xi}-1)~{},~{}~{}~{}U=e^{\frac{S_{\\!{{\rm
dS}}}}{2h^{2}}}\,h^{-\frac{\xi+2}{2}}~{}.$ (A.9)

It is convenient to introduce the differential operator

${\mathcal{L}}_{H}=U^{-1}\,{\mathcal{L}}\,U~{}.$ (A.10)

Assuming that $P$ vanishes at the boundaries of the physical field range at
all times, the operator ${\mathcal{L}}_{H}$ is self-adjoint as can be easily
shown by integration by parts. By construction, ${\mathcal{L}}_{H}$ has the
same eigenvalue spectrum of ${\mathcal{L}}$

$({\mathcal{L}}_{H}-{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})=0~{},~{}~{}~{}p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=U\,\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})~{}.$ (A.11)

Since ${\mathcal{L}}_{H}$ is self-adjoint, we conclude that the FPV
eigenvalues ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}$ must be
real.

For eigenfunctions $\Phi$ normalised to one, we obtain

${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}=\int
d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,{\mathcal{L}}_{H}\,\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})=\int
d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,\left[3S_{\\!{{\rm
dS}}}(h^{\xi}-1)\left|\Phi\right|^{2}-\Big{|}h^{\frac{\xi+2}{2}}U\frac{\partial}{\partial{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}(U^{-1}\Phi)\Big{|}^{2}\right]$
(A.12)

where we have assumed vanishing boundary terms. Therefore, we find that the
maximum eigenvalue satisfies

${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}\leq
3S_{\\!{{\rm dS}}}\int
d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,(h^{\xi}-1)\left|\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max})\right|^{2}~{}.$
(A.13)

In $e$-folding gauge ($\xi=0$), eq. (A.13) shows that the eigenvalues must be
non-positive and ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm
max}\leq 0$. For a general gauge with $0\leq\xi\leq 1$, we obtain

${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}\leq
3\xi\,v_{\rm max}~{},$ (A.14)

where $v_{\rm max}$ is the maximum of the dimensionless potential $v$ in the
field range under consideration. We have assumed $v_{\rm max}\geq 0$, which is
always possible after an appropriate redefinition of $H_{0}$.

Due to the linearity of the FPV, each spectral mode evolves in time
independently. The value of ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{\rm max}$ is especially important because it
corresponds to the fastest inflating mode, which describes the asymptotic
behaviour of $P$ at large times

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=e^{3H_{0}t+{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm
max}\tau}\,p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max})~{}~{}~{}~{}~{}({\rm
for}~{}t\\!\to\\!\infty)~{}.$ (A.15)

###  A.4 Boundary Conditions

Using the results obtained in sect. A.3, we can write the most general FPV
solution as

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\tau+\frac{S_{\\!{{\rm
dS}}}}{2h^{2}}}\,h^{-\frac{\xi+2}{2}}\left[g_{1}({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\Phi_{1}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})+g_{2}({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\Phi_{2}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\right]~{},$ (A.16)

where $g_{1,2}({\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})$ are two
arbitrary functions and
$\Phi_{1,2}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})$ are two linearly-independent
solutions of the differential equation in Sturm-Liouville form

$\left(h^{\xi+2}\Phi^{\prime}\right)^{\prime}+(q-{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\Phi=0~{},~{}~{}~{}~{}q=S_{\\!{{\rm
dS}}}h^{\xi-1}h^{\prime\prime}-S_{\\!{{\rm
dS}}}^{2}h^{\xi-4}{h^{\prime}}^{2}+3S_{\\!{{\rm dS}}}(h^{\xi}-1)~{},$ (A.17)

where we have dropped negligible corrections
${\mathcal{O}}(H^{2}/\pi^{2}M_{P}^{2})$. The FPV solution is uniquely
determined once $g_{1,2}({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})$ are computed from given boundary and initial
conditions.

The boundary conditions are two time-dependent equations that determine
$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\uparrow\downarrow}},t)$,
where ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\uparrow}}$ and
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\downarrow}}$ are the
upper and lower endpoints of the field range, respectively. Particularly
convenient are the absorbing or reflecting boundary conditions

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\uparrow\downarrow}},t)=0~{}~{}~{}~{}{\rm(absorbing~{}boundary~{}conditions)},$
(A.18)
$P^{\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\uparrow\downarrow}},t)=0~{}~{}~{}~{}{\rm(reflecting~{}boundary~{}conditions)}.$
(A.19)

In either case, these boundary conditions determine (i) the ratio
$g_{2}/g_{1}$ and (ii) the eigenvalue spectrum, which is discrete and bounded
from above for bounded potentials. For absorbing boundary conditions, we find

$\frac{g_{2}({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})}{g_{1}({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})}=-\frac{\Phi_{1}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\downarrow}},{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})}{\Phi_{2}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\downarrow}},{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})}~{},$ (A.20)

while the eigenvalue spectrum is given by the discrete set of solutions of the
equation

$\Phi_{1}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\uparrow}},{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,\Phi_{2}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\downarrow}},{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})=\Phi_{2}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\uparrow}},{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,\Phi_{1}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{{\downarrow}},{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})~{}.$ (A.21)

Analogous equations can be written for reflecting boundary conditions.

In particular, absorbing or reflecting boundary conditions determine the
maximum eigenvalue ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm
max}$, which fully specifies the behaviour of the asymptotic solution at large
times. For any linear combination of absorbing and reflecting boundary
conditions, initial conditions affect only an overall constant of the
distribution $P$ and therefore the asymptotic state is effectively independent
of initial conditions. This may no longer be true for boundary conditions
which are non-homogeneous in $P$ or its derivative.

###  A.5 Initial Conditions

The initial condition is a field-dependent equation that determines
$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,0)$ at $t=0$. While
boundary conditions determine $g_{2}/g_{1}$ and the eigenvalue spectrum, the
initial condition fixes the remaining combination of $g_{1}$ and $g_{2}$, thus
fully specifying a unique FPV solution.

If the potential is continuous and differentiable everywhere in the field
range (a hypothesis that does not always hold in the examples we will
consider), the Sturm-Liouville theorem ensures that, for absorbing or
reflecting boundary conditions (or for any linear combination of the two),
there is a single eigenfunction
$\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})$ for each eigenvalue, and
together they form an orthonormal set

$\int
d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{1})\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{2})=\delta_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{1}{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{2}}~{}~{}~{}\forall\,{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{1,2}~{}.$ (A.22)

With the help of this relation, we can determine $g({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})$ for any initial condition of $P$ at $t=0$

$g({\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})=\int
d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,e^{-\frac{S_{\\!{{\rm
dS}}}}{2h^{2}}}\,\,h^{\frac{\xi+2}{2}}\,\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,0)~{}.$
(A.23)

A special case is when the field range is unbounded, with absorbing boundary
conditions set at infinite field values and with an initial condition such
that the scalar field is infinitely localised at a value
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}$

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,0)=\delta({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi-{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0})~{}.$
(A.24)

For this initial condition, the FPV solution is

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\tau+\frac{S_{\\!{{\rm
dS}}}}{2}(\frac{1}{h^{2}}-\frac{1}{h_{0}^{2}})}\,\left(\frac{h_{0}}{h}\right)^{\frac{\xi+2}{2}}\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0},{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\Phi({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})~{},$ (A.25)

where $h_{0}=h({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0})$.

###  A.6 Schrödinger Form

An alternative form of the FPV is obtained by making a field redefinition in
eq. (A.7) such that the coefficient of the second derivative with respect to
the field becomes constant and the one of the first derivative vanishes

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=e^{\frac{S_{\\!{{\rm
dS}}}}{2h^{2}}}\,\,h^{-\frac{3(\xi+2)}{4}}\,\Psi(x,t)~{},~{}~{}~{}~{}\frac{dx}{d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}=h^{-\frac{\xi+2}{2}}~{},$
(A.26) $\frac{\partial\Psi}{H_{0}\,\partial
t}=\frac{\partial^{2}\Psi}{S_{\\!{{\rm dS}}}\,\partial
x^{2}}+{\hat{V}}(x)\Psi~{},~{}~{}~{}~{}{\hat{V}}(x)=\big{(}h^{\xi-1}h^{\prime\prime}-S_{\\!{{\rm
dS}}}h^{\xi-4}{h^{\prime}}^{2}+3h^{\xi}\big{)}_{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi={\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi(x)}~{},$
(A.27)

where primes denote derivatives with respect to
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ and corrections
${\mathcal{O}}(H^{2}/\pi^{2}M_{P}^{2})$ have been neglected. Equation (A.27)
exhibits an obvious analogy with the Schrödinger equation in Euclidean time.
Its general solution can be written as a spectral expansion in stationary
modes, following the same procedure described in sect. A.3.

###  A.7 Perturbative Expansion

In many physical applications, we are interested in the behaviour of the FPV
solutions in a neighbourhood of a generic field point which, with an
appropriate coordinate shift, can be chosen to be
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi=0$ with $V(0)=V_{0}$.
The perturbative domain is defined as the neighbourhood around
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi=0$ in which the
variation of the energy density due to its field dependence is only a small
perturbation of $V_{0}$ (i.e.
$|V({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)-V_{0}|\ll V_{0}$).
Therefore, the conditions for perturbative expansion, together with slow roll,
are

$v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi),v^{\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi),v^{\prime\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\ll
S_{\\!{{\rm dS}}}~{}.$ (A.28)

In the perturbative domain, we can expand eq. (A.7) in powers of
$v/S_{\\!{{\rm dS}}}$ and, at leading order, we obtain

$t_{S}\Big{(}\frac{\partial}{\partial
t}-3H_{0}\Big{)}P=P^{\prime\prime}+(2v^{\prime}P)^{\prime}+3\xi vP~{},$ (A.29)

where $v(0)=0$. Here we have dropped the ${\mathcal{O}}(v/S_{\\!{{\rm dS}}})$
correction to the diffusion term, which is a subleading effect in our
perturbative expansion, but we have kept the ${\mathcal{O}}(v)$ contribution
to the volume term, which is a truly leading effect since the field-
independent constant can be simply reabsorbed in the definition of $P$.

It is instructive to rewrite for a moment eq. (A.29) in terms of the original
dimensionful variables

$\Big{(}\frac{\partial}{H_{0}\partial t}-3\Big{)}P=(R_{Q}+R_{C}+R_{V})\,P~{},$
(A.30)
$R_{Q}=\frac{\hbar\,H_{0}^{2}}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}~{},~{}~{}~{}R_{C}=\frac{\partial}{\partial\phi}\frac{V^{\prime}}{3H_{0}^{2}}~{},~{}~{}~{}R_{V}=\frac{\xi(V-V_{0})}{2H_{0}^{2}M_{P}^{2}}~{},$
(A.31)

where $R_{Q,C,V}$ represent, at leading order in the perturbative expansion,
the differential operators of the quantum, classical and volume terms,
respectively. This manifestly shows the physical origin of the three terms.
Working at fixed de Sitter radius, we note that $R_{Q}$ is a purely quantum
effect (as it vanishes for $\hbar\to 0$), $R_{C}$ derives from classical
mechanics (as it is independent of $\hbar$ and $M_{P}$) and $R_{V}$ is a
purely gravitational effect related to space expansion (as it vanishes for
$M_{P}\to\infty$). In other words, the FPV dynamics originates from an
intimate interplay of different physical effects, when quantum mechanics,
classical physics and general relativity come together in the description of
phenomena occurring during the inflationary epoch.

In terms of the spectral modes $p$ defined in eq. (A.8), the perturbative FPV
equation in eq. (A.29) becomes

$({p}^{\prime}+2v^{\prime}{p})^{\prime}+(3\xi v-{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})p=0~{}.$ (A.32)

This equation describes the stationary modes of the FPV distribution for a
scalar field whose potential gives only a small field-dependent modulation of
the background vacuum energy. All local information related to the microscopic
dynamics of the scalar field is encoded in the function $v$, while the
eigenvalues ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}$ contain
information related to physics beyond the range of the effective theory, which
is described by the boundary conditions at the field endpoints.

If a mode $p$ has a local Gaussian peak at
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi={\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$
(such that
${p}^{\prime}({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})=0$)
with width
$\sigma^{2}\equiv-{p}({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})/{p}^{\prime\prime}({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})$,
then eq. (A.32) gives a relation between the peak parameters and the
corresponding eigenvalue ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}$

$3\xi
v({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})+2v^{\prime\prime}({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})={\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}+\frac{1}{\sigma^{2}}~{}.$ (A.33)

###  A.8 Junction Conditions

#### Potentials with kinks

In applications to critical phenomena, one is confronted with scalar
potentials that are not differentiable at a critical point
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c}$

$v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=v_{-}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\,\Theta({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c}-{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)+v_{+}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\,\Theta({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi-{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})~{}.$
(A.34)

Here $v_{\pm}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)$ describe
the potential on either side of the critical point and $\Theta$ is the
Heaviside step function

$\Theta(x)=\left\\{\begin{array}[]{l}1~{}~{}{\rm for}~{}x>0\\\ 0~{}~{}{\rm
for}~{}x<0\end{array}\right.~{}.$ (A.35)

We will first consider the case in which the potential is continuous at the
critical point,
$v_{+}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})=v_{-}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})$,
while its first derivative is not,
$v^{\prime}_{+}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})\neq
v^{\prime}_{-}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})$.

Since the potential is continuous, the FPV distribution and its time
derivative must be continuous as well, but its field derivative may not. The
corresponding junction condition at the critical point, for each spectral mode
$p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})$, is obtained by integrating
eq. (A.32) in a neighbourhood of
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c}$

$\lim_{\epsilon\to
0}\int_{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c}-\epsilon}^{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c}+\epsilon}d{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,({p}^{\prime}+2v^{\prime}{p})^{\prime}=0~{}~{}~{}\Rightarrow~{}~{}~{}\frac{\Delta
p^{\prime}}{p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})}=-2\Delta
v^{\prime}~{},$ (A.36)

where $\Delta v^{\prime}\equiv
v_{+}^{\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})-v_{-}^{\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})$
and $\Delta p^{\prime}\equiv
p^{{{{{{(+)}}}}}\,\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})-p^{{{{{{(-)}}}}}\,\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})$,
with $p^{{{{{{(\pm)}}}}}}$ being the spectral modes on either side of the
critical point.

On both sides of the critical point, the spectral modes can be written as a
linear combination of two independent solutions of eq. (A.32) for the
corresponding potentials $v_{\pm}$

$p^{{{{{{(\pm)}}}}}}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=e^{-v_{\pm}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)}\left[g_{{{1}}}^{{{{{{(\pm)}}}}}}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,\Phi_{{{1}}}^{{{{{{(\pm)}}}}}}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})+g_{{{2}}}^{{{{{{(\pm)}}}}}}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,\Phi_{{{2}}}^{{{{{{(\pm)}}}}}}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\right]~{}.$ (A.37)

While one pair of functions $g_{{{1,2}}}^{{{{{{(\pm)}}}}}}$ is fixed by the
boundary conditions at the endpoints of the field range and by initial
conditions, as explained in sects. (A.4) and (A.5), the other pair is
determined by the junction condition in eq. (A.36) together with the
continuity condition $\Delta
p({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})=0$, which give

$\\!\\!\\!g_{{{1,2}}}^{{{{{{(+)}}}}}}({\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})=\left.\frac{g_{{{1,2}}}^{{{{{{(-)}}}}}}W[\Phi_{{{1,2}}}^{{{{{{(-)}}}}}},\Phi_{{{2,1}}}^{{{{{{(+)}}}}}}]+g_{{{2,1}}}^{{{{{{(-)}}}}}}W[\Phi_{{{2,1}}}^{{{{{{(-)}}}}}},\Phi_{{{2,1}}}^{{{{{{(+)}}}}}}]+(g_{{{1,2}}}^{{{{{{(-)}}}}}}\Phi_{{{1,2}}}^{{{{{{(-)}}}}}}+g_{{{2,1}}}^{{{{{{(-)}}}}}}\Phi_{{{2,1}}}^{{{{{{(-)}}}}}})\Phi_{{{2,1}}}^{{{{{{(+)}}}}}}\,\Delta
v^{\prime}}{W[\Phi_{{{1,2}}}^{{{{{{(+)}}}}}},\Phi_{{{2,1}}}^{{{{{{(+)}}}}}}]}\right|_{({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c},{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})}$ (A.38)

where the Wronskian is defined as
$W[\Phi_{1},\Phi_{2}]=\Phi_{1}\Phi_{2}^{\prime}-\Phi_{1}^{\prime}\Phi_{2}$.
Note that $W[\Phi_{{{1}}}^{{{{{{(\pm)}}}}}},\Phi_{{{2}}}^{{{{{{(\pm)}}}}}}]$
is independent of ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$
because $\Phi_{{{1,2}}}^{{{{{{(\pm)}}}}}}$ satisfy linear second-order
differential equations without first derivatives.

#### Multivalued potentials

Another example of junction conditions, which is relevant to physical
applications, is the case of a multivalued potential with two branches
corresponding to two different phases. The potentials
$v_{a,b}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)$ on the two
branches are both continuous and differentiable, but branch $b$ has a
termination point and exists only for
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\geq{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c}$.
Assume that the field cannot make any transition between the two branches,
except at the critical point
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c}$ where the field
undergoes an instantaneous phase transition from $b$ to $a$, and take
$v_{b}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})>v_{a}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})$.

This physical setup imposes the junction condition

$p^{(b)}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})=0~{}.$
(A.39)

The junction conditions on the $a$ branch are obtained from considering the
sum of the two FPV in eq. (A.32) on the two branches

$(p^{(a)\,\prime}+p^{(b)\,\prime}+2v^{\prime}_{a}p^{(a)}+2v^{\prime}_{b}p^{(b)})^{\prime}+3\xi(v_{a}p^{(a)}+v_{b}p^{(b)})-{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}(p^{(a)}+p^{(b)})=0~{}.$ (A.40)

By integrating this equation in a neighbourhood of the critical point, as
before, and exploiting the continuity of $p^{(a)}$ and $p^{(a)}+p^{(b)}$, we
find the junction conditions

$\Delta
p^{(a)\,\prime}+p^{(b)\,\prime}({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{c})=0~{},~{}~{}~{}~{}\Delta
p^{(a)}=0~{},$ (A.41)

where $\Delta p^{(a)\,\prime}$ is the discontinuity across the critical point,
defined as before.

Equation (A.41) describes flux conservation. Although the FPV does not have a
conserved current, there is an effective conservation law because the volume
term does not enter the discontinuity.

##  Appendix B Solutions for Linear and Quadratic Potentials

Linear and quadratic potentials offer interesting tools for FPV studies
because (i) they allow for exact analytical solutions and (ii) they serve as
prototypes or reference points for broader classes of theories. In particular,
linear potentials are valid local approximations of any monotonic potential
and give the leading term of the general class of EFT potentials considered in
this paper. Quadratic potentials provide simple models to describe phenomena
around minima or maxima of more general potentials.

###  B.1 Linear Potential

Consider a potential linear in the field with coupling constant $\Gamma$

$V(\phi)=V_{0}+\Gamma\,\phi~{},~{}~{}~{}v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=\gamma\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi~{},~{}~{}~{}\gamma\equiv\frac{\Gamma
S_{\\!{{\rm dS}}}}{6M_{P}H_{0}^{2}}~{},$ (B.1)

where, with no loss of generality, we take $\Gamma>0$. The perturbative
regime, where ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ gives
only a small modulation of the energy density, is valid for field excursions
around ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi=0$ of size
$\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ with

$\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\ll\frac{S_{\\!{{\rm
dS}}}}{\gamma}~{},$ (B.2)

while the slow-roll condition requires $\gamma<S_{\\!{{\rm dS}}}$. In this
range, we can use the perturbative FPV in eq. (A.32), which becomes

${p}^{\prime\prime}+2\gamma\,{p}^{\prime}+(3\xi\,\gamma{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi-{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})p=0~{}.$ (B.3)

Following the EFT analysis (see sect. B.5), we define the following regimes
for the strength of the coupling constant $\gamma$.

$\begin{array}[]{lc}{\mbox{\bf Classical (C)
:}}&~{}~{}~{}\gamma\gg\xi\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\\\[5.0pt]
{\mbox{\bf Quantum+Volume (QV)
:}}&~{}~{}~{}\sqrt{\xi}\ll\gamma\ll\xi\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\\\[5.0pt]
{\mbox{\bf Quantum{\boldmath${}^{2}$}+Volume (Q{\boldmath${}^{2}$}V)
:}}&~{}~{}~{}\frac{1}{\xi\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{3}}\ll\gamma\ll\sqrt{\xi}\\\\[5.0pt]
{\mbox{\bf Quantum (Q)
:}}&~{}~{}~{}\gamma\ll\frac{1}{\xi\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{3}}\end{array}$

The most general solution of eq. (B.3) is (see sect. C.1 for definitions)

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau-\gamma{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\left[g_{1}({\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\,{\rm Ai}(x)+g_{2}({\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\,{\rm Bi}(x)\right]~{},$ (B.4)
$x=\frac{\gamma^{2}+{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}{c^{2}}-c\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi~{},~{}~{}~{}c=(3\xi\gamma)^{1/3}~{}.$
(B.5)

The first solution has a peak at
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi={\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$
with width $\sigma$ such that

${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}=\frac{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}{3\xi\gamma}~{},~{}~{}~{}~{}\sigma=\sqrt{\frac{2}{3\xi}}~{}~{}~{}({\rm
C~{}or~{}QV~{}regime})~{},$ (B.6)
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}=\frac{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}{3\xi\gamma}-\frac{a_{1}^{\prime}}{c}~{},~{}~{}~{}~{}\sigma=\frac{1}{\sqrt{-a_{1}^{\prime}}\,c}~{}~{}~{}({\rm
Q}^{2}{\rm V~{}regime})~{},$ (B.7)

and turns negative, entering a subsequent oscillatory regime, for

${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi>\frac{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}{3\xi\gamma}+\frac{\gamma}{3\xi}-\frac{a_{1}}{c}~{},$
(B.8)

where $a_{1}=-2.34$ and $a_{1}^{\prime}=-1.02$ are the first zeros of the
function ${\rm Ai}(x)$ and its derivative. Because of the bound on the maximum
eigenvalue in eq. (A.14), the peak location
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ is always below
the upper endpoint of the field range under consideration. In the C and QV
regimes, the peak is well approximated by a Gaussian with a universal width
$\sigma$, independent of the eigenvalue ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}$ and the coupling constant $\gamma$. In the Q2V
regime, the peak corresponds to the first oscillation of ${\rm Ai}(x)$ before
the function turns negative.

The second solution is monotonically decreasing with
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ until it turns
negative entering a subsequent oscillatory regime for

${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi>\frac{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}{3\xi\gamma}+\frac{\gamma}{3\xi}-\frac{b_{1}}{c}~{},$
(B.9)

where $b_{1}=-1.17$ is the first zero of the function ${\rm Bi}(x)$.

In the case of $e$-folding gauge ($\xi=0$), the parameter $c$ vanishes and the
solution is

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau-\gamma{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\left[g_{1}({\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\,e^{-\sqrt{\gamma^{2}+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}+g_{2}({\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\,e^{\sqrt{\gamma^{2}+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\right]~{}.$
(B.10)

####  B.1.1 ${\boldmath\delta}$-function Initial Condition

Consider absorbing boundary conditions set at infinite field values

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!\pm\infty,t)=0~{}~{}~{}~{}~{}{\rm(boundary~{}conditions)}$
(B.11)

and choose an initial condition with a $\delta$-function as in eq. (A.24). The
FPV solution is (see identities in sect. C.1)

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\int\frac{d{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}{c}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau-\gamma({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi-{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0})}\,{\rm
Ai}(x_{0}){\rm Ai}(x)~{},$ (B.12)

where $x_{0}=x({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0})$.
The boundary conditions in eq. (B.11) leave the eigenvalue spectrum
unconstrained and therefore the summation over ${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}$ is replaced here by an integral.

Setting boundary conditions at infinity is inconsistent with our perturbative
expansion, which is valid only in the finite field range defined by eq. (B.2).
The choice in eq. (B.11) is an approximation that allows for a simpler
analytical treatment of the realistic case in which the absorbing boundary
conditions are set at finite field values
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi=\pm{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}$.
This approximation gives sensible results as long as the FPV solution is
localised in a field region
$|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi|\ll{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}$
and vanishes exponentially well before reaching the endpoints, thus remaining
insensitive to the field value where the absorbing boundary condition is set.
Whenever this is not the case, the approximation of using eq. (B.11) breaks
down and its results can be trusted only up to a time cutoff. To obtain the
distribution at times larger than this cutoff, boundary conditions at finite
field values must be reinstated. This caveat about setting boundary conditions
at infinity applies also to other examples of potentials that we study in
following sections.

The spectral integral in eq. (B.12), which can be performed using eq. (C.17),
gives a Gaussian function peaked at
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}(t)$, with
variance $\sigma(t)$ and normalisation factor $\chi(t)$, defined such that
$e^{3H_{0}t+\chi}$ measures how much the inflationary expansion makes $P$
differ from a true probability distribution,

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[3H_{0}t+\chi-\frac{({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi-{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})^{2}}{2\sigma^{2}}\right]~{},$
(B.13)
$\sigma^{2}=2\tau~{},~{}~{}~{}~{}{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}={\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}-\gamma\tau(2-3\xi\tau)~{},~{}~{}~{}~{}\chi=3\xi\gamma\tau[{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}+\gamma\tau(\xi\tau-1)]~{}.$
(B.14)

where the dimensionless time coordinate $\tau$ is defined in eq. (A.3).

The variance $\sigma$ grows as the square root of time, with the familiar
random-walk behaviour $\langle\phi^{2}\rangle=H_{0}^{3}t/(2\pi)^{2}$. In
$e$-folding gauge ($\xi=0$), the peak of the distribution
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ slides down the
potential, tracking the classical equation of motion. The same happens in the
C regime for a general gauge ($\xi\neq 0$), with the peak rolling down until
the time
$\tau\sim{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}/\gamma\ll
1/\xi$, when the approximation of using eq. (B.11) breaks down. The field
value at which the peak eventually stops depends on the exact location of
boundary conditions.

In the QV and Q2V regimes,
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ starts by
rolling down but, when $\sigma$ reaches the value $\sqrt{2/3\xi}$ (and for
$|{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{0}-\gamma/3\xi|\ll{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}$),
the peak of the distribution reverses its motion and climbs back the slope.
This is the effect of the volume term, which favours patches with large
$\phi$, since they inflate faster. The time for
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ to get back
where it started is

$t_{\rm idle}=\frac{2{t_{S}}}{3\xi}~{}.$ (B.15)

Note that eq. (B.15) fully depends on initial conditions, and this result
corresponds to the choice of a $\delta$-function. In practice, $t_{\rm idle}$
can be made arbitrarily short by starting with a broader field distribution.
For $t>t_{\rm idle}$, the peak of the distribution continues to climb the
potential, eventually growing with constant acceleration, until the
approximation breaks down at
$\tau\sim\sqrt{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}/\xi\gamma}$.
The final location of the peak depends on the boundary conditions and
therefore lies beyond the validity of the approximation.

The time $\Delta t$ for
${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ to travel a
distance $\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$,
measured after the end of the idling phase for the quantum regime, is

$\frac{\Delta
t}{t_{S}}\approx\left\\{\begin{array}[]{cc}\frac{\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}{2\gamma}&({\rm
C~{}regime})\vspace{0.2cm}\\\
\sqrt{\frac{\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}{3\xi\gamma}}&({\rm
QV/Q}^{2}{\rm V~{}regimes})\end{array}\right.~{}.$ (B.16)

When the motion is in the C regime
($\gamma\gg\xi\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$),
the time $\Delta t$ for the peak to explore the field range
$\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ is smaller than
$t_{S}$. When the motion is in the QV or Q2V regime
($\gamma\ll\xi\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$),
the time $\Delta t$ is larger than $t_{S}$.

The approximation of eq. (B.11) allows for a simple analytical treatment of
the early stages of the time evolution, starting from a sharply localised
distribution. It makes manifest the difference between the dynamics in the
classical and quantum regimes. In the classical regime, the dynamics follows
the corresponding FP behaviour, with the peak of the distribution tracking the
classical equation of motion and its width growing as the square root of time.
The typical timescale of the evolution is smaller than $t_{S}$. In the quantum
regime, there is a sharp change of behaviour at $t\sim t_{S}$, when the volume
term in the FPV starts playing a crucial role and the peak steers in a
direction opposite to classical motion, towards maximising the potential
energy. The hallmark in the change of behaviour is the width, with
$\sigma\geq\sqrt{2/3\xi}$ (i.e. having super-Planckian size, when measured in
physical units) signalling the new dynamical FPV regime. Therefore, the time
scale to reach the asymptotic configuration is larger than $t_{S}$.

While the approximation of eq. (B.11) gives us information about early
dynamics, it is unable to describe the approach to the asymptotic state in the
case of a linear potential. The reason is that, for a linear potential with
absorbing boundary conditions, the asymptotic state is always affected by the
endpoints of the field range (by the lower endpoint for the classical regime
and the upper endpoint for the quantum regime). To determine the asymptotic
distribution, we need to reinstate the boundary conditions at a finite field
point, as done in the following section.

####  B.1.2 Bounded Field Range

Consider a bounded field range with absorbing boundary conditions at the
endpoints

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!\pm{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E},t)=0~{}~{}~{}~{}~{}{\rm(boundary~{}conditions)},$
(B.17)

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.

The most general FPV solution that satisfies the boundary conditions in eq.
(B.17) is

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau-\gamma{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,g({\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})\left[\frac{{\rm Ai}(x)}{{\rm
Ai}(x_{\\!E_{-}})}-\frac{{\rm Bi}(x)}{{\rm Bi}(x_{\\!E_{-}})}\right]~{},$
(B.18) $x=\frac{\gamma^{2}+{\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}{c^{2}}-c{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi~{},~{}~{}~{}x_{\\!E_{\pm}}=x({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\\!=\\!\pm{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E})~{},$
(B.19)

where $g({\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda})$ are arbitrary
constants and the eigenvalues are determined by the equation ${\rm
Ai}(x_{\\!E_{+}})/{\rm Ai}(x_{\\!E_{-}})={\rm Bi}(x_{\\!E_{+}})/{\rm
Bi}(x_{\\!E_{-}})$.

As long as we are not in the Q regime, we can consider the limit
$c\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}\\!\gg\\!1$,
in which eq. (B.18) becomes

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{n=1}^{\infty}e^{3H_{0}t+{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}\tau-\gamma{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\,g_{n}\left[2\,{\rm
Ai}(x_{n})-e^{-\frac{4}{3}x_{\\!E_{-}}^{3/2}}\,{\rm Bi}(x_{n})\right]~{},$
(B.20)
$x_{n}=a_{n}+c\,({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)~{},~{}~{}~{}x_{\\!E_{-}}=a_{n}+2c\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}~{},$
(B.21)

where, for the eigenvalues that satisfy
$|a_{n}|\\!\ll\\!c\,{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}$
and up to corrections exponentially suppressed in $x_{\\!E_{-}}$,

${\mkern 0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}_{n}=a_{n}(3\xi\gamma)^{\frac{2}{3}}+3\xi\gamma{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-\gamma^{2}~{},$
(B.22)

with $a_{n}$ being the zeroes of the Airy function ${\rm Ai}(x)$.

Whenever the distribution is localised away from the lower endpoint, the
contribution from ${\rm Bi}(x)$ in eq. (B.20) is negligible and can be safely
dropped. Effectively, this corresponds to setting the lower endpoint of the
field range to minus infinity, without affecting the result. Only when the
distribution is sizeable in the neighbourhood of the lower endpoint, the
location of the boundary condition matters.

The asymptotic state at large times is obtained by retaining in eq. (B.20)
only the contribution from the maximum eigenvalue, which corresponds to
$n\\!=\\!1$ ($a_{1}\\!=\\!-2.34$). The asymptotic distribution is peaked at
the field value ${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$,
which is given by the solution of the equation

$\frac{{\rm Ai}^{\,\prime}({\bar{x}})}{{\rm
Ai}({\bar{x}})}=-\frac{\gamma}{c}+\Big{(}\frac{{\rm
Bi}^{\,\prime}({\bar{x}})}{{\rm
Bi}({\bar{x}})}+\frac{\gamma}{c}\Big{)}\frac{{\rm
Bi}({\bar{x}})\,e^{-\frac{4}{3}x_{\\!E}^{3/2}}}{2\,{\rm
Ai}({\bar{x}})}~{},~{}~{}~{}~{}{\bar{x}}=a_{1}+c\,({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})~{}.$
(B.23)

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

$\sigma^{2}=\frac{1}{\gamma^{2}-c^{2}\,{\bar{x}}}~{}.$ (B.24)

Whenever ${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}$ is far
from the lower endpoint, one can simply take the limit
$\exp({-\frac{4}{3}x_{\\!E}^{3/2}})\to 0$. The solution behaves differently in
the three following regimes.

#### Q2V regime

For $\gamma\ll\sqrt{\xi}$, the peak of the asymptotic distribution is such
that

${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}={\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-\frac{a_{1}^{\prime}-a_{1}}{(3\xi\gamma)^{\frac{1}{3}}}~{},~{}~{}~{}\sigma=\frac{1}{\sqrt{-a_{1}^{\prime}}\,(3\xi\gamma)^{\frac{1}{3}}}~{}.$
(B.25)

As expected from eq. (A.33), this result satisfies $3\xi
v({\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})={\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{\rm max}+\sigma^{-2}$. As the
potential gets shallower, the peak moves away from the endpoint, although its
width grows proportionally. As a result, in the Q2V regime, the peak is always
located within a fixed-width distance from the endpoint, since
$({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})/\sigma\approx
1.3$.

#### QV regime

For
$\sqrt{\xi}\ll\gamma\ll\xi{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}$,
the peak of the asymptotic distribution is such that

${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}={\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-\frac{\gamma}{3\xi}~{},~{}~{}~{}\sigma=\sqrt{\frac{2}{3\xi}}~{},$
(B.26)

which also satisfies eq. (A.33). The width $\sigma$ is independent of the
coupling $\gamma$ and has Planckian size in physical units. Note that the
asymptotic value of $\sigma$ is smaller than the maximum value of the width
$\sigma\sim({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}/\xi\gamma)^{1/4}$
that was found in sect. B.1.1 from the time evolution following a
$\delta$-function initial condition. A numerical analysis of eq. (B.18) with a
localised initial condition indeed shows that $\sigma$ grows as the square
root of time, reaching super-Planckian values, but eventually decreases and
converges to its asymptotic value given in eq. (B.26).

The peak location moves away from the endpoint as the potential gets steeper.
However, its relative distance from the upper endpoint of the field range can
be made arbitrarily small in the QV regime, as long as $\gamma$ is much
smaller than ${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}$.
Nonetheless, the peak is always separated from the endpoint by a distance that
is parametrically larger then the width, since
$({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}-{\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi})/\sigma={\mathcal{O}}(\gamma/\\!\sqrt{\xi})\gg
1$.

#### C regime

As we increase the value of $\gamma$ and enter the C regime
($\gamma\gg\xi{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}$),
the peak slides down the potential and the result becomes sensitive on how we
set boundary conditions at the lower endpoint of the field range. In this
case, the contribution from ${\rm Bi}(x)$ in eq. (B.20) is important and peak
of the asymptotic distribution is such that

${\bar{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}=-{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi_{\\!E}+\frac{1}{\gamma}~{},~{}~{}~{}~{}\sigma=\frac{1}{\gamma}~{}.$
(B.27)

The peak gets closer to the lower endpoint as the potential becomes steeper
and it is always only 1-$\sigma$ away from it. However, our definition of
$\sigma$ describes only the local property of the peak at its maximum, while
in reality the distribution is asymmetric and more spread than $1/\gamma$ on
the side opposite to the lower endpoint.

###  B.2 Quadratic Potential (Positive Curvature)

Consider a potential quadratic in the field with positive curvature
($M^{2}>0$)

$V(\phi)=V_{0}+\frac{M^{2}\phi^{2}}{2}~{},~{}~{}~{}v({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)=\frac{m^{2}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}~{},~{}~{}~{}m^{2}\equiv\frac{S_{\\!{{\rm
dS}}}M^{2}}{6H_{0}^{2}}~{}.$ (B.28)

The perturbative regime, where $\phi$ gives only a small modulation of the
energy density, is valid for field excursions around
${\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi=0$ of size
$\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$ with

$\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\ll\sqrt{\frac{S_{\\!{{\rm
dS}}}}{m^{2}}}~{},$ (B.29)

while the slow-roll condition requires $m^{2}<S_{\\!{{\rm dS}}}$, which
corresponds to $M<H_{0}$. In this range, we can use the perturbative FPV in
eq. (A.32), which becomes

$p^{\prime\prime}+2m^{2}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi\,p^{\prime}+\Big{(}\frac{3}{2}\,{\xi\,m^{2}{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}+2m^{2}-{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}\Big{)}p=0~{}.$ (B.30)

Following the EFT analysis (see sect. B.5), we define four different regimes
for $m^{2}$.

$\begin{array}[]{lcc}{\mbox{\bf Classical (C) :}}&~{}~{}~{}\xi\ll
m^{2}\ll\frac{S_{\\!{{\rm
dS}}}}{\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}&\\\\[5.0pt]
{\mbox{\bf Quantum+Volume (QV)
:}}&~{}~{}~{}\frac{\sqrt{\xi}}{\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}\ll
m^{2}\ll\xi&({\rm with~{}}\frac{\xi}{S_{\\!{{\rm dS}}}}\ll m^{2})\\\\[5.0pt]
{\mbox{\bf Quantum{\boldmath${}^{2}$}+Volume (Q{\boldmath${}^{2}$}V)
:}}&~{}~{}~{}\frac{1}{\xi{\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{4}}}\ll
m^{2}\ll\frac{\sqrt{\xi}}{\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi}&\\\\[5.0pt]
{\mbox{\bf Quantum (Q)
:}}&~{}~{}~{}m^{2}\ll\frac{1}{\xi{\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{4}}}&\end{array}$

Combining the definition of the QV regime with eq. (B.29), we find that
$m^{2}$ must necessarily satisfy $\xi/S_{\\!{{\rm dS}}}\ll m^{2}\ll\xi$,
whenever the system is in the QV regime. Therefore, when
$m^{2}<\xi/S_{\\!{{\rm dS}}}$ or equivalently $M<\sqrt{3\xi/2}\,\hbar
H_{0}^{3}/(4\pi^{2}M_{P}^{2})$, the system must necessarily be in the Q2V
regime, regardless of the field range
$\Delta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi$.

The most general FPV solution, for $m^{2}>3\xi/2$, is (see sect. C.2 for
definitions)

$P({\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi,t)=\sum_{{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}\,e^{3H_{0}t+{{\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda}}\tau-m^{2}(1+R)\frac{{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi^{2}}{2}}\Big{[}g_{1}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,H_{\nu}(\zeta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)+g_{2}({\mkern
0.75mu\mathchar
22\relax\mkern-9.75mu\lambda})\,{H}_{\nu}(-\zeta{\mathrel{\raisebox{-1.7pt}{\\_}}}\mkern-6.6mu\varphi)\Big{]}~{},$
(B.31)
$\zeta=\sqrt{{m^{2}R}}~{},~{}~{}~{}R=\sqrt{1-\frac{3\xi}{2m^{2}}}~{},~{}~{}~{}\nu=\frac{m^{2}(1-R)-{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}}{2m^{2}R}~{}.$ (B.32)

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