CERN-TH-2021-077

Self-Organised Localisation

Gian F. Giudice,a Matthew McCullough,${}^{a,b^{*}}$ Tevong Youa,b,c

aCERN, Theoretical Physics Department, Geneva, Switzerland
bDAMTP, University of Cambridge, Cambridge, UK
cCavendish Laboratory, University of Cambridge, Cambridge, UK

Abstract

> We describe a new phenomenon in quantum cosmology: self-organised
> localisation. When the fundamental parameters of a theory are functions of a
> scalar field subject to large fluctuations during inflation, quantum phase
> transitions can act as dynamical attractors. As a result, the theory
> parameters are probabilistically localised around the critical value and the
> Universe finds itself at the edge of a phase transition. We illustrate how
> self-organised localisation could account for the observed near-criticality
> of the Higgs self-coupling, the naturalness of the Higgs mass, or the
> smallness of the cosmological constant.

∗On leave.

###### Contents

  1. 1 Introduction
  2. 2 Scalar Fields in an Inflationary Universe
    1. 2.1 The Stochastic Approach
    2. 2.2 On Time, Eternal Inflation and Quantum Gravity
    3. 2.3 The Measure Problem
    4. 2.4 EFT Potential
    5. 2.5 Boundary Conditions and Eigenvalues
    6. 2.6 Positivity
    7. 2.7 Peak Properties
    8. 2.8 Linear Potential
    9. 2.9 The Classical Regime
    10. 2.10 Junction Conditions
  3. 3 Self-Organised Localisation
    1. 3.1 Localisation in the Pyramid Scheme
    2. 3.2 Microscopic Features of the Phase Transition
    3. 3.3 Localisation in the Waterfall Scheme
    4. 3.4 Determining Fundamental Parameters with SOL
    5. 3.5 SOL Post-Inflationary Dynamics
    6. 3.6 EFT Parameters and SOL
  4. 4 Near-Criticality of the Higgs Self-Coupling from SOL
    1. 4.1 Phase Diagram
    2. 4.2 SOL Predictions
  5. 5 Higgs Naturalness from SOL
    1. 5.1 Phase Diagram
    2. 5.2 SOL Prediction
    3. 5.3 Reconciling the Prediction with the SM Higgs
  6. 6 A Small Cosmological Constant from SOL
    1. 6.1 Theoretical Setup
    2. 6.2 SOL Dynamics
    3. 6.3 SOL Prediction
  7. 7 Summary and Conclusions
  8. Appendix
  9. A The Volume-Weighted Fokker-Planck Equation
    1. A.1 Structure of the Equation
    2. A.2 Time Reparametrisation
    3. A.3 Spectral Representation
    4. A.4 Boundary Conditions
    5. A.5 Initial Conditions
    6. A.6 Schrödinger Form
    7. A.7 Perturbative Expansion
    8. A.8 Junction Conditions
  10. B Solutions for Linear and Quadratic Potentials
    1. B.1 Linear Potential
      1. B.1.1 ${\boldmath\delta}$-function Initial Condition
      2. B.1.2 Bounded Field Range
    2. B.2 Quadratic Potential (Positive Curvature)
      1. B.2.1 ${\boldmath\delta}$-function Initial Condition
    3. B.3 Quadratic Potential (Negative Curvature)
      1. B.3.1 ${\boldmath\delta}$-function Initial Condition
    4. B.4 Linear-Linear Potential
      1. B.4.1 Pyramid Potential
      2. B.4.2 Cliff Potential
    5. B.5 EFT Notation
  11. C Useful Identities of Some Special Functions
    1. C.1 Airy’s Equation
    2. C.2 Hermite’s Equation

##  1 Introduction

The paradigm of symmetry underlies the construction of the Standard Model and
General Relativity (SM+GR) and thus, arguably, the most successful scientific
effective field theory ever created. It is jarring that, after decades of
symmetry-based speculation, the paradigm has thus far failed to uncover the
microscopic origins of SM+GR. Once we add the experimental information about
the value of the cosmological constant and the lack of new physics at the LHC,
it becomes natural to conclude that perhaps it is time for the paradigm to
shift. If so, the questions we seek to answer remain, but the way we hope to
answer them must change, and not adiabatically. The trail beyond symmetry has
few signposts and must be forged anew. There are, however, some breadcrumbs.
The Higgs naturalness problem, the cosmological constant problem, and the
peculiar feature of Higgs vacuum metastability all share one feature in
common: criticality. The measurement of the Higgs mass, together with a high-
energy extrapolation of the Standard Model, have revealed that the Universe
may exist in a near-critical state, with two phases subsisting side-by-side,
as if we lived in water while ice is but just a fluctuation away. The
hierarchy problem also exhibits the hallmark of criticality, with a mass
parameter seemingly tuned arbitrarily close to the symmetry-breaking point, as
if the Universe couldn’t decide between an ordered or disordered phase,
finally opting to live in a state as close to indecision as possible, with
just a smidgen of order. So too for the cosmological constant, where we find
ourselves teetering inexplicably close to the critical boundary between
explosion and implosion, apparently opting, by a slither, for the former. All
of these observations suggest criticality, viewed as a coexistence of phases,
as a potential signpost towards a new paradigm.

If these examples are not the product of symmetry, then criticality must be
arrived at by other means. If, in some competing paradigm, criticality emerges
out of a non-critical state in a form of self-organised criticality [1], then
both a timeline and a mechanism for the organisation are required. It is
natural to suppose that cosmological evolution could furnish both. In this
work we suggest that fundamental criticality, as classified by Ehrenfest [2]
in terms of finding a Universe balanced on the precipice between two different
phases of existence, may actually be a generic cosmological occurrence. We
will show that, in relatively typical circumstances, the cosmological
evolution of empirical parameters is towards criticality, not from it.

The emergence of order out of disorder is not a new concept, rather it has
inspired the development of some of the most universal conceptual advancements
in theoretical physics. The prototypical ferromagnet furnishes a familiar and
cherished example. As the temperature of the system is lowered below some
critical value, order spontaneously emerges in the form of a bulk
magnetisation, with the prevalence of the order increasing to completeness at
zero temperature. The nature of the classical transition between the two
phases is diagnosed by a change in the symmetry of the ground state and also
by the analytic properties of the magnetisation as a function of temperature.
A first-order phase transition occurs for a discontinuity in the magnetisation
and second-order for a discontinuity in its derivative.

Phase transitions need not be classical. In a quantum phase transition it is
not the temperature which varies, but some background field such as the
magnetic field. The system may be held at a constant temperature, but a
transition between two distinct phases occurs as the background field $\phi$
is varied across some critical threshold. In practice, in the scalar
potential, $\phi$ is coupled as $V=(\phi-\phi_{c})\,\mathcal{O}$ to some
operator $\mathcal{O}$ whose expectation value changes as $\phi$ passes
through some critical value $\phi_{c}$. If $\langle\mathcal{O}\rangle$ is
discontinuous across the critical point, as in a first-order phase transition,
then $V^{\prime}=\partial V/\partial\phi$ is discontinuous. If, on the other
hand, $\langle\mathcal{O}\rangle$ changes continuously from
$\langle\mathcal{O}\rangle=0$ for $\phi>\phi_{c}$ to
$\langle\mathcal{O}\rangle\propto\phi_{c}-\phi$ for $\phi<\phi_{c}$, then
$V^{\prime\prime}$ is discontinuous at the critical point. In particle physics
parlay, it is the tadpole (first-order) and the mass (second-order) of the
scalar field, respectively, which are discontinuous. In this work, we seek to
study the role of quantum phase transitions in the early Universe, with a
particular focus on first-order transitions.

Classical phase transitions have long been studied in cosmology and are known
to have been instrumental for the evolution of the Universe, as exemplified by
the QCD and the electroweak phase transitions, not to mention the many
speculative hypotheses about other critical phenomena related to inflation,
grand unification or quantum gravity. While work remains to be done, the
theory of classical cosmological phase transitions is well established, with
the pattern of symmetry breaking and derivative discontinuities, manifesting
itself at a grander scale in the laboratory of the cosmos.

An obvious difference with cosmology is that in the laboratory one has the
freedom to increase or decrease temperatures, background fields, or other
parameters at will, and sit as close to, or far from, the critical points of
classical or quantum phase transitions as desired. On the other hand, the
inevitable march of the Hubble rate to lower values means that we may not
reverse cosmological evolution to replay, or tune parameters to sit
arbitrarily close to any possible cosmological phase transition. In the case
of classical cosmological phase transitions, our knowledge about the thermal
history of the Universe allows us to reconstruct with reasonable confidence
the conditions for their occurrence. However, the situation is more intricate
for quantum phase transitions, as their occurrence depends on the cosmological
evolution and fate of the $\phi$ background field.

In this work we establish some aspects of the theory of cosmological quantum
phase transitions. Unexpectedly we find that, under circumstances which are
not atypical, inflation in general relativity, quantum fluctuations of scalar
fields seeded by the inflating geometry and the discontinuities present at a
quantum critical point conspire in a special interplay that leads to a
universal phenomenon. When these features borrowed from the main pillars of
modern physics are combined together, they cooperate to localise scalar fields
exponentially close to the critical point.

The feature of a critical point acting as an attractor is closely akin to
self-organised criticality [1]. For this reason, we will call Self-Organised
Localisation (SOL) the phenomenon discussed in this paper, in which a period
of inflation induces a dynamical attraction towards the critical point of a
quantum phase transition. A very interesting, and related, point of view has
been presented in refs. [3, 4, 5]. Although the approach followed in refs. [3,
4, 5] is different from ours, an important common aspect is the use of methods
in quantum statistical mechanics and critical phenomena as central elements
for the description of physical processes in eternal inflation.

The prediction of near-criticality based on SOL has a probabilistic nature.
This is not surprising, as we are dealing with a quantum phenomenon. Just as
physicists a hundred years ago had to resign to the fact that phenomena at the
atomic level must be described probabilistically, so too predictions in the
domain of quantum cosmology may have a probabilistic nature. The important
difference is that, while in atomic physics the statistical sample is provided
by many laboratory measurements, in our setting the statistical sample is
given by different patches of the Universe not necessarily in causal contact.
The information about the statistical ensemble is distributed in the spacetime
geography of the full multiverse, even beyond the event horizon of our
observable patch. This causes several well-know interpretation problems that
will also be discussed in this paper.

The consequences of SOL for predicting physical parameters are profound, as
SOL represents a radical change of perspective with respect to the usual
intuition based on Effective Field Theories (EFT). When physical parameters in
an EFT are functions of more fundamental parameters that belong to an
underlying UV theory, they are expected to obey a specific structure dictated
by quantum mechanics and symmetry properties. In particular, the smallness of
an EFT parameter should be accounted for by the selection rules defined by
symmetry or else it is destabilised by large quantum fluctuations. SOL defies
this logic without modifying the power counting based on symmetry quantum
numbers and without introducing new-physics energy scales. The SOL reasoning
is that some EFT parameters, which are functions of scalar fields belonging to
an underlying theory, are attracted towards their critical values as the
result of the evolution during inflation of the fields governing the dynamics
of the corresponding quantum phase transition.

The mechanism of SOL may have various applications in the interface between
particle physics and cosmology. In this paper we start the exploration of the
phenomenon and discuss its implications for the determination of Standard
Model parameters. In sect. 2 we give some introductory material about the
dynamics of light scalars in an inflationary background. In sect. 3 we
describe the general features of SOL, studying specific examples of potentials
that exhibit critical behaviour. Next, we discuss how SOL could address some
of the outstanding open questions in fundamental physics: near-criticality of
the Higgs self-coupling (sect. 4), Higgs naturalness (sect. 5) and the
smallness of the cosmological constant (sect. 6). In sect. 7 we give a
comprehensive summary of our study, presenting all the key results. Finally,
the appendix, written in a self-contained form, outlines some general
properties of the stochastic equation and gives a compendium of analytical
solutions.

##  2 Scalar Fields in an Inflationary Universe

###  2.1 The Stochastic Approach

During inflation, scalar fields are subject to quantum fluctuations fuelled by
the Hubble rate. In the stochastic approach [6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20], the field evolution is governed by a Langevin-like
equation where quantum fluctuations correspond, qualitatively, to a random
walk. Starting from an initial distribution of field values in a patch of the
Universe, the distribution changes as the patch grows and the field values
fluctuate. After a sufficiently long time, with respect to a given foliation,
the distribution will take a stationary form which is universal, in the sense
that is independent of initial conditions but is specified only by field
boundary conditions. This asymptotic distribution can in principle be used to
determine the relative likelihood of different values of $\phi$ at the end of
inflation.

To determine this distribution one could calculate the average of a large
number of random walks using the Langevin approach. Equivalently, one may
instead use a Fokker-Planck (FP) equation which describes the average
behaviour of solutions to the Langevin problem [8, 10]. The relevant FP
equation, in proper time and with Ito ordering for the diffusion term [21,
22], is

$\frac{\partial}{\partial\phi}\left[\frac{\hbar}{8\pi^{2}}\frac{\partial(H^{3}\,P_{\rm
FP})}{\partial\phi}+\frac{V^{\prime}\,P_{\rm FP}}{3H}\right]=\frac{\partial
P_{\rm FP}}{\partial t}~{},$ (2.1)

where $P_{\rm FP}(\phi,t)$ is a normalised time-evolving probability
distribution, $H$ is the Hubble rate, $V(\phi)$ is the scalar potential and
$V^{\prime}$ its first derivative.

Instead of considering the probability distribution $P_{\rm FP}$, for our
purposes we are interested in determining the volume-weighted distribution of
field values, found from calculating $\langle e^{3Ht}\rangle$ averaged over
the random walks. This distribution will be referred to as $P(\phi,t)$ and, as
a function of proper time, is a solution to the volume-weighted Fokker-Planck
(FPV) equation [23, 12, 15, 19]

$\frac{\partial}{\partial\phi}\left[\frac{\hbar}{8\pi^{2}}\frac{\partial(H^{3}\,P)}{\partial\phi}+\frac{V^{\prime}\,P}{3H}\right]+3HP=\frac{\partial
P}{\partial t}~{}.$ (2.2)

Here $P(\phi,t)$ is _not_ a normalised time-evolving probability distribution,
but instead can be thought of as describing the volume fraction distribution
for $\phi$ obtained after averaging over a large number of field trajectories.
Once initial and boundary conditions for $P(\phi,t)$ are set, the evolution of
the distribution may thus be determined.

It is important to be clear on how to interpret the distributions $P_{\rm FP}$
and $P$. The function $P_{\rm FP}$ describes the probability distribution of
measuring the field value $\phi$ after some proper time when $\phi$ has
undertaken a random walk, as dictated by the Langevin equation. The function
$P$ is essentially reporting the probability distribution multiplied by the
volume expansion factor corresponding to the field value $\phi$. As a result,
in the multiverse we may assign a sort of probabilistic interpretation to this
quantity. For simplicity, suppose we begin with a large number of patches, all
with the same initial field value, and let them undertake their random walk,
which we stop at some specific proper time. We now add up the volume of all
patches, with their associated final field value, chop the entire multiverse
up into equal-sized portions and make a histogram of the field value in each
portion. This is the distribution reported by $P$.

We may not, as individual observers within our own Hubble patch, measure the
distribution of field values in the multiverse and this leads to several open
questions about the probabilistic interpretation of $P$ as an ‘observable’. In
the following section, we will discuss some of these conceptual difficulties
about the probabilistic interpretation of the solutions to the FPV equation.

###  2.2 On Time, Eternal Inflation and Quantum Gravity

We wish to calculate the volume distribution of $\phi$ values and ascribe to
it a probabilistic interpretation. One could take an abstract view of $P$,
removed from questions of observers, and interpret it probabilistically at any
timescale. Indeed, this is the approach often taken to consider important
dynamical questions such as the onset of eternal inflation and the approach to
steady states. On the other hand, as also often noted, if we wish to interpret
$P$ as representing the likelihood of measurements of fundamental parameters
performed by observers we must define observers more carefully. Importantly,
these two subtly different interpretations of $P$ may have profoundly
different outcomes if inflation is eternal.

Since observers can only exist on patches of the multiverse which have
reheated after inflation has terminated, a meaningful statistical sample to
compare probabilities across the multiverse is given by patches that live on
the 3-volume hypersurface of all reheating events in spacetime, which is
called the ‘reheating surface’ (for a discussion see, e.g., chapter 6 of ref.
[24]).

In particular, this implies that the field $\phi$, which governs the SOL
dynamics, in no sense can play the role of the inflaton. By definition,
inflation would terminate at a specific $\phi$ value and thus all ‘observers’
would measure an essentially unique value of $\phi$, determined by the
reheating process, instead of being part of a statistical distribution $P$
which carries the fingerprints of SOL. However, note that this does not
preclude the possibility that $\phi$ could give the dominant contribution to
the vacuum energy density in certain field regions.

While the reheating surface is the bridge between the stochastic approach
during inflation and probabilistic predictions of observables in inhabitable
universes, its calculation requires knowledge of the specific model of
inflation. Given a certain model we could, in principle, determine the joint
$\phi$ and inflaton distribution, from which we could derive the probabilistic
distribution of physical observables. However, given that the true model of
inflation is still unknown, in this work we treat inflation as a fixed
background hosting the $\phi$ dynamics, with the goal of determining the
inflaton-independent properties of the $\phi$ distribution.

The problem of identifying the reheating surface is further exacerbated by the
fact that, for the parameters chosen in all the phenomenological applications
that we will consider in this work, the SOL dynamics occurs in a regime where
inflation is eternal [6, 25, 26, 27]. This does not, however, imply that all
applications of SOL necessarily require eternal inflation as a fundamental
ingredient. In non-eternal inflation the definition of the reheating surface
is unambiguous: all spacetime trajectories eventually reach the reheating
stage and inflation ends everywhere within a finite time. The most likely
field value measured by an observer corresponds to the patch that occupies the
largest volume fraction of the total reheating surface. Instead, in eternal
inflation, the reheating surface is infinite because at all times there exist
trajectories that are still inflating. This leads to the well-known ‘measure
problem’ (as reviewed, for example, in refs. [24, 28]), which afflicts any
prediction from eternal inflation.

In eternal inflation, probabilities obtained with any volume-based measure are
independent of the initial conditions and instead correspond to a stationary
state which expands, in some time foliation, uniformly at a constant rate [29,
15, 19, 18, 30, 31]. This stationarity follows automatically from the eternal
nature of inflation and is independent of choice of cut-off or time foliation
(for a discussion see again chapter 6 of ref. [24]), as it can be understood
from the volume-weighting, which exponentially favours observers at the latest
possible times. As a result, in practical terms the fraction of all observers
who have a finite proper time in their past light cone is zero, even though
their spacetime is past-incomplete (see e.g. [32, 33, 34, 35, 36, 37] for
discussions on these aspects, including the distinction between technical and
practical finiteness). This renders any question of dynamics from the
perspective of observers moot, since only a stationary solution remains at
infinite time. However, this needs not be necessarily the case for an
interpretation of $P$ without observers, where dynamics of the solution can
occur at finite times.

Problematically, the specific form of this stationary distribution is not
unambiguous. As mentioned, in practice the ‘clock’ which is stopped when
inflation ends is the inflaton itself when it arrives at the reheating point
on the potential. Absent a model of inflation we have opted to consider a
time-like variable as a proxy for the inflaton, noting that this ansatz may or
may not be valid for different inflationary models. Moreover, the appropriate
choice of time parametrisation depends on how the infinite reheating surface
is regularised. As the solution $P$ depends on the prescription of time
foliation we say it has a residual gauge dependence. Physically, this can be
understood by observing with a specific example that, instead of computing the
$\phi$ distribution that enters a steady-state with respect to proper time,
one could also consider a distribution whose $\phi$ functional form becomes
independent of the number of $e$-foldings that have occurred.

To make this explicit, following refs. [21, 38], we define a one-parameter
family of gauges $\xi$ (with $0\leq\xi\leq 1$) characterised by the time
coordinate $t_{\xi}$ such that $dt_{\xi}/dt=(H/H_{0})^{1-\xi}$, where $H_{0}$
is a reference constant value of the Hubble rate. In terms of the new time
coordinate, eq. (2.2) becomes

$\frac{\partial}{\partial\phi}\left[\frac{\hbar}{8\pi^{2}}\frac{\partial(H^{2+\xi}\,P)}{\partial\phi}+\frac{V^{\prime}\,P}{3H^{2-\xi}}\right]+3H^{\xi}P=H_{0}^{\xi-1}\,\frac{\partial
P}{\partial t_{\xi}}~{},$ (2.3)

where we have redefined $P\to(H_{0}/H)^{1-\xi}P$. Proper-time gauge
corresponds to $\xi=1$ and $e$-folding gauge to $\xi=0$, with values in
between interpolating between the two.

After expanding $H$ around a background inflationary value $H_{0}$ and
dropping sub-dominant terms, the only remaining effect of $\xi$ is to multiply
the volume term. The residual $\xi$-dependence is therefore a characteristic
of the FPV extension of the usual Fokker-Planck treatment. The parameter $\xi$
encodes the ambiguity coming from our inability to determine the reheating
surface and, therefore, the statistical sample of universes over which
probabilities are computed. As we will show in the following, many of the
consequences of SOL are fairly robust against variations of $\xi$, with
exponential localisation persisting as long as $\xi\neq 0$. In particular, in
all applications that we will consider, the exponential localisation is lost
only if $\xi=0$ with an extraordinary accuracy, a case which corresponds to a
strict choice of $e$-folding gauge or scale factor measure [39, 40, 41, 42,
43, 44], for which volume-growth effects are absent. Contrary to the case
$\xi=0$, in measures for which volume effects can dominate at late times a
number of paradoxes arise, particularly the ‘Youngness Paradox’. The extent to
which this paradox afflicts interpretations of the $\phi$ distributions
calculated here will be discussed in detail in sect. 2.3.

A different concern, detailed in refs. [45, 46, 47, 48], is that an observer
in an asymptotically flat region can only measure a limited number of
inflationary modes, bounded by the entropy of the surrounding dS space. This
argument limits the amount of dS space outside the horizon whose modes would
eventually enter the horizon, giving the following bound on the number of
$e$-foldings possible in non-eternal inflation

$N<{S_{\\!{{\rm dS}}}}=\frac{8\pi^{2}M_{P}^{2}}{\hbar\,H^{2}}~{}.$ (2.4)

In our phenomenological applications of SOL, the bound in eq. (2.4) is grossly
violated. However, this only reiterates SOL’s need for eternal inflation when
applied to questions of phenomenological relevance to fundamental physics. In
eternal inflation an observer can never access all of the spacetime volume
after reheating, thus eq. (2.4) is not a reason for concern.

The presence of eternal inflation raises the question of whether the semi-
classical stochastic approach to the dynamics of $\phi$ remains valid
eternally. Indeed, in [49, 50] it was argued that, when the total number of
$e$-foldings violates the bound of eq. (2.4), the semi-classical treatment
breaks down due to the large occupation number of gravitons. It is not clear,
however, in what way the stochastic treatment should break down. For instance,
in a QFT it is possible for UV effects to have a significant impact on long
timescales without the EFT approach ever breaking down.111An EFT can undergo a
change of regime after a sufficiently long timescale, with UV quantum effects
dominating over IR classical effects. However, this does not necessarily imply
a breakdown of the effective theory, as long as the energies involved are much
smaller than the cutoff. To illustrate this point with an example, consider
proton scattering processes at nuclear energies $E_{IR}$, which are
appropriately described by a low-energy effective theory. In general, the low-
energy theory must include higher-dimensional operators (e.g. generated by GUT
interactions at the mass scale $M_{UV}$) that mediate proton decay with a rate
$\Gamma\sim E_{IR}^{5}/M_{UV}^{4}$. The effects of UV interactions are
negligible for normal scattering experiments, but they become dominant after a
time $t\sim 1/\Gamma$ when they completely change the state of the system as
protons disappear from the colliding beams. Although UV interactions dominate
the outcome of the experiment, there is no breakdown of the effective-theory
description. In other words, as long as we scatter protons at low energies,
the processes are adequately described by the effective theory even without
detailed knowledge of the UV microscopic dynamics. Of course, it is not
guaranteed that these EFT-based considerations should hold for gravity as
well, but let us suppose that this is the case. The FPV solutions show a
change of regime after a time $t\sim 1/\Gamma$, where $\Gamma\sim
H^{3}/M_{P}^{2}$ is the typical rate of quantum-gravity processes. Just like
in the GUT example, beyond this critical time the dynamics is dominated by UV
quantum effects. Nonetheless, as long as the curvature ($\sim
H^{2}/M_{P}^{2}$) is small, we posit that the GR effective theory gives an
appropriate description and no knowledge of the full quantum-gravity dynamics
is needed. Exceeding the critical timescale only gives enough time for the
system to become dominated by UV quantum effects, but does not necessarily
imply a breakdown of the effective-theory validity. Returning to the
inflationary case, recent calculations appear to support the validity of the
stochastic approach even at long times (see, for instance, [46, 47, 48, 51,
52, 53] for related discussions and in particular [54] for a recent discussion
of the validity of the stochastic approach in eternal inflation). Thus, while
the phase transition to eternal inflation does occur when eq. (2.4) is
satisfied, which is a very nontrivial outcome, it is not clear that the
stochastic description of $\phi$ dynamics should be called into question. As a
result, in our study we will take for granted that, even in an eternally
inflating background, the dynamics of $\phi$ can be described through the
semi-classical stochastic approach.

A final question concerns the Swampland conjectures (see e.g. ref. [55] for a
review). There are two facets to consider. The first is inflation. The
Swampland de Sitter conjecture is [56]

$|\nabla V|>c\frac{V}{M_{P}}~{},$ (2.5)

where $c$ is an $\mathcal{O}(1)$ parameter. This suggests that the slow-roll
parameter $\epsilon$ should also be an $\mathcal{O}(1)$ parameter, in
contradiction with slow-roll itself. Thus a naïve contradiction with the
Swampland conjectures arises already within the context of generic, but not
necessarily all, slow-roll inflationary models. Furthermore, the condition for
eternal inflation, written in terms of the Swampland de Sitter conjecture
[57], is

$\frac{H}{M_{P}}\gtrsim 2\pi c~{}.$ (2.6)

Thus, unless the Hubble rate is parametrically close to the Planck scale, on
the edge of the validity of a semi-classical approach, the eternally inflating
backgrounds we will assume throughout are likely in the Swampland. Similarly,
the $\phi$ potential required for SOL will also typically violate both the
Swampland de Sitter Conjecture and the Distance Conjecture [58, 59], thus it
would seem that both the inflationary sector and SOL sectors are firmly in the
Swampland. Given the incertitude of these conjectures we will not attempt to
address this tension here, although it should be kept in mind that the
fundamental ingredients for SOL appear, at least at first glance, to be at
odds with UV-completion within string theory. For further discussion on this
see, for example, [56, 60, 61, 62, 63, 64, 65].

###  2.3 The Measure Problem

Many solutions have been proposed to the measure problem. One of the earliest
attempts was the proper time cut-off measure[18, 19]. For finite proper time,
the reheating surface is finite. As this time cut-off is taken to infinity the
ratio of volume distributions remains constant. Predictions in proper time
cut-off measure therefore correspond to the asymptotically stationary volume
distribution in proper time. Unfortunately, this measure is in conflict with
observation due to an exponential preference for younger vacua over older ones
such as ours [66, 25, 67, 28] As well as predicting apparent violations of the
Copernican principle that our location in the Universe is not special [67,
68].222An additional potential problem relates to ‘Boltzmann Brains’. However,
since our models all predict reheated patches which must ultimately decay to
AdS, either by tunnelling or $\phi$ rolling to AdS, it is not clear to what
extent this problem would apply to our setup.

#### The Youngness Paradox

To see this Youngness Paradox, consider, for example, a metastable false
vacuum for the inflaton $\chi$ at $\chi_{A}$ whose potential energy density is
greater than that of a stable vacuum at $\chi_{B}$, where we assume reheating
occurs relatively soon after tunnelling to vacuum B. The Hubble expansion rate
at $\chi_{A}$ is greater than that at $\chi_{B}$, $H_{A}>H_{B}$, but in
equilibrium the ratio of volume distributions $P_{A,B}(t)\equiv P(\chi,t)$ at
$A$ and $B$ is constant so they must be expanding at the same rate,

$P_{A}(t)=P_{A}(0)e^{3H_{A}t}\quad,\quad P_{B}(t)=P_{B}(0)e^{3H_{A}t}\,.$
(2.7)

This is due to a vacuum decay flux from $A$ to $B$ which maintains the steady
state. The pre-existing $B$ vacuum from $t=0$ has a volume
$P_{B}^{\text{old}}(t)=P_{B}(0)e^{3H_{B}t}$ after a proper time $t$. The new
vacuum created by $A$ decays to $B$ is
$P_{B}^{\text{new}}(t)=P_{B}(t)-P_{B}^{\text{old}}(t)$. The volume ratio of
new to old vacua is then

$\frac{P_{B}^{\text{new}}(t)}{P_{B}^{\text{old}}(t)}=e^{3(H_{A}-H_{B})t}-1\,.$
(2.8)

We therefore see that, in steady state where everything expands at the same
rate, there is an exponential preference for newly created vacua over older
ones. This result generalises to slow-roll inflationary potentials. In this
case, reheating occurs whenever the inflaton reaches the reheating point on
the inflaton potential. However, on the trajectories leading up to that event,
the inflaton will typically be much further up the potential with
significantly higher Hubble rate.

In this work, the field $\phi$ is not the inflaton and the $\phi$-distribution
is sampled at the moment of reheating, which is unrelated to the $\phi$
dynamics. There are three important differences between $\phi$ and the
inflaton that must be kept in mind regarding questions such as the Youngness
Paradox.

  * •

The $\phi$ potential is extremely shallow and the field range exponentially
larger than the Planck scale. Thus if we repeat the above exercise the
furthest up the potential that $\phi$ could have explored relative to the
reheated patch is $|\Delta\phi|\lesssim H_{B}H_{A}t$. For $t\sim 1/H_{B}$, as
in our Universe at the present day, this becomes $|\Delta\phi|\lesssim H_{A}$.
In all applications we will consider the $\phi$-potential is so shallow that
the difference in Hubble rate for $|\Delta\phi|\sim H_{A}$ is significantly
smaller than the present day Hubble constant, thus there is no Youngness
Paradox pertaining to $\phi$ in this regard.

  * •

In all of our applications we will also find that the $\phi$ distribution is
highly localised in a field range over which the change in Hubble constant
changes by an amount smaller than the present day Hubble constant, so no
Youngness Paradox is associated with the stationary distribution of $\phi$
which is sampled.

  * •

In all of our applications the $\phi$ potential is so shallow that subsequent
to reheating $\phi$ rolls by a distance which changes the Hubble constant by
an amount smaller than the present day value of the Hubble constant, so no
Youngness Paradox is associated with the subsequent dynamics of $\phi$ after
inflation, as compared to patches that reheated more recently.

Therefore, as regards the dynamics of $\phi$, patches of the Universe which
are 13.7 billion years old have effectively the same value of $\phi$ and
$V(\phi)$ as patches which have just reheated, hence there is no Youngness
Paradox associated with $\phi$, even if we work in proper time gauge.

This suggests that the Youngness Paradox is decoupled from the detailed form
of the stationary $\phi$ distribution. However, it remains to understand if
working in a proper-time-like gauge for $\phi$ dynamics could implicitly mean
a Youngness Paradox is reintroduced in the inflaton sector by proxy. To see
this, we now consider two proposed measures that avoid the Youngness Paradox
for the inflaton and discuss their application when including the $\phi$
distribution.

#### Scale factor measure

In our first example, we may employ scale factor cut-off measure in e-folding
gauge for the inflationary sector, where e-foldings in the simple false vacuum
model above is $N_{A}=H_{A}t$. Working with this variable is a constant
relabelling of proper time as regards $\phi$ dynamics and the form of the
stationary distribution for $\phi$ is unchanged. The inflationary sector in
this measure does not suffer from a youngness problem, yet the stationary
$\phi$ distribution is the same as in proper-time gauge and there is no
Youngness Paradox stemming from the $\phi$ measure. We therefore see that
stationarity for the $\phi$ distribution in one gauge does not necessarily
imply stationarity for inflaton dynamics in the same gauge, hence questions
concerning the Youngness Paradox are relegated to the assumptions made for the
measure applied to the inflaton dynamics.

More generally, the decoupling of the measure paradoxes between $\phi$ and
inflationary sectors can become more pronounced in asynchronous gauges which
weight volumes according to spacetime events. Such a prescription seems
plausible given that the effective past-eternity of eternal inflation for the
majority of observers may suggest that the important clock registers events
according to the termination of inflation, rather than at some fixed time at
which it began. In this case conditioning on reheating gives the inflationary
sector a special prominence as compared to the $\phi$ dynamics.

#### Stationary measure

In our second example, one may consider the stationary measure for both the
inflationary _and_ $\phi$ sectors whose prescription circumvents the Youngness
Paradox in proper time for the inflationary sector [69, 70]. Contrary to the
previous discussion, in this case the _same measure_ is assumed for both
fields. The toy model of eternal inflation in ref. [38] is similar to the
previous false vacuum setup, but after the inflaton $\chi$ tunnels from $A$ to
$B$, it slow-rolls along a shallow potential lasting a fixed
$N_{B}=H_{B}t_{B}$ e-foldings before reheating. In addition, there are two
such shallow potentials: one in domain $B_{-}$ and the other in domain
$B_{+}$, both with the same energy density. Reheating occurs at $\chi_{-}$ for
the $B_{-}$ vacuum and at $\chi_{+}$ for $B_{+}$. They are expanding with the
same Hubble rate $H_{B}$, but due to the difference in slow-roll distance to
reach the point of reheating one has a smaller number of e-foldings than the
other, $N_{B-}<N_{B^{+}}$.

In this example setup we wish to compare the volume distributions in the two
domains $B_{-}$ and $B_{+}$ at reheating (the post-reheating evolution is
identical for the two so we neglect this here). Following [38], this can be
estimated as follows. The volume distribution at $A$ grows by inflationary
expansion and shrinks due to decay to either $B_{+}$ or $B_{-}$ with equal
rate $r<3H_{A}/2$ per unit time,

$\frac{dP_{A}(t)}{dt}=(3H_{A}-2r)P_{A}(t)\quad\Rightarrow\quad
P_{A}(t)=P_{A}(0)e^{(3H_{A}-2r)t}\,.$ (2.9)

The total volume at $B_{\pm}$ increases by the rate $r$ from $A$ vacuum decay
with an initial volume $P_{A}(t_{A})$ determined by the time $t_{A}$ spent in
the metastable vacuum $A$. There is also a fixed expansion factor from the
slow-roll phase in $B$ that lasts $N_{B^{\pm}}=H_{B}t_{B^{\pm}}$ e-foldings.
The total time is then $t=t_{A}+t_{B^{\pm}}$. We have

$\displaystyle\frac{dP_{B^{\pm}}(t)}{dt}$
$\displaystyle=re^{3H_{B}t_{B^{\pm}}}P_{A}(t-t_{B^{\pm}})\,$ (2.10)
$\displaystyle\Rightarrow\,P_{B^{\pm}}(t)$
$\displaystyle=\frac{rP_{A}(0)}{3H_{A}-2r}e^{3N_{B^{\pm}}}\exp\left[\frac{(3H_{A}-2r)}{H_{B}}(N-N_{B^{\pm}})\right]\,,$
(2.11)

where we have dropped negligible corrections of $\mathcal{O}(H_{B}/H_{A})$ in
eq. (2.10). The ratio of volume distributions at $B_{+}$ and $B_{-}$ then
tends to a constant,

$\lim_{t\to\infty}\frac{P_{B^{+}}}{P_{B^{-}}}=e^{3(N_{B^{+}}-N_{B^{-}})}\exp\left[-\frac{(3H_{A}-2r)}{H_{B}}(N_{B^{+}}-N_{B^{-}})\right]\,.$
(2.12)

We see that the volume distribution at $B_{+}$ is exponentially suppressed
with respect to $B_{-}$. Proper time cut-off measure therefore favours the
vacuum with a smaller number of e-foldings, $N_{B^{-}}<N_{B^{+}}$. The reason
is that less e-foldings means more time spent inflating at a higher expansion
rate in the metastable vacuum $A$.

Now, in stationary measure we wish to compare the volume distributions at
reheating in $B_{\pm}$, not from the total time $t$ but synchronised from the
point where they were respectively created after tunnelling. This means
undoing the volume expansion from sitting in the metastable vacuum $A$, since
the time before tunnelling is irrelevant. The physically meaningful comparison
is between the two vacua when they were created and expanded at the same rate.
The stationary measure volume distribution modifies eq. (2.11) to become

$\displaystyle\left.P_{B^{\pm}}(t)\right|_{\text{stationary measure}}$
$\displaystyle=\exp\left[-\frac{(3H_{A}-2r)}{H_{B}}(N-N_{B^{\pm}})\right]P_{B^{\pm}}(t)$
(2.13) $\displaystyle=\frac{rP_{A}(0)}{3H_{A}-2r}e^{3N_{B^{\pm}}}\,,$ (2.14)

so the volume ratio is

$\lim_{t\to\infty}\frac{P_{B^{+}}}{P_{B^{-}}}=e^{3(N_{B^{+}}-N_{B^{-}})}\,,$
(2.15)

which now exponentially favours vacuum $B_{+}$ with longer e-foldings since
there is no longer any gain from inflating in $A$.

If we add a scalar field $\phi$ to the above toy model then the volume
distribution becomes a function of both $\phi$ and the inflaton $\chi$,
$P(\chi,\phi,t)$. However, just like the post-inflationary evolution is
identical at $B_{+}$ and $B_{-}$ and so does not affect the stationary measure
procedure, so are the dynamics and distribution of $\phi$ the same for both
domains. In practise this means we can write $P(\chi,\phi,t)\simeq
P(\chi,t)P(\phi,t)$, assuming a negligible dependence of the $\phi$
distribution on the varying inflationary Hubble rate that is treated as
approximately constant for the FPV dynamics of $\phi$. Moreover, the
stationary measure mandates synchronisation when in the stationary regime with
respect to all fields and processes, which implies $P(\phi,t)$ must be the
asymptotically stationary distribution for $\phi$,
$P(\phi)\equiv\lim_{t\to\infty}P(\phi,t)$. The main point is that the
synchronisation procedure of stationary measure concerns only the inflaton
sector and is independent of $\phi$. The inflationary volume expansion that is
undone by the stationary measure is that of the inflaton which from the point
of view of $P(\phi)$ is just an overall rescaling. Similarly, from the
inflaton’s point of view the $\phi$ distribution and energy density is a
constant background that is sampled identically in both domains.

In summary we see that, while the various measure problems remain in eternal
inflation, they apply in a decoupled manner to the inflationary and $\phi$
sectors. Due to the localisation of the stationary $\phi$ distribution and the
shallow $\phi$-potential there is no Youngness Paradox associated with $\phi$,
even though some additional prescription must presumably be applied to resolve
the measure problem paradoxes associated with the inflaton sector. The above
two examples illustrate how this may be applied in the context of SOL. As a
result of these considerations we expect that when the correct understanding
and/or framework is found for eternal inflation resolving the various
paradoxes, the qualitative features of SOL will remain.

With all of these independent, but interrelated, aspects in mind we may now
proceed to detailed calculations of $P$.

###  2.4 EFT Potential

To illustrate the nature of the dynamics of the field distribution, and the
ultimate stationary configurations, we must solve the FPV equation explicitly.
We will consider the following class of potentials

$V=3H_{0}^{2}M_{P}^{2}+g_{\epsilon}^{2}f^{4}\omega(\varphi)~{},~{}~{}~{}~{}\omega(\varphi)=\sum_{n=1}^{\infty}\frac{c_{n}}{n!}\varphi^{n}~{},~{}~{}~{}~{}\varphi\equiv\frac{\phi}{f}~{},~{}~{}~{}~{}\omega(0)=0~{},$
(2.16)

which is characterised by three quantities: (i) a constant background vacuum
energy $V_{0}=3H_{0}^{2}M_{P}^{2}$ which drives the inflationary expansion;
(ii) a scale $f$ which defines the field range of $\phi$ where we trust the
EFT approach, such that $|\varphi|\leq 1$; (iii) a coupling constant
$g_{\epsilon}$ which defines an overall size of the $\phi$ potential. The
parameters $f$ and $g_{\epsilon}$ encode the physical information about $\phi$
interactions, while the Wilson coefficients $c_{n}$ are $\mathcal{O}(1)$
numbers with no large hierarchies, implying that $\omega(\varphi)$ and all of
its derivatives are also $\mathcal{O}(1)$ quantities across the field slice.

We employ the EFT in the regime

$V_{0}=3H_{0}^{2}M_{P}^{2}\gg g_{\epsilon}^{2}f^{4}~{}.$ (2.17)

This could result from dynamics in which the vacuum energy density associated
with $\phi$ gives only a small modulation of the constant background value
$V_{0}$ that is primarily responsible for the inflationary process. But eq.
(2.16) could also be viewed as an expansion around a generic field point
(taken at $\varphi=0$ with an appropriate coordinate choice) where
$V_{0}=V(\varphi\\!=\\!0)$ is the leading constant term. In this case, $f$ is
interpreted as the largest field excursion such that $|V(\phi)-V_{0}|\ll
V_{0}$. For this reason, we will call perturbative range the field region in
which eq. (2.17) is satisfied.

The structure in eq. (2.16) encompasses a general class of scalar theories,
but in our applications we will be particularly interested in the case in
which $g_{\epsilon}\ll 1$ and $f\gg M_{P}$. The hypothesis $g_{\epsilon}\ll 1$
is naturally realised when $\phi$ is a Goldstone boson coming from an
underlying spontaneously broken global symmetry, while $g_{\epsilon}$ measures
the amount of explicit symmetry breaking.

For potentials of the form in eq. (2.16), we can use the condition in eq.
(2.17) and expand the FPV equation at leading order in $g_{\epsilon}$

$\frac{\alpha}{2}\frac{\partial^{2}P}{\partial\varphi^{2}}+\frac{\partial\left(\omega^{\prime}P\right)}{\partial\varphi}+\beta\omega
P=\frac{\partial P}{\partial T}~{}.$ (2.18)

The dimensionless parameters $\alpha$, $\beta$ and $T$ are defined as

$\alpha=\frac{3\hbar
H_{0}^{4}}{4\pi^{2}g_{\epsilon}^{2}f^{4}}~{},~{}~{}~{}~{}\beta=\frac{3\xi
f^{2}}{2M_{P}^{2}}~{},~{}~{}~{}~{}T=\frac{t}{t_{R}}~{},~{}~{}~{}~{}t_{R}=\frac{3H_{0}}{g_{\epsilon}^{2}f^{2}}~{}.$
(2.19)

We also reabsorbed the constant part of the expansion term through a
redefinition

$P\to e^{3H_{0}t}P~{}.$ (2.20)

By separation of variables, we can write $P$ as a linear combination of the
stationary distributions $p$, whose field functional dependence does not
change form with time,

$P(\varphi,T)=\sum_{\lambda}e^{\lambda T}p(\varphi,\lambda)~{}~{}.$ (2.21)

Since any initial state is built out of the eigenfunctions $p$, the one with
the greatest eigenvalue ($\lambda=\lambda_{\rm max}$) will eventually come to
dominate the distribution in steady state, for any physical initial condition.

Let us consider the parameters entering eq. (2.18) and their interpretation.
First take $t_{R}$. The typical gradient of the EFT potential is
$V^{\prime}\sim g_{\epsilon}^{2}f^{3}$ and the slow-roll velocity is
$\dot{\phi}=V^{\prime}/3H_{0}$. Thus the typical time taken to classically
evolve along the potential is $\Delta t\sim 3H_{0}\Delta\phi/V^{\prime}\sim
t_{R}$. If the dynamics is dominantly quantum, in the sense that quantum
fluctuations exceed the rate of classical evolution, then relaxation is much
slower. Thus $t_{R}$ represents an estimate of the minimum time it takes for
relaxation to a stationary state. Indeed, explicit calculations show that the
evolution time of the system is $\alpha\beta\,t_{S}$ for $\alpha\beta\ll 1$,
and $\sqrt{\alpha\beta}\,t_{S}$ for $\alpha\beta\gg 1$, where
$t_{S}={S_{\\!{{\rm dS}}}}/H_{0}$ is the largest timescale allowed by the
entropy bound, see eq. (2.4). Thus, for $\alpha\beta\gg 1$, one is implicitly
discussing relaxations on timescales so long that eternal inflation is an
inevitable assumption.

The parameter $\alpha$ is proportional to $\hbar$ and thus it describes a
quantum effect. The smaller $\alpha$ the more classical the evolution, with
$\alpha\to 0$ furnishing the classical limit. This result can also be
understood by estimating, on dimensional grounds, the ‘thermal’ dS corrections
to the scalar potential as $\Delta V\propto T_{\rm dS}^{4}$ where $T_{\rm
dS}=H_{0}/2\pi$ is the Gibbons-Hawking temperature [71]. As a result we see
that $\alpha\sim T_{\rm dS}^{4}/V$ parameterises the scale of ‘thermal’ dS
corrections to the potential relative to the overall height of the potential.
Thus, if $\alpha\ll 1$, the evolution in the FP equation will be dominantly
classical. However, the condition that the $\phi$ potential gives only a small
contribution to the background vacuum energy, see eq. (2.17), gives the lower
bound $\alpha\gg 1/{S_{\\!{{\rm dS}}}}$.

Finally, $\beta$ parameterises the field range in Planck units. It also
encodes all the information about the $\xi$-dependence in the perturbative
FPV. In our applications, we will be interested in the case $\beta\gg 1$, with
a large number of $e$-foldings during which the field makes super-Planckian
field excursions. Although the energy density always remains sub-Planckian,
the regime $\beta\gg 1$ is in conflict with the Swampland ‘Distance
Conjecture’ [58].

More relevant for this work, however, is that $\beta$ sets the scale of the
volume effects, allowing to identify a third regime of parameter space by
treating effects on purely perturbative grounds. As discussed, if $\alpha\ll
1$ then the dynamics is dominantly classical. However, a quantum fluctuation
combined with volume growth will have effects which scale, perturbatively, as
$\alpha\beta$. Thus, if $\alpha\beta\gg 1$ then, despite the fact that quantum
fluctuations are small, we should expect to enter a regime in which the
combination of quantum and volume effects dominates the dynamics. It may seem
that we are in presence of a strongly-coupled regime, but one can still trust
the result because the FPV follows directly from the FP solution, which is
entirely calculable also in this case. Going further, if $\alpha^{2}\beta\gg
1$ then NLO quantum effects combined with volume growth will also be greater
than classical evolution, suggesting a transition to another regime of
solutions. In sect. 2.6 we will show more precisely how these changes of
regime occur and what form the solutions take in each.

###  2.5 Boundary Conditions and Eigenvalues

To calculate the stationary state at large times one must supplement the FPV
equation with boundary conditions at the endpoints of the field range.
Boundary conditions encode information of the dynamics beyond the EFT field
range and their choice requires knowledge of the UV completion. From an EFT
perspective, all we can do is to make reasonable guesses. A simple, but
relatively generic, assumption is to take absorbing boundary conditions at the
field endpoints $P(\phi_{\text{min},\text{max}})=0$, which correspond to
discarding every path which ventures beyond these boundaries. Physically, this
means that one considers only trajectories which remain within the EFT.
Alternatively, one could choose reflecting boundary conditions,
$P^{\prime}(\phi_{\text{min},\text{max}})=0$, implying that UV physics
repulses any trajectory incident on the boundary.

If the potential $V$ and its first derivative are continuous functions of the
scalar field then, for any linear combination of absorbing or reflecting
boundary conditions, the Sturm-Liouville theorem ensures that the eigenvalue
spectrum is uniquely determined. Hence, the asymptotic stationary state is
fully determined by the choice of boundary conditions and any information
about the initial condition is erased.

###  2.6 Positivity

Volumes are positive quantities and probabilities are too, thus any physical
volume-weighted field distribution is strictly positive. In particular, the
eigenfunction $p(\lambda_{\rm max})$ must be positive everywhere because, up
to a time-dependent (but field-independent) volume expansion factor, it is
equal to the asymptotic physical distribution. This property offers a powerful
criterion for identifying a physical stationary state without solving the full
eigenvalue problem because positivity typically singles out $p(\lambda_{\rm
max})$ among all possible FPV stationary solutions satisfying given boundary
conditions.

Let us consider the FPV equation for a monotonically increasing potential $V$,
whose form is completely general and not restricted to the EFT structure nor
to the perturbative range. We use the spectral representation

$P(\phi,t)=\sum_{\lambda_{\scriptscriptstyle
H}}\,e^{3\lambda_{\scriptscriptstyle
H}^{\xi}H_{0}t}\,p(\phi,\lambda_{\scriptscriptstyle H})~{},$ (2.22)

where $H_{0}$ is a constant reference value of the Hubble rate. The
parametrisation of the eigenvalues has been chosen to conform with the overall
expansion rate of each steady-state solution, which is given by
$3H_{0}(H/H_{0})^{\xi}$. The FPV for the spectral modes is

$\frac{\hbar\,H^{2}}{8\pi^{2}}\,p^{\prime\prime}+\frac{M_{P}^{2}V^{\prime}}{V}\,p^{\prime}+\left[3-3\Big{(}\frac{\lambda_{\scriptscriptstyle
H}H_{0}}{H}\Big{)}^{\xi}+(\xi-2)\epsilon+\eta\right]p=0~{},$ (2.23)
$\epsilon=\frac{M_{P}^{2}\,V^{\prime
2}}{2\,V^{2}}~{},~{}~{}~{}\eta=\frac{M_{P}^{2}\,V^{\prime\prime}}{V}~{},$
(2.24)

where $\epsilon$ and $\eta$ must be smaller than one in slow-roll regime and
therefore can be neglected in eq. (2.23) anywhere outside the field region
where $H$ is approximately equal to $\lambda_{\scriptscriptstyle H}H_{0}$.

Suppose that the distribution $p$ is peaked around a field value
${\bar{\phi}}$ (such that $p^{\prime}({\bar{\phi}})=0$) with width
$\sigma^{2}\equiv-p({\bar{\phi}})/p^{\prime\prime}({\bar{\phi}})$. Then eq.
(2.23) in slow-roll regime yields the relation

$\Big{(}\frac{\lambda_{\scriptscriptstyle
H}H_{0}}{\bar{H}}\Big{)}^{\xi}=1-\frac{\hbar\,{\bar{H}}^{2}}{24\pi^{2}\sigma^{2}}~{},~{}~{}~{}~{}{\bar{H}}\equiv
H({\bar{\phi}})~{}.$ (2.25)

As long as $\sigma\gg{\bar{H}}$, we find that $\lambda_{\scriptscriptstyle
H}$, which measures the expansion rate of the stationary solution, is
proportional to the Hubble rate evaluated at the peak and
$\lambda_{\scriptscriptstyle H}={\bar{H}}/{H_{0}}$.

In analogy with an algebraic quadratic equation, we can define a
‘discriminant’ of eq. (2.23)

$D=2\epsilon+\frac{3\hbar\,H^{2}}{2\pi^{2}M_{P}^{2}}\bigg{[}\left(\frac{\lambda_{\scriptscriptstyle
H}H_{0}}{H}\right)^{\xi}-1\bigg{]}~{},$ (2.26)

working at leading order in slow-roll parameters. The ‘discriminant’ $D$
provides a useful means for identifying when a stationary solution becomes
negative since the FPV solutions become oscillatory in the vicinity of a field
point where $D$ turns negative. This can be understood by comparing the
second-order linear differential equation to an algebraic quadratic equation
and using the criterion for determining when the roots are real or imaginary.
Since negative discriminant implies oscillatory, hence negative, solutions we
infer that the solution with the largest eigenvalue is the one which remains
positive-definite over the greatest field range.

Equation (2.26) shows that positivity is always satisfied below the peak
($H\ll\lambda_{\scriptscriptstyle H}H_{0}$). However, well above the peak
($H\gg\lambda_{\scriptscriptstyle H}H_{0}$), the solution remains positive
only if

$V^{\prime}>H^{3}~{}.$ (2.27)

We recognise this inequality as the familiar Classical-beats-Quantum (CbQ)
condition. We stress that positivity requires CbQ only above the peak and not
in the full field range.

If CbQ does not hold, then eq. (2.26) gives us information about the location
of possible peaks in the stationary solution. Indeed, positivity requires

$V({\bar{\phi}})>V_{f}\,\bigg{(}1-\frac{2\pi^{2}M_{P}^{6}V_{f}^{\prime
2}}{\hbar\,V_{f}^{3}}\bigg{)}^{\frac{2}{\xi}}~{}~{}\Rightarrow~{}~{}{\bar{\phi}}>f-\frac{4\pi^{2}M_{P}^{2}V_{f}^{\prime}}{\hbar\,9\xi\,H_{f}^{4}}~{},$
(2.28)

where $f$ is the upper endpoint of the field range and $V_{f}=V(f)$ is the
maximum of the monotonic potential. Equation (2.28) shows that, when CbQ does
not hold, any peak (if it exists) must be close to the maximum of the
potential.

The extreme case is when $D$ becomes negative before $\phi$ reaches $f$, but
the solution remains positive because the peak is given by the first
oscillation of $p$. In this case, the solution of eq. (2.23) develops an
imaginary part in the argument of the exponential and, in the limit $\phi\to
f$ and for absorbing boundary condition at the upper endpoint, we find

$p=e^{\frac{S_{f}V^{\prime}_{f}(f-\phi)}{2V_{f}}}\,\sin\left[\frac{S_{f}\sqrt{-D}(f-\phi)}{2M_{P}}\right]~{},~{}~{}~{}D=-\frac{12\xi
H_{f}^{\prime}(f-{\bar{\phi}})}{S_{f}H_{f}}~{},$ (2.29)
$S_{f}=\frac{8\pi^{2}M_{P}^{2}}{\hbar\,H_{f}^{2}}~{},~{}~{}~{}~{}H_{f}\equiv
H(f)~{}.$ (2.30)

The peak is located at the point where the argument of the sine is $\pi/2$ and
therefore

${\bar{\phi}}=f-\bigg{(}\frac{\hbar\,M_{P}^{2}H_{f}^{4}}{16\,\xi
V_{f}^{\prime}}\bigg{)}^{\frac{1}{3}}~{}.$ (2.31)

In this case, the peak has a distance from the endpoint of the same size as
its width.

In conclusion, just based on the positivity requirement, we can identify three
regimes for possible distribution peaks, which depend on the behaviour of the
monotonic potential near the top of the field range.

  * •

Classical (C): $H_{f}^{3}<V_{f}^{\prime}<M_{P}H_{f}^{2}$. In this regime, the
CbQ and slow-roll conditions are satisfied near the top of the potential. The
peak, if it exists, is generically far from the upper endpoint with a location
determined by the eigenvalue $\lambda_{\scriptscriptstyle H}$.

  * •

Quantum+Volume (QV): $\sqrt{\xi}\,H_{f}^{4}/M_{P}<V_{f}^{\prime}<H_{f}^{3}$.
In this case, if allowed by boundary conditions, the peak must be close to the
upper endpoint, at a distance $M_{P}^{2}V_{f}^{\prime}/\xi H_{f}^{4}$. The
peak is well separated from the endpoint, since its displacement is greater
than its width.

  * •

Quantum2+Volume (Q2V): $V_{f}^{\prime}<\sqrt{\xi}\,H_{f}^{4}/M_{P}$. In this
case the peak, if it exists, is close to the upper endpoint, at a distance
$(M_{P}^{2}H_{f}^{4}/\xi V_{f}^{\prime})^{1/3}$ with a separation comparable
to the width.

###  2.7 Peak Properties

We can further investigate the nature of local peaks in stationary volume-
weighted field distributions by studying the case of EFT potentials within a
perturbative field range. When expressed in terms of the modes
$p(\varphi,\lambda)$, the FPV in eq. (2.18) becomes

$\frac{\alpha}{2}\,p^{\prime\prime}+\omega^{\prime}p^{\prime}+\left(\omega^{\prime\prime}+\beta\omega-\lambda\right)p=0~{},$
(2.32)

where $\omega(\varphi)$ is a monotonically increasing function such that
$\omega(0)=0$ and $\omega(\pm 1)=\pm 1$. The corresponding ‘discriminant’ is

$D=\omega^{\prime 2}+2\alpha(\lambda-\beta\omega-\omega^{\prime\prime})~{}.$
(2.33)

Let us consider absorbing boundary conditions at the endpoints $\varphi=\pm
1$. The ratio of the coefficients of the two general solutions of eq. (2.32)
can be determined by the boundary condition at $\varphi=-1$, while the
boundary condition at $\varphi=1$ can be used to determine the eigenvalue
spectrum. In order to cross zero at the upper endpoint and be positive
elsewhere, the solution corresponding to $\lambda_{\rm max}$ must enter the
oscillating regime at $\varphi=1$. As a result, $\lambda_{\rm max}$ can be
estimated as the solution of $D=0$ at $\varphi=1$. Using the expression of the
discriminant in eq. (2.33) together with $\alpha\ll 1$ and $\beta\gg 1$, we
find

$\lambda_{\rm max}=\beta-\frac{\omega_{1}^{\prime\,2}}{2\alpha}~{},$ (2.34)

where $\omega_{1}^{\prime}\equiv\omega^{\prime}(1)$ is a number of order
unity.

Now consider a hypothetical localised peak at $\varphi\\!=\\!{\bar{\varphi}}$
with width $\sigma$ defined as
$\sigma^{2}=-p({\bar{\varphi}})/p^{\prime\prime}({\bar{\varphi}})$, in analogy
with a Gaussian distribution. From eq. (2.32), we find

$\lambda=\beta\omega({\bar{\varphi}})+\omega^{\prime\prime}({\bar{\varphi}})-\frac{\alpha}{2\sigma^{2}}~{}.$
(2.35)

This shows that, for $\alpha\ll 1$ and $\beta\gg 1$, the eigenvalue $\lambda$
measures the height of the potential at the peak location. Assuming an
absorbing boundary condition at $\varphi=1$, we can combine this result with
eq. (2.34), finding that the peak location is

$\omega({\bar{\varphi}})=1-\frac{\omega_{1}^{\prime\,2}}{2\alpha\beta}~{}.$
(2.36)

This gives a simple, and general, illustration of the two different parameter
regimes. For a classically dominated solution ($\alpha\beta\ll 1$) the implied
peak slides down the potential, as expected for classical evolution. The
classical tendency to minimise the potential energy can only be prevented by
imposing appropriate boundary conditions at $\varphi=-1$.

On the other hand, if $\alpha\beta\gg 1$ the peak is located near the top of
the potential, with the distribution settling as high as possible consistently
with boundary conditions. This neatly demonstrates the general condition under
which the volume distribution will preferentially be peaked atop a potential,
with $1/\alpha\beta$ governing the distance from the true maximum. This
confirms the expectation that departures from classical evolution occur when
$\alpha\beta\gg 1$.

We can also estimate the width $\sigma$ of the peak by taking the derivative
of eq. (2.32) evaluated at ${\bar{\varphi}}$. In the limit $\alpha\ll 1$ and
$\beta\gg 1$, we find

$\sigma=\sqrt{\frac{1}{\beta}}~{}.$ (2.37)

This corresponds to a width of Planckian size for the dimensionful field
$\phi$.

The result in eq. (2.37) breaks down if the distance between the position of
the peak and the top of the potential ($\sim 1/\alpha\beta$) is smaller than
the implied width ($\sim 1/\sqrt{\beta}$), which occurs whenever
$\alpha^{2}\beta\gg 1$. Note that, from a perturbative perspective, this
simply reflects the fact that NLO quantum effects combined with volume growth
now dominate over classical evolution, and thus one is in a highly quantum
regime of the theory. In this case the width can be calculated directly from
the discriminant condition, since the peak will be the first oscillation after
entering the oscillatory regime, in order to satisfy the boundary condition.
From the frequency arising as the square root of the discriminant we find that
in the oscillatory regime the solution assuming absorbing boundary conditions
must be of the form

$p=e^{\frac{\omega_{1}^{\prime}}{\alpha}(1-\varphi)}\,\sin\Big{[}\sqrt{\frac{2\beta\Delta\omega}{\alpha}}\,(1-\varphi)\Big{]}~{},~{}~{}~{}\Delta\omega\equiv
1-\omega({\bar{\varphi}})=\omega_{1}^{\prime}(1-{\bar{\varphi})}$ (2.38)

and the peak is such that

$1-{\bar{\varphi}}\approx\Big{(}\frac{\alpha}{\beta}\Big{)}^{1/3}~{},~{}~{}~{}~{}\sigma\approx\Big{(}\frac{\alpha}{\beta}\Big{)}^{1/3}~{}.$
(2.39)

Thus we see that the nature of the peak in this case arises solely as a result
of entering the oscillatory regime.

Figure 1: The different regimes of the FPV dynamics for an EFT potential
parametrised by $\alpha$ and $\beta$. For $\alpha\ll 1/{S_{\\!{{\rm dS}}}}$
the field range is non-perturbative and for
$\alpha^{2}\beta\ll\xi/{S_{\\!{{\rm dS}}}^{2}}$ the dynamics is no longer in
slow-roll regime (shown for $\xi=1$). The Q and FP regimes are relatively
uninteresting for SOL because the volume term is only a subleading effect.

Armed with these estimates we may now sketch the parameter space for EFT
potentials of the kind in eq. (2.16).

  * •

Classical (C): $\alpha\beta\ll 1$. In this case the FPV distribution naturally
evolves down the potential, as much as it is allowed by boundary conditions.

  * •

Quantum+Volume (QV): $\alpha\beta\gg 1$, $\alpha^{2}\beta\ll 1$. In this case
the FPV distribution naturally climbs the potential. The resulting asymptotic
peak will settle at a distance $1/\alpha\beta$ from the top of the potential
with a width $\sigma\approx(1/\beta)^{1/2}$.

  * •

Quantum2+Volume (Q2V): $\alpha\beta\gg 1$, $\alpha^{2}\beta\gg 1$. Also in
this case the FPV distribution naturally climbs the potential. Since NLO
quantum effects are large, the asymptotic peak will be as close to the top of
the potential as possible, with a displacement of comparable magnitude to the
width of the peak. The width follows from the discriminant and is
$\sigma\approx(\alpha/\beta)^{1/3}$.

The different regimes are summarised graphically in fig. 1. For $\alpha\ll
1/S_{\\!{{\rm dS}}}$, the perturbative expansion of the FPV breaks down. When
the slow-roll parameter $\epsilon=3\xi\omega^{\prime\,2}/(S_{\\!{{\rm
dS}}}^{2}\alpha^{2}\beta)$ is larger than one, the FPV description can no
longer be trusted as the contribution from kinetic energy becomes important.
When $\beta<1$, the FPV solutions are approximately given by the corresponding
FP solutions up to a time-dependent, but nearly field-independent, factor. In
the Q regime ($1<\beta<\alpha$), the dynamics is dominated by quantum
fluctuations. In both the FP and Q regimes, the width of the distribution
grows in time, eventually occupying the full field range. Since the asymptotic
distribution is essentially uniform in field space, the FPV dynamics does not
lead to interesting effects for our applications. In this work, we focus
instead on C, QV and Q2V, which are the three regimes where the volume term
characterises the distinguishing features of FPV dynamics.

We can now show how the three regimes of the EFT coincide with the
corresponding regimes defined for a general potential in sect. 2.6. The EFT
potential can be written as

$V=V_{0}\left[1+\frac{2\omega(\varphi)}{{S_{\\!{{\rm dS}}}}\alpha}\right]~{}.$
(2.40)

Replacing this expression in eq. (2.26), we find

$D=\frac{3\xi}{S_{\\!{{\rm dS}}}^{2}\,\alpha^{2}\beta}\,\Big{(}\omega^{\prime
2}(\varphi)+2\alpha\beta\left[\omega({\bar{\varphi}})-\omega(\varphi)\right]\Big{)}~{},$
(2.41)

which, in slow-roll approximation, agrees with eq. (2.33) up to an arbitrary
constant. From this, it is immediate to see that the regimes defined in sect.
2.6 from positivity arguments are in perfect correspondence with those defined
in the EFT directly from the FPV. We note that simply translating the CbQ
condition in terms of the parameters $\alpha$ and $\beta$ would lead to an
incorrect answer because it does not properly take into account the
perturbative expansion of the Hubble rate.

###  2.8 Linear Potential

To elucidate the estimates and the different regimes found in sect. 2.7 we
compare them here with the case of a linear potential
$\omega(\varphi)=\varphi$, in which eq. (2.32) allows for the following
analytic solution (see appendix)

$p(\varphi,\lambda)=e^{-\frac{\varphi}{\alpha}}\left[a(\lambda)\,{\rm
Ai}(x)+b(\lambda)\,{\rm
Bi}(x)\right]~{},~{}~{}~{}x=\frac{1+2\alpha(\lambda-\beta\varphi)}{(2\alpha^{2}\beta)^{2/3}}~{},$
(2.42)

where ${\rm Ai}(x)$ and ${\rm Bi}(x)$ are the Airy functions, while $a$ and
$b$ are two arbitrary coefficients. Since we are interested in cases where the
volume term is important for the dynamics, we take $\beta>1$ and
$\beta>\alpha$.

The first solution in eq. (2.42) has a peak at $\varphi={\bar{\varphi}}$ with
width $\sigma^{2}=-p({\bar{\varphi}})/p^{\prime\prime}({\bar{\varphi}})$,

${\bar{\varphi}}=\left\\{\begin{array}[]{c}\frac{\lambda}{\beta}\\\
\frac{\lambda}{\beta}-a_{1}^{\prime}\big{(}\frac{\alpha}{2\beta}\big{)}^{1/3}\end{array}\right.~{},~{}~{}~{}\sigma=\left\\{\begin{array}[]{c}\frac{1}{\sqrt{\beta}}\\\
\frac{1}{\sqrt{-a_{1}^{\prime}}}\big{(}\frac{\alpha}{2\beta}\big{)}^{1/3}\end{array}\right.~{}~{}~{}\begin{array}[]{c}({\rm
C~{}or~{}QV~{}regime})\\\\[7.0pt] ({\rm Q}^{2}{\rm V~{}regime})\end{array}$
(2.43)

where $a_{1}^{\prime}=-1.02$ is the first zero of ${\rm Ai}^{\,\prime}(x)$.
The location of the peak grows with $\lambda$, but it always remains below the
upper perturbative endpoint (${\bar{\varphi}}\leq 1$) because of the bound on
the maximum eigenvalue $\lambda_{\rm max}\leq\beta$ (see eq. (A.14) in the
appendix). However, for $\lambda<-\beta$, the peak disappears below the lower
endpoint of the perturbative field range. In the C and QV regimes, the local
behaviour of the solution around the peak is well approximated by a Gaussian
function with a variance $\sigma$ which is universal, in the sense that is
independent of the eigenvalue $\lambda$ and the coupling constant of the
linear potential. It has a typical Planckian size in physical units (as long
as $\xi\neq 0$). In the Q2V regime, the peak is non-Gaussian, with a
characteristic width $\sigma$ which is independent of $\lambda$ and grows as
the coupling constant decreases.

The first solution in eq. (2.42) turns negative, entering a subsequent
oscillatory phase, for $\varphi>\varphi_{\rm pos}$ with

$\varphi_{\rm pos}=\frac{1+2\alpha\lambda-
a_{1}(2\alpha^{2}\beta)^{2/3}}{2\alpha\beta}~{},$ (2.44)

where $a_{1}=-2.34$ is the first zero of ${\rm Ai}(x)$. In the C regime, the
oscillatory phase exists only when the peak has vanished below the lower
endpoint of the field range. In the quantum regimes, the separation between
the peak and the end of the positivity range is small and independent of the
eigenvalue $\lambda$,

$\varphi_{\rm
pos}-{\bar{\varphi}}=\left\\{\begin{array}[]{c}\frac{1}{2\alpha\beta}\\\
(a_{1}^{\prime}-a_{1})\big{(}\frac{\alpha}{2\beta}\big{)}^{1/3}\end{array}\right.~{}~{}~{}\begin{array}[]{c}({\rm
QV~{}regime})\\\\[7.0pt] ({\rm Q}^{2}{\rm V~{}regime})\end{array}$ (2.45)

In the QV regime, the peak is well within the positivity region ($\varphi_{\rm
pos}-{\bar{\varphi}}\gg\sigma$) while, in the Q2V regime, is at the edge
($\varphi_{\rm pos}-{\bar{\varphi}}\approx\sigma$). Indeed, the peak in the
Q2V regime is not an isolated Gaussian, as in the case of C and QV, but
corresponds to the first oscillation of ${\rm Ai}(x)$ before the function
turns negative. Positivity of the asymptotic solution, together with the upper
bound on $\lambda_{\rm max}$, imply that the asymptotic location of the peak
(${\bar{\varphi}}_{\infty}$) must be in the range

$\displaystyle\beta\Big{(}1-\frac{1}{2\alpha\beta}\Big{)}<\lambda_{\rm
max}<\beta$ $\displaystyle\Rightarrow$ $\displaystyle
1-\frac{1}{2\alpha\beta}<{\bar{\varphi}}_{\infty}<1~{}~{}~{}({\rm QV})$
$\displaystyle\beta\Big{[}1+a_{1}\Big{(}\frac{\alpha}{2\beta}\Big{)}^{1/3}\Big{]}<\lambda_{\rm
max}<\beta$ $\displaystyle\Rightarrow$ $\displaystyle
1-(a_{1}^{\prime}-a_{1})\Big{(}\frac{\alpha}{2\beta}\Big{)}^{1/3}<{\bar{\varphi}}_{\infty}<1~{}~{}~{}({\rm
Q}^{2}{\rm V})$ (2.46)

This means that, both in the QV and Q2V regimes, the location of the peak is
parametrically close to the upper endpoint, independently of the choice of
boundary conditions. On the other hand, in the C regime, the peak location is
entirely a matter of boundary conditions.

The second solution in eq. (2.42) is monotonically decreasing until it turns
negative, entering a subsequent oscillatory phase, for

$\varphi>\varphi_{\rm
pos}-(b_{1}-a_{1})\Big{(}\frac{\alpha}{2\beta}\Big{)}^{\frac{1}{3}}~{},$
(2.47)

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

The previous considerations were independent of boundary conditions. Let us
now focus on the case of absorbing boundary conditions at the endpoints of the
field range, $p(\varphi\\!=\\!\pm 1)=0$. The solution in eq. (2.42) becomes,
up to an overall constant,

$p=e^{-\frac{\varphi}{\alpha}}\left[\frac{{\rm Ai}(x_{n})}{{\rm
Ai}(x_{\\!E})}-\frac{{\rm Bi}(x_{n})}{{\rm
Bi}(x_{\\!E})}\right]~{},~{}~{}~{}~{}\lambda_{n}=\beta-\frac{1}{2\alpha}+a_{n}\,\Big{(}\frac{\alpha\beta^{2}}{2}\Big{)}^{1/3}~{},$
(2.48)
$x_{n}=a_{n}+\Big{(}\frac{2\beta}{\alpha}\Big{)}^{1/3}(1-\varphi)~{},~{}~{}~{}~{}x_{\\!E}=x_{n}(\varphi\\!=\\!-1)~{},$
(2.49)

where $a_{n}$ are the zeros of ${\rm Ai}(x)$. The asymptotic distribution at
large times corresponds to the largest eigenvalue, which is obtained for
$n=1$. The asymptotic distribution in eq. (2.48) has the following behaviours
in the three different regimes.

#### C regime

For $\alpha\beta\ll 1$, we find

${\bar{\varphi}}=-1+\alpha~{},~{}~{}~{}~{}\sigma=\alpha~{}.$ (2.50)

In the C regime, the peak of the distribution slides down the potential and,
being supported only by the absorbing boundary condition, it settles at a
distance $\alpha$ from the lower endpoint of the field range. The steeper the
potential, the closer to the endpoint is the peak, but it always remains
1-$\sigma$ away from it. Note, however, that $\sigma$ measures only the local
property of the distribution at the maximum. Since the peak in the C regime is
fairly asymmetric, its spread away from the endpoint is actually larger than
$\alpha$.

#### QV regime

For $\alpha\beta\gg 1$ and $\alpha^{2}\beta\ll 1$, we find

${\bar{\varphi}}=1-\frac{1}{2\alpha\beta}~{},~{}~{}~{}~{}\sigma=\sqrt{\frac{1}{\beta}}~{}.$
(2.51)

The width of the peak is independent of the coupling constant in the
potential. The peak is located a short distance away from the upper endpoint,
but slides down as the potential gets steeper. Although being relatively near
to the upper endpoint, the peak is always separated from it by a distance much
larger than the width.

#### Q2V regime

For $\alpha\beta\gg 1$ and $\alpha^{2}\beta\gg 1$, we find

${\bar{\varphi}}=1-(a_{1}^{\prime}-a_{1})\,\Big{(}\frac{\alpha}{2\beta}\Big{)}^{1/3}~{},~{}~{}~{}~{}\sigma=\frac{1}{\sqrt{-a_{1}^{\prime}}}\,\Big{(}\frac{\alpha}{2\beta}\Big{)}^{1/3}~{}.$
(2.52)

The peak is located near the upper endpoint, but it moves away from it the
shallower the potential. However, its distance from the endpoint remains
constant in units of width, being always equal to about 1.3 $\sigma$.

These analytic results for the linear potential confirm the general EFT
expectations for the three parameter regimes introduced in sect. 2.7. The FPV
distribution in eq. (2.48) is plotted in fig. 2 for parameter values
corresponding to the three regimes discussed, varying $\beta$ keeping $\alpha$
fixed on the left and vice versa on the right.

Figure 2: The asymptotic volume-weighted field distribution for a linear
potential with absorbing boundary conditions at $\varphi=\pm 1$, for fixed
$\alpha=10^{-2}$ (left panel) with $\beta=10$ (C regime), $10^{3}$ (QV
regime), $2\times 10^{4}$ (Q2V regime), and for fixed $\beta=10^{4}$ (right
panel) with $\alpha=2.5\times 10^{-5}$ (C regime), $10^{-3}$ (QV regime),
$10^{-1}$ (Q2V regime).

###  2.9 The Classical Regime

We now want to show that the classical regime allows for a full
characterisation of the FPV solutions for a general potential, beyond EFT or
perturbative field range.

Consider a monotonically increasing potential $V(\phi)$ and define the
following special field points

$V(\phi_{\rm
AdS})=0~{},~{}~{}~{}V(0)=V_{0}~{},~{}~{}~{}V(\phi_{P})=2V_{0}~{},~{}~{}~{}V(f)=M^{4}/g_{*}^{2}~{}.$
(2.53)

The point $\phi_{\rm AdS}$ is where the potential crosses zero. It is the
absolute lower endpoint of the field range that we can consider, since the FPV
is defined only in dS space. The point $\phi=0$ is defined, after an
appropriate coordinate shift, to be a generic field value around which we will
expand perturbatively in a field region such that $|V(\phi)-V_{0}|\ll V_{0}$.
Therefore, $\phi_{P}$ gives the maximum field value where we can trust a
perturbative expansion, which is valid in the range

$\phi_{\rm AdS}\ll\phi\ll\phi_{P}~{}.$ (2.54)

The point $\phi=f$ defines the absolute upper endpoint of the field range that
we can consider, since $M$ describes the energy cutoff at which the EFT
description breaks down, while $g_{*}$ is a typical coupling constant of the
UV theory. Because of the onset of quantum gravity at Planckian energies,
$M/g_{*}$ must be smaller than $M_{P}$.

#### The diffusionless solution

Consider the diffusionless limit ($\hbar\to 0$) of the FPV in eq. (2.23). Up
to an overall normalisation, the solution is

$p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}=\frac{V}{V^{\prime}}\Big{(}\frac{V_{0}}{V}\Big{)}^{\frac{\xi}{2}}\,\exp\frac{3}{M_{P}^{2}}\int
dV\,\frac{V}{V^{\prime 2}}\left[\left(\lambda_{\scriptscriptstyle
H}\sqrt{\frac{V_{0}}{V}}\right)^{\xi}-1\right]~{}.$ (2.55)

The solution $p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ in eq. (2.55) has a peak at
${\bar{\phi}}$ with width $\sigma$ such that

${\bar{H}}\equiv H({\bar{\phi}})=H_{0}\,\lambda_{\scriptscriptstyle
H}~{},~{}~{}~{}\sigma=\sqrt{\frac{2}{3\xi}}\,M_{P}$ (2.56)

at leading order in the slow-roll parameters $\epsilon$ and $\eta$.

By replacing eq. (2.55) into eq. (2.23), we can compare the relative size of
the diffusion ($Q$) and drift ($C$) terms. At leading order in slow-roll
parameters, their ratio is

$\frac{Q}{C}=\frac{\hbar\,27}{8\pi^{2}}\left(\frac{H^{3}}{V^{\prime}}\right)^{2}\,\left[\Big{(}\frac{\bar{H}}{H}\Big{)}^{\xi}-1\right]~{}.$
(2.57)

This shows that the diffusionless approximation is valid when the potential
satisfies the CbQ condition in the field region above the peak and the
stronger condition $V^{\prime}>H^{3-\xi/2}{\bar{H}}^{\xi/2}$ below. The
diffusion term can be important at the initial stages of the time evolution to
create a spread of the distribution. However, in the classical regime, it
becomes irrelevant once the width has reached Planckian size since the
subsequent evolution is largely determined by drift and volume terms alone.

The behaviour of $p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ well above the peak is
obtained by taking $V/V_{0}\gg\lambda_{\scriptscriptstyle H}^{2}$ in eq.
(2.55) and we find, at leading order in slow-roll parameters,

$\frac{p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}^{\prime}}{p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}}=-\frac{3V}{M_{P}^{2}V^{\prime}}~{}~{}~{}~{}({\rm
for~{}large}~{}\phi)~{}.$ (2.58)

As $\phi\to\phi_{\rm AdS}$, eq. (2.55) shows that the diffusionless solution
goes like $p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}\propto V^{1-{\xi}/{2}}$ and
therefore it vanishes exactly at the AdS boundary. The behaviour below the
peak ($V/V_{0}\ll\lambda_{\scriptscriptstyle H}^{2}$) is

$\frac{p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}^{\prime}}{p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}}=\frac{3V}{M_{P}^{2}V^{\prime}}\Big{(}\frac{\bar{V}}{V}\Big{)}^{\frac{\xi}{2}}+\Big{(}1-\frac{\xi}{2}\Big{)}\frac{V^{\prime}}{V}~{}~{}~{}~{}({\rm
for~{}small}~{}\phi)~{},$ (2.59)

where the first term dominates in the slow-roll regime.

For a linear potential $V=V_{0}+V^{\prime}\phi$ with constant $V^{\prime}$, we
can perform the integral in eq. (2.55) and obtain

$p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}=\frac{V}{V^{\prime}}\Big{(}\frac{V_{0}}{V}\Big{)}^{\frac{\xi}{2}}\,\exp\frac{3V^{2}}{2M_{P}^{2}V^{\prime
2}}\left[\frac{4}{4-\xi}\left(\lambda_{\scriptscriptstyle
H}\sqrt{\frac{V_{0}}{V}}\right)^{\xi}-1\right]~{}.$ (2.60)

It can be verified that, in a perturbative field range around $\phi=0$, the
solution $p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ coincides with the first of the
two solutions in eq. (2.42), $p=\exp(-\varphi/\alpha){\rm Ai}(x)$.

Just for illustration consider, as a second example, an exponential potential

$V=V_{0}\left[1+a\,\big{(}e^{\frac{\kappa\phi}{M_{P}}}-1\big{)}\right]~{},~{}~{}~{}a=\Big{(}1-e^{\frac{\kappa\phi_{\rm
AdS}}{M_{P}}}\Big{)}^{-1}~{}.$ (2.61)

In this case, the integral in eq. (2.55) gives, for $\xi=1$,

$p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}={\scriptstyle\sqrt{V}\,\exp\big{[}{\frac{3}{\kappa^{2}}\big{(}\frac{\lambda_{\scriptscriptstyle
H}}{\sqrt{a-1}}\arctan\frac{r}{\sqrt{a-1}}-\frac{a-1+\lambda_{\scriptscriptstyle
H}r}{a}\,e^{-\frac{\kappa\phi}{M_{P}}}\big{)}-\big{(}\kappa+\frac{3}{\kappa}\big{)}\,\frac{\phi}{M_{P}}}\big{]}~{},~{}~{}~{}r=\sqrt{\frac{V}{V_{0}}}}~{},$
(2.62)

while, for a general $\xi$, $p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ can be
expressed in terms of hypergeometric functions.

#### The Gibbs solution

Consider the equation

$\frac{\hbar\,H^{2}}{8\pi^{2}}\,p^{\prime\prime}+\frac{M_{P}^{2}V^{\prime}}{V}\,p^{\prime}+(\eta-4\epsilon)p=0~{},$
(2.63)

which can be conveniently rewritten as

$p^{\prime\prime}=({\mathcal{V}}^{\prime}p)^{\prime}~{},~{}~{}~{}~{}{\mathcal{V}}=\frac{24\pi^{2}M_{P}^{4}}{\hbar\,V}~{}.$
(2.64)

One of its two solutions is

$p_{G}=\exp\left(\frac{24\pi^{2}M_{P}^{4}}{\hbar\,V}\right)~{},$ (2.65)

which is the stationary Gibbs distribution corresponding to the potential
${\mathcal{V}}$.

Equation (2.63) properly accounts for the first and second derivative terms in
the FPV and differs from eq. (2.23) by terms which are small when evaluated on
$p_{G}$ if the following conditions are satisfied

$\displaystyle H^{3}<V^{\prime}<V/M_{P}~{}~{}$ $\displaystyle~{}~{}{\rm
for}~{}V>\lambda_{\scriptscriptstyle H}^{2}V_{0}~{},$ $\displaystyle
H^{3-\xi/2}(\lambda_{\scriptscriptstyle
H}H_{0})^{\xi/2}<V^{\prime}<V/M_{P}~{}~{}$ $\displaystyle~{}~{}{\rm
for}~{}V<\lambda_{\scriptscriptstyle H}^{2}V_{0}~{}.$ (2.66)

These are exactly the same classical slow-roll conditions under which the
diffusionless approximation gives a valid solution to the FPV equation.

The Gibbs distribution in eq. (2.65) is monotonically decreasing, it explodes
in the region of small $V$ and satisfies

$\frac{p_{G}^{\prime}}{p_{G}}=-\frac{V_{0}{S_{\\!{{\rm
dS}}}}V^{\prime}}{V^{2}}~{}.$ (2.67)

Therefore its relative rate of decrease at large $\phi$ is faster than for
$p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ because

$\frac{|{p_{G}^{\prime}}/{p_{G}}|}{|{p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}^{\prime}}/{p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}}|}\sim\left(\frac{V^{\prime}}{H^{3}}\right)^{2},$
(2.68)

which is larger than one in the CbQ regime. For a linear potential in a
perturbative field range around $\phi=0$, the solution $p_{G}$ coincides with
the second solution in eq. (2.42), $p=\exp(-\varphi/\alpha){\rm Bi}(x)$.

#### Summary

The FPV solutions can be fully characterised in the classical slow-roll
regime, which is defined by eq. (2.66). One solution, called
$p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ and given in eq. (2.55), corresponds to
the case in which the diffusion term is negligible. The drift and volume terms
in the FPV balance each other and create a peak in the stationary solutions
with Planckian width and location at the field point where the potential
height matches the global expansion rate measured by the eigenvalue
$\lambda_{\scriptscriptstyle H}$.

The second solution, called $p_{G}$ and given in eq. (2.65), is monotonically
decreasing and gives an exponential preference towards the lowest possible
value of the potential, up to small quantum fluctuations. The proper
combination of $p_{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}$ and $p_{G}$ which gives
the physical FPV solution can be determined only with knowledge of the
boundary conditions.

###  2.10 Junction Conditions

Consider the cosmological fate of the scalar field $\phi$ in the case that, at
some critical value $\phi_{c}$, it triggers a first-order quantum phase
transition. Since the potential is continuous, the FPV solution $P$ and its
time derivative are also continuous. However, when the gradient of the
potential is discontinuous at $\phi_{c}$, the gradient of the solution
$\partial P/\partial\phi$ will not be continuous. We may understand the
dynamics at the critical point by integrating the FPV in eq. (2.2) across the
critical point, with continuity of the equation leading to the constraint

$\lim_{\epsilon\to
0}\int_{\phi_{c}-\epsilon}^{\phi_{c}+\epsilon}d\phi\,\frac{\partial}{\partial\phi}\left[\frac{V^{\prime}P}{3H}+\frac{\hbar}{8\pi^{2}}\frac{\partial}{\partial\phi}(H^{3}P)\right]=0~{}.$
(2.69)

Since this is a total derivative, the integral is directly given by the
boundary terms and we find

$\frac{\Delta P^{\prime}}{P(\phi_{c})}=-\frac{24\pi^{2}M_{P}^{4}\,\Delta
V^{\prime}}{\hbar\,V^{2}(\phi_{c})}~{},$ (2.70)

where the discontinuities for any quantity $Q$ is defined by

$\Delta Q=\lim_{\epsilon\to
0}\left[Q(\phi_{c}+\epsilon)-Q(\phi_{c}-\epsilon)\right]~{}.$ (2.71)

In eq. (2.70) we have dropped a correction proportional to $V/M_{P}^{4}$ which
is always negligible as long as the dynamics is not quantum-gravity dominated.

The junction condition in eq. (2.70) does not rely on any assumption on the
form of the potential nor any approximations beyond those implicit in the use
of the stochastic approach. It is also valid for all times and therefore it
holds for each $p(\lambda)$, mode by mode. In the approximation of eq. (2.17)
and for the EFT potential, the junction condition in eq. (2.70) becomes

$\frac{\Delta
P^{\prime}}{P(\varphi_{c})}=-\frac{2\Delta\omega^{\prime}}{\alpha}~{}.$ (2.72)

Since potential gradient discontinuities with
$\Delta\omega^{\prime}=\mathcal{O}(1)$ are the hallmark of a first-order phase
transition, we see that the phase transition imprints itself upon the field
distribution as a discontinuity with the opposite sign. The sign of the
discontinuity $\Delta\omega^{\prime}$ is negative for phase transitions, thus
$\Delta P^{\prime}$ must be positive. The negativity of the gradient
discontinuity follows from the fact that, in thermal equilibrium, the lower of
the two available states is occupied. Thus, the potential of the equilibrium
state as a function of $\varphi$ will track the lowest of any branches of
vacua, stable or not, that may exist at any $\varphi$ value in different
quantum phases.

Another example of junction conditions relevant to our applications is the
case of a multivalued potential with two branches corresponding to two
different phases. The potentials $V_{a,b}(\phi)$ on the two branches are both
continuous and differentiable, but branch $b$ has a termination point at
$\phi_{c}$, where $V_{b}(\phi_{c})>V_{a}(\phi_{c})$, and therefore exists only
for $\phi\geq\phi_{c}$. If the field undergoes a sudden phase transition from
$b$ to $a$ at the critical point $\phi_{c}$, then the junction condition on
the $b$ branch must be

$P_{b}(\phi_{c})=0~{}.$ (2.73)

The junction conditions on the $a$ branch are obtained by considering the sum
of the two FPV on the branches. By integrating this equation in a
neighbourhood of the critical point, following the same procedure as in eq.
(2.69), and exploiting the continuity of $P_{a}$ and $P_{a}+P_{b}$, we find
the junction conditions

$\Delta P_{a}^{\prime}=-P_{b}^{\prime}(\phi_{c})~{},~{}~{}~{}~{}\Delta
P_{a}=0~{},$ (2.74)

where the discontinuities across the critical point are defined as in eq.
(2.71). Equation (2.74) 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.

##  3 Self-Organised Localisation

An essential element of SOL is the localisation of the volume-weighted field
distribution triggered by a quantum phase transition. We will illustrate the
mechanism using some simple examples that exhibit different features of the
SOL phenomenon.

###  3.1 Localisation in the Pyramid Scheme

As a first prototype example, let us consider the ‘pyramid’ potential defined
by $\omega(\varphi)=-|\varphi|$. The discontinuity of the potential gradient
at $\varphi_{c}=0$ reflects an underlying first-order phase transition. Taking
absorbing boundary conditions at the endpoints of the field range,
$p(\varphi\\!=\\!\pm 1)\\!=\\!0$, and the junction condition in eq. (2.72),
the solution of eq. (2.32) for the mode corresponding to the largest
eigenvalue is (for $\beta\gg\alpha$ and up to an overall constant)

$p=e^{\frac{|\varphi|}{\alpha}}\left[2{\rm
Ai}(x)-e^{-\frac{4}{3}x_{\\!E}^{3/2}}{\rm
Bi}(x)\right]~{},~{}~{}~{}~{}\lambda_{\rm
max}=a\,\Big{(}\frac{\alpha\beta^{2}}{2}\Big{)}^{1/3}-\frac{1}{2\alpha}~{},$
(3.1)
$x=a+\Big{(}\frac{2\beta}{\alpha}\Big{)}^{1/3}|\varphi|~{},~{}~{}~{}~{}x_{\\!E}=x(|\varphi|\\!=\\!1)~{},$
(3.2)

where $a$ is the largest solution of the equation

$\frac{{\rm Ai}^{\,\prime}(a)}{{\rm
Ai}(a)}=(2\alpha^{2}\beta)^{-1/3}~{}~{}~{}\Rightarrow~{}~{}~{}a=\left\\{\begin{array}[]{cc}a_{1}&({\rm
for~{}}\alpha^{2}\beta\ll 1)\\\ a_{1}^{\prime}&({\rm for~{}}\alpha^{2}\beta\gg
1)\end{array}\right.~{}.$ (3.3)

Therefore, $a$ always lies in the range $-2.34\\!<\\!a\\!<\\!-1.02$. The
distribution in eq. (3.1) exhibits a pair of symmetric peaks whose locations
${\bar{\varphi}}_{\pm}$ and width $\sigma$ have different behaviours in the
three following regimes.

#### C regime

For $\alpha\beta\ll 1$, we find

${\bar{\varphi}}_{\pm}=\pm(1-\alpha)~{},~{}~{}~{}~{}\sigma=\alpha~{}.$ (3.4)

The asymptotic distribution is localised in the proximity of the field
endpoints because the boundary conditions prevent any further descent.

#### QV regime

For $\alpha\beta\gg 1$ and $\alpha^{2}\beta\ll 1$, we find

${\bar{\varphi}}_{\pm}=\pm\frac{1}{2\alpha\beta}~{},~{}~{}~{}~{}\sigma=\sqrt{\frac{1}{\beta}}~{}.$
(3.5)

The peaks are located relatively close to the critical point, although they
are always well separated in units of width since
$|{\bar{\varphi}}_{+}\\!-\\!{\bar{\varphi}}_{-}|/\sigma\gg 1$.

Figure 3: The asymptotic volume-weighted field distribution for a pyramid
potential with absorbing boundary conditions at $\varphi=\pm 1$, for
$\alpha=10^{-2}$ and $\beta=10$ (C regime), $10^{3}$ (QV regime), $2\times
10^{4}$ (Q2V regime). The figure shows the occurrence of SOL and of a peak at
or near the critical point for the latter two regimes, zoomed in on the right.

#### Q2V regime

For $\alpha\beta\gg 1$ and $\alpha^{2}\beta\gg 1$, we find

${\bar{\varphi}}_{\pm}=\mp\frac{1}{a_{1}^{\prime}}\,\Big{(}\frac{2}{\alpha\beta^{2}}\Big{)}^{1/3}~{},~{}~{}~{}~{}\sigma=\frac{1}{\sqrt{-a_{1}^{\prime}}}\,\Big{(}\frac{\alpha}{2\beta}\Big{)}^{1/3}~{}.$
(3.6)

The two peaks are much broader than their separation, since
$|{\bar{\varphi}}_{+}\\!-\\!{\bar{\varphi}}_{-}|/\sigma\ll 1$, and therefore
in practice they form a single peak centred at the critical point.

The asymptotic FPV distribution in eq. (3.1) is plotted in fig. 3 for a
representative choice of parameters, showing the narrow peak located at the
critical point in the $Q^{2}V$ regime, as well as solutions in the $C$ and
$QV$ regimes (left panel). Zooming in closely on the critical point (right
panel) we see that the $Q^{2}V$ distribution has a local minimum, as predicted
by the junction condition, even though this behaviour persists only for a
short field range. Globally, the distribution is peaked around the critical
point, as expected.

This calculation illustrates a simple realisation of SOL, where the volume-
weighted field distribution becomes localised at the critical point of a
first-order quantum phase transition during a long period of inflation. The
pyramid example has been chosen because it allows for a simple analytical
treatment. However, the SOL mechanism remains qualitatively the same for a
general class of potentials where a negative $\Delta g^{\prime}$ creates a
cusp, corresponding to a local maximum of the potential at the critical point,
and even for discontinuous potentials at the maximum.

###  3.2 Microscopic Features of the Phase Transition

The example of SOL presented in the previous section relies only on the nature
of the dynamical regimes on either side of the critical point. Implicitly we
have assumed that the system immediately undergoes the phase transition to the
true minimum, neglecting the microscopic details of the transition and any
associated timescale. In order to illustrate how these details can matter, we
will study a toy model inspired by Landau theory.

Consider the scalar potential

$V=\frac{\lambda}{4}\left(\psi^{2}-\rho^{2}\right)^{2}+\kappa\phi\psi~{}~{},$
(3.7)

where $\psi$ is the microscopic scalar field and $\phi$ plays a role analogous
to the $Z_{2}$-breaking external magnetic field in the Ising model. The $\psi$
vacuum, as a function of $\phi$, has two configurations

$\langle\psi\rangle_{\pm}=\pm\rho\,C(\phi/\phi_{\pm})~{}~{}~{}~{}~{}{\rm
for}~{}\phi\,\,\stackanchor[0.5pt]{\scalebox{0.8}{$<$}}{\scalebox{0.8}{$>$}}\,\,\phi_{\pm}$
(3.8)
$\phi_{\pm}=\pm\frac{2\lambda\rho^{3}}{3\sqrt{3}\kappa}~{},~{}~{}~{}C(x)=\left\\{\begin{array}[]{ll}\frac{2}{\sqrt{3}}\,\cos\left[\frac{\arccos(-x)}{3}\right]&{\rm
for}~{}|x|<1\vspace{0.2cm}\\\ \frac{2}{\sqrt{3}}\,\cosh\left[\frac{{\rm
arcosh}(-x)}{3}\right]&{\rm for}~{}x<-1\end{array}\right.~{}.$ (3.9)

The two branches of the $\phi$ potential on these $\psi$ configurations are

$V(\phi,\langle\psi\rangle_{\pm})=\frac{\lambda\rho^{4}}{4}\left[1+2C^{2}(\phi/\phi_{\pm})-3C^{4}(\phi/\phi_{\pm})\right]~{}~{}~{}~{}~{}{\rm
for}~{}\phi\,\,\stackanchor[0.5pt]{\scalebox{0.8}{$<$}}{\scalebox{0.8}{$>$}}\,\,\phi_{\pm}~{},$
(3.10)

which are shown in fig. 4. For $\phi$ large and negative, the true minimum of
$\psi$ is at $\psi_{+}$. As $\phi$ increases, the minimum is lifted and, at
$\phi=0$, it becomes degenerate with the configuration $\psi_{-}$. However,
$\psi_{+}$ persists as a local minimum in a ‘supercooled’ phase beyond the
critical point $\phi_{c}=0$ until it becomes classically unstable at
$\phi=\phi_{+}$.

In the previous section we have used a simplified treatment assuming immediate
tunnelling to the true vacuum, with a sudden transition at $\phi_{c}$
generating the first-order discontinuity. In practice, this means assuming
that the dynamical evolution of the field $\phi$ strictly follows the lower,
true, vacuum. However, in reality the tunnelling rate to this vacuum at a
given $\phi$-point depends on the microscopic physics. Parametrically, at a
generic value for $\phi$ it scales as $\Gamma\sim e^{-\zeta/\hbar\lambda}$,
where $\zeta$ is a numerical coefficient which must be evaluated on a case-by-
case basis. If the timescale for this tunnelling is greater than the other
cosmological timescales, the assumption of immediate tunnelling will break
down.

Figure 4: Vacuum structure in the Landau model, where $\phi$ plays an
analogous role to an external magnetic field. Left panel: $V$ as a function of
$\psi$ for different values of $\phi$. Right panel: $V$ as a function of
$\phi$ with $\psi$ fixed at its two possible minima.

As shown in fig. 4, when $\phi$ increases beyond $\phi_{+}$, the metastable
vacuum eventually disappears and $\psi$ rolls directly to the true vacuum,
with no tunnelling required. If the typical gradient of the $\psi$ potential
exceeds the slow roll condition, this rolling is essentially instantaneous
from a cosmological perspective. Depending on the dynamical timescales, $\psi$
may tunnel to the true vacuum long before $\phi_{+}$ has been reached, or the
tunnelling process may be so slow that $\phi$ evolves along the supercooled
branch all the way up to $\phi_{+}$ before proceeding to the true vacuum. A
complete modelling of this dynamical process would require a random walk which
can explore metastable branches of the vacuum structure and take into account
the probability of tunnelling to the true vacuum from the excited state.
However, we may provide a simplified treatment by noting that, since the
tunnelling rate depends exponentially on the vacuum energy difference, there
will be a fairly abrupt change of behaviour at a given field point
$\phi_{\star}$ (with $0<\phi_{\star}<\phi_{+}$) such that tunnelling to the
true vacuum is essentially forbidden for $\phi<\phi_{\star}$, whereas it
becomes inevitable for $\phi>\phi_{\star}$. In the supercooled state, this
corresponds to an absorbing boundary condition at $\phi_{\star}$ because once
$\phi$ has ventured beyond this point it essentially vanishes from this
branch, appearing on the lower one. On the lower branch the boundary condition
must conserve flux, including the flux from the upper branch, and also be
continuous. As a result, at $\phi_{\pm}$ one must impose an absorbing boundary
condition on the upper branch, and a boundary condition on the lower branch
for which the derivatives of the distribution have zero sum, as discussed in
sect. 2.10.

One can now picture a random walk on a vacuum structure such as in fig. 4. In
the classical regime the field will roll down the potential. However in the
quantum regimes it will climb the potential, drop off the end of a branch, and
then start climbing again. The final steady state will reflect the preference
for high altitudes, but also the fact that one has essentially absorbing
boundary conditions at the top of a branch. With this in mind we see that the
field will be found in proximity to the highest values of the potential, with
a distribution given by the standard widths in the QV or Q2V regimes. Explicit
calculations confirm this expectation. If the width is greater than the length
of the supercooled branch, the presence of the metastable states is of little
significance, since the distribution will encompass the entire region.
However, if the width is smaller, then the distribution will be found
localised at a point on the supercooled branch away from the degenerate point
and will remain stable relative to the timescales relevant for the dynamics.
Typically, since quantum tunnelling will often not lead to a phase transition
which can complete through percolation, as in old inflation, the supercooled
branch can be effectively stable. In the applications to follow, these aspects
will be important.

###  3.3 Localisation in the Waterfall Scheme

Let us consider a multivalued potential describing two phases of the
underlying system. In phase $v$ (to be identified with the vacuum in the
‘visible’ configuration of the system), we take a generic, but monotonically
increasing, potential defined within a field range
$\varphi_{E}^{-}<\varphi<\varphi_{E}^{+}$ in which the scalar field gives only
a small modulation of the background vacuum energy. In phase $h$
(corresponding to a ‘hidden’ configuration), we take a flat potential and we
choose coordinates such that the two phases are degenerate in energy at the
point $\varphi=\varphi_{c}$. Nonetheless, a phase transition occurs only at
the point $\varphi=\varphi_{T}$, as illustrated in fig. 5 (left panel). We
will refer to this setup as the ‘waterfall potential’.

The question at hand is to determine the final stationary volume distribution
of $\varphi$ as it fluctuates and explores the vacuum structure of the
waterfall potential. In particular, we are interested in whether the field can
become exponentially localised in phase $v$ at the point of energy degeneracy
between the two vacua. This would constitute a self-organisation of the vacuum
energy in phase $v$ to be as close to the one in phase $h$ as the width of the
distribution will tolerate.

Since the crossing point of the two phases is not a local maximum and in both
quantum regimes the field prefers to localise at maxima, we must consider a
classical regime if we are to have localisation at the point of degeneracy.
Furthermore, as found in sect. 2.6, the greatest separation one can find
between the location of a peak and the point at which the solution no longer
satisfies positivity is $\Delta\varphi\sim 1/(\alpha\beta)$. Thus if the
asymptotic solution in phase $v$ is to be positive everywhere and localised at
the degenerate point, we must have $\alpha\beta\lesssim 1$. This is the
classical regime.

Figure 5: Left panel. A sketch of the scalar potential of the waterfall
scheme. The two branches $v$ and $h$ are degenerate in energy at a generic
field value $\varphi_{c}$, but transitions between them occur only at
$\varphi=\varphi_{T}$. Right panel. The asymptotic volume-weighted
distribution in phase $v$ (in black and multiplied by a reduction factor of
$10^{-213}$) and phase $h$ (in red) for $\alpha=10^{-5}$, $\beta=10^{3}$,
$\varphi_{T}=-0.5$, $\varphi_{c}=0$, $\varphi_{E}^{+}=1$ and $k_{v}=k_{h}=1$
(the value of $\varphi_{E}^{-}$ is inconsequential for the result). The
distribution is dominantly found in phase $v$, peaked near
$\varphi=\varphi_{c}$.

It remains to specify the boundary conditions. In the following, with no loss
of generality, we make a coordinate choice such that $\varphi_{c}=0$ and
$\varphi_{E}^{\pm}=\pm 1$ (although the result is insensitive to the location
of the lower endpoint). For phase $h$ we choose an absorbing boundary
condition at the upper endpoint, $P_{h}(1)=0$, but this is unimportant for our
conclusions and analogous results are obtained for a reflecting boundary
condition. For phase $v$ we also choose an absorbing boundary condition at the
lower endpoint, $P_{v}(-1)=0$. However, at the upper endpoint, we set a non-
vanishing boundary condition for the derivative of the distribution,
$P_{v}^{\prime}(1)\neq 0$. We assume this is due to a UV sector in the
landscape which provides a source of flux, possibly due to tunnelling. This
will not in general be constant, however once steady state is reached this
will become a constant, up to the overall inflationary contribution common to
all sectors. Factoring this out we assume a value of the derivative equal to
$-k_{v}$, with $k_{v}$ being a positive constant that parametrises these UV
properties beyond our EFT knowledge. In practice, the chosen boundary
condition is a way to enforce that, in the far UV, the distribution
asymptotically approaches the diffusionless solution rather than the Gibbs one
discussed in sect. 2.9.

Since boundary conditions are determined by the UV completion, they are
incalculable within the EFT and we have no way to assign any meaningful
measure of their likelihood in theory space. However it is important to assess
whether fine-tuning of boundary conditions was required to realise the
diffusionless solution on the visible branch. This solution is realised for
$k_{v}$ above a certain value $k_{v}^{{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}}$,
defined such that for $k_{v}<k_{v}^{{D\\!\\!\\!\\!\\!\not\,\,\,\,\,}}$ the
solution is predominantly Gibbs-like and no peak is generated. Since the
absolute normalisation of the combined distribution is unphysical (there is a
degeneracy in rescaling simultaneously $k_{v}$ and the parameter $k_{h}$ that
normalises the solution in the $h$ phase, defined below) one cannot attach any
meaning to the absolute numerical values of $k_{v}$. Furthermore, on phase $v$
the solutions behave exponentially whereas on phase $h$ they do not, so a
numerical comparison between $k_{v}$ and $k_{h}$ is unlikely to provide any
insight. More practically, once $k_{v}$ is in the diffusionless window, we may
consider the logarithmic derivative of the position or width of the peak with
respect to $k_{v}$ as a fine-tuning criterion. Our calculation shows that,
once in this basin of attraction, fine-tuning of $k_{v}$ is not necessary.

A complementary perspective comes from considering the inflationary physics.
In slow-roll the dominant contribution of a field to the vacuum energy
density, and hence inflationary rate, is given by the corresponding height of
the scalar potential. As a result, on physical grounds, for a given eigenvalue
(i.e. inflationary rate) we expect the field to be located at the
corresponding position on the scalar potential. This is precisely the case for
the diffusionless solution, giving some physical motivation for the family of
boundary conditions that realise this solution. On the other hand, for Gibbs-
like solutions the field is not located at the corresponding height on the
potential, but very far from it. This means that the inflationary rate is not
being supported by the vacuum energy of a slowly-rolling scalar, but instead
by a large diffusion term, even though the dynamics is in the C regime. This
would seem to imply that for the Gibbs-like class of solutions individual
trajectories must be outside of the slow-roll regime and so we wish to avoid
Gibbs-like solutions in the case that they correspond to implausible or fine-
tuned boundary conditions. It would, however, be valuable to further
investigate the possible UV motivations for the different ranges of IR
boundary conditions and how they delineate the corresponding classes of
solutions.

The junction conditions at $\varphi=\varphi_{T}$ are dictated by the structure
of the phase transition and follow from eqs. (2.73)–(2.74)

$P_{h}(\varphi_{T})=0~{},~{}~{}~{}\Delta
P_{v}(\varphi_{T})=0~{},~{}~{}~{}\Delta
P_{v}^{\prime}(\varphi_{T})=-P_{h}^{\prime}(\varphi_{T})~{}.$ (3.11)

With these boundary and junction conditions in mind, we can easily determine
the properties of the stationary solutions. The mode that satisfies positivity
in the $h$ phase must be of the form

$p_{h}\propto\sin\left[\sqrt{\frac{-2\lambda}{\alpha}}\,(\varphi-\varphi_{T})\right]~{},~{}~{}~{}~{}\lambda=-\frac{\pi^{2}\alpha}{2(1-\varphi_{T})^{2}}~{}.$
(3.12)

Now consider what this implies for the system in the $v$ phase. For $\beta\gg
1$, eq. (2.35) shows that, if a peak of the distribution exists, its location
${\bar{\varphi}}$ must correspond to a potential height

$\omega_{v}({\bar{\varphi}})=-\frac{\pi^{2}\alpha}{2(1-\varphi_{T})^{2}\beta}~{},$
(3.13)

which can be very small in the regime we are considering ($\alpha\beta\ll 1$,
$\beta\gg 1$).

This is a highly non-trivial result. In isolation, the field value
corresponding to $\omega\approx 0$ is not special, thus there is no reason for
the stationary solution in phase $v$ to be peaked at this point. Yet, as a
result of a steady state being reached in phase $h$, where the vacuum may
decay to vacuum $v$, a peak is formed in phase $v$ arbitrarily close to the
point where the two vacua are degenerate, even though the tunnelling between
the two vacua at this specific point is exponentially suppressed.

These considerations are confirmed by the analytical solution of the FPV,
which can be obtained for an exactly linear potential in phase $v$, taking
$\omega_{v}=\varphi$. The asymptotic solution in the perturbative C regime is

$\left\\{\begin{array}[]{lc}{\displaystyle
P_{v}=\alpha\,e^{\frac{\varphi_{T}-\varphi}{\alpha}}\left(a_{\pm}\,{A}_{\varphi}+b_{\pm}\,{B}_{\varphi}\right)}&({\rm
for~{}}\varphi\,\stackanchor[0pt]{$>$}{$<$}\,\varphi_{T})\\\\[5.0pt]
{\displaystyle
P_{h}=\frac{(1-\varphi_{T})k_{h}}{\pi}\,\sin\Big{[}\frac{\pi(\varphi-\varphi_{T})}{(1-\varphi_{T})}\Big{]}}&({\rm
for~{}}\varphi>\varphi_{T})\end{array}\right.$ (3.14)
${A}_{\varphi}=\frac{{\rm Ai}(x)}{{\rm
Ai}(x_{T})}~{},~{}~{}~{}{B}_{\varphi}=\frac{{\rm Bi}(x)}{{\rm Bi}(x_{T})}~{},$
(3.15)
$\begin{array}[]{llll}a_{-}=\frac{k_{h}}{2}+a_{+}~{},&b_{-}=-z\,a_{-}~{},&a_{+}=\frac{k_{v}z_{v}-k_{h}z_{h}}{\alpha\beta}~{},&b_{+}=\frac{k_{h}}{2}+b_{-}~{},\end{array}$
(3.16) $z=\frac{{A}_{-}}{{B}_{-}}\approx
e^{-\frac{2(1+\varphi_{T})}{\alpha}}~{},~{}~{}~{}z_{v}=\frac{e^{\frac{1-\varphi_{T}}{\alpha}}}{{A}_{+}}\,\approx
e^{\frac{\beta}{2}(1-\varphi_{T}^{2})}~{},~{}~{}~{}z_{h}=\frac{{B}_{+}}{{A}_{+}}\approx
e^{-\frac{2(1-\varphi_{T})}{\alpha}}~{},$ (3.17)

where $x$ is defined in eq. (2.42), $x_{T}=x(\varphi_{T})$, and ${A}_{\pm}$ or
${B}_{\pm}$ are the functions ${A}_{\varphi}$ or ${B}_{\varphi}$ evaluated at
$\varphi=\pm 1$. Finally, $k_{h}$ is a normalisation constant determined by
initial conditions. In eq. (3.16), corrections ${\mathcal{O}}(\alpha\beta)$
have been neglected, consistently with the approximation of the C regime. The
expressions in eq. (3.17) after the symbols $\approx$ are given only to show
the size of the corresponding quantities, but should not be used in evaluating
the distribution since they neglect sizeable corrections.

Whenever $k_{v}$ is sufficiently larger than $k_{h}$, the term proportional to
$a_{+}$ dominates the behaviour of $P_{v}$ and the distribution shows a peak
with location and width given by

${\bar{\varphi}}=-\frac{\pi^{2}\alpha}{2(1-\varphi_{T})^{2}\beta}~{},~{}~{}~{}~{}\sigma=\sqrt{\frac{1}{\beta}}~{}.$
(3.18)

This result is in agreement with eq. (3.13). The solution given in eq. (3.14)
is shown in fig. 5 (right panel) for a representative choice of parameters.

In a neighbourhood of the peak location ${\bar{\varphi}}$, we can expand the
solution and obtain

$P_{v}=\alpha\,a_{+}\,e^{\frac{\beta}{2}[\varphi_{T}^{2}-(\varphi-{\bar{\varphi}})^{2}]}\approx\frac{k_{v}}{\beta}\,e^{\frac{\beta}{2}[1-(\varphi-{\bar{\varphi}})^{2}]}~{},$
(3.19)

where the last expression is valid as long as $k_{v}$ is not exceedingly
small. Therefore, the relative probability for the system to reside
asymptotically in the two phases is

$\frac{\int d\varphi\,P_{v}}{\int
d\varphi\,P_{h}}=\sqrt{\frac{\pi^{5}}{2\beta^{3}}}\,\frac{k_{v}\,e^{\frac{\beta}{2}}}{k_{h}(1-\varphi_{T})^{2}}~{}.$
(3.20)

The relative probability depends on the ratio $k_{v}/k_{h}$. If this ratio is
large enough to ensure that $a_{+}>b_{+}$, then phase $v$ is favoured by an
exponentially large factor, since $\int P_{v}/\int
P_{h}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}\sqrt{\alpha^{2}/\beta}\,\exp(\beta\varphi_{T}^{2}/2)$.
If the term proportional to $a_{+}$ dominates the behaviour of $P_{v}$ only up
to the crossing point $\varphi=0$, then phase $h$ is exponentially favoured
since $\int P_{v}/\int
P_{h}\sim\sqrt{\alpha^{2}/\beta}\,\exp(\varphi_{T}/\alpha)$, where
$\varphi_{T}$ is negative.

Note that stationarity in this instance is a form of equilibrium. For any non-
vanishing stationary configuration, as enforced by the boundary conditions,
the solutions on both branches must inflate at precisely the same rate, and
this is the fundamental reason for SOL in this case. There is a form of
equilibrium between the two branches, much as two thermally-coupled boxes of
gas will reach equilibrium even if their microphysics is radically different.
Indeed, since the Hubble scale can be interpreted as a form of temperature of
de Sitter space this equality of constant Hubble scales is very much analogous
to thermal equilibrium.

The waterfall scheme illustrates a striking form of SOL. Thanks to the
interplay between two different phases, the vacuum energy is dynamically self-
tuned to the right value to guarantee degeneracy. Localisation in phase $a$
occurs at a point which is not an extremum of the potential nor special for an
observer confined to this phase. The hidden property of the localisation point
is its degeneracy in energy with a different phase of the theory, which
communicates with phase $a$ only through transitions that occur in a field
region far from the localisation point.

###  3.4 Determining Fundamental Parameters with SOL

One of the most interesting applications of SOL is the determination through
cosmological evolution of fundamental parameters of a microscopic theory,
which we will identify here with the SM, but which could equally be any of its
field-theoretical extensions. The speculation that coupling constants could be
determined by multiple point criticality, which corresponds to the condition
for coexistence of different phases, has already been proposed in ref. [72],
although without offering any concrete theoretical realisation. Conceptual
connections between criticality and cosmology have also been suggested in
broader contexts [73]. In the case of SOL, the basic assumption is that the SM
parameters can be promoted to dynamical variables scanned by the cosmological
evolution of a new scalar field $\phi$, which will be called
apeiron333Apeiron, a Greek word for “boundless,” is the central element of
Anaximander’s vision of cosmology: it is the origin of cosmic order and
balance of forces. Similarly, the apeiron is central in our theory to
determine fundamental parameters through cosmological self-organisation. The
word apeiron is not new in physics, but was previously introduced in 1944 as a
concept in statistical mechanics by Max Born et al. [74]. and which has a
nearly-flat potential because of an underlying approximate shift symmetry.

While scanning SM parameters, the apeiron explores a field range $f$. The SM
sector has an energy cutoff $M$, where new physics appears and whose
interactions are generically described by a coupling constant $g_{*}$ which,
by naïve dimensional analysis, must be smaller than $4\pi$. Based on its
symmetry properties, the general form of the EFT potential is

$V=\frac{M^{4}}{g_{*}^{2}}\,\omega(\varphi)-\varphi\,{\mathcal{O}}_{\rm
SM}+V_{\rm SM}~{},$ (3.21)

where $\varphi=\phi/f$ varies in the range $|\varphi|\leq 1$ and $\omega$ is a
generic function of order unity, assumed to be monotonic with
$\omega^{\prime}>0$.

It is natural to interpret the apeiron as a Goldstone boson emerging from a
spontaneously broken global symmetry in a hidden sector, with a small amount
of explicit symmetry breaking which is responsible for the $\phi$ potential.
In this case, $f$ is not to be necessarily identified with the Goldstone
periodicity because we are interested in an EFT field range where
$\omega(\varphi)$ is a monotonic function. The scale $f$ has to be interpreted
as the field excursion for the apeiron to scan some physical parameter. As a
result, the Goldstone decay constant must be larger than, or at best
comparable with, the scale $f$. The shallowness of the potential in practise
requires $f\gg M_{P}$, which implies that the Goldstone decay constant must be
super-Planckian. This is not necessarily in conflict with a field-theoretical
treatment as long as the energy densities involved are sub-Planckian but, as
discussed in Sec. 2.2, it raises concerns about the embedding of the theory in
a quantum-gravity completion. This problem about super-Planckian field
excursions, which is common in many cosmological setups, can be circumvented
by assuming that the largeness of the scale $f$ is only a mirage created by an
underlying non-trivial monodromy [75] or clockwork mechanism [76, 77, 78].
Alternatively, one could interpret $\phi$ as a nearly flat direction of a non-
compact modulus, protected by an effective supersymmetry residing in the
hidden sector. For our applications, we will simply use the EFT expansion in
eq. (3.21), but we will not need to specify the microscopic nature of $\phi$.

In eq. (3.21), $V_{\rm SM}$ describes the SM potential and ${\mathcal{O}}_{\rm
SM}$ is an operator made of SM fields with a coupling constant scanned by the
apeiron in a range of order one. We have made a field redefinition such that
the apeiron coupling to the SM operator is linear in $\varphi$, but the
generalisation to couplings with other functional dependences and multiple
scanning parameters can be contemplated.

Note that the overall normalisation of the $\varphi$ potential in eq. (3.21)
is dictated by the SM cutoff $M$. This is consistent with naturalness
arguments based on stability under quantum corrections. Loops amount to
replacing ${\mathcal{O}}_{\rm SM}$ with $M^{4}/(4\pi g_{*})^{2}$ and do not
affect the structure of the EFT potential in eq. (3.21). In the notations of
eq. (2.16), the symmetry-breaking coupling $g_{\epsilon}$ is given by

$g_{\epsilon}=\frac{M^{2}}{g_{*}f^{2}}~{},$ (3.22)

and the EFT parameters $\alpha$ and $\beta$ measure the Hubble rate in units
of the SM cutoff ($\alpha\sim H_{0}^{4}/M^{4}$) and the field range in
Planckian units ($\beta\sim f^{2}/M_{P}^{2}$), respectively.

During inflation, the value of $\phi$ undergoes large fluctuations, causing
the SM parameters to vary widely. At certain critical values, the SM vacuum
structure may suddenly change, shifting from a low-field to a high-field
vacuum. In other words, the $\phi$-space can exhibit a critical point with a
first-order quantum phase transition. Much as a cooled ferromagnet
spontaneously and discontinuously flips internal spins as an external magnetic
field is varied to change orientation, so the SM can spontaneously transition
between two vacua once the field $\phi$ evolves past the critical point.

The phase diagram of the theory is such that $\varphi$ triggers a phase
transition in the SM sector at a critical point which can be chosen to be
$\varphi=0$ with an appropriate field shift. The corresponding order parameter
is

$\langle{\mathcal{O}}_{\rm SM}\rangle=\left\\{\begin{array}[]{llc}v_{{\rm
IR}}&{\rm for}~{}\varphi<0&{\rm(IR~{}phase)}\\\ v_{{\rm UV}}&{\rm
for}~{}\varphi>0&{\rm(UV~{}phase)}\end{array}\right.$ (3.23)

with $v_{{\rm IR}}\ll v_{{\rm UV}}\sim M^{4}/g_{*}^{2}$ describing the two
different quantum phases.

As $\varphi$ crosses the critical point, the SM sector backreacts generating a
new contribution to the apeiron effective potential. This process dynamically
drives fundamental parameters towards critical surfaces, providing predictions
of physical quantities that defy the EFT logic, since their values do not
correspond to enhanced symmetries. The role of the apeiron is to drive the SM
to conditions under which two different vacua coexist and are on the verge of
a phase transition. This situation of vacua critical co-existence is the
smoking gun of the SOL mechanism.

###  3.5 SOL Post-Inflationary Dynamics

SOL can only make a prediction about the SM parameters at the end of
inflation, but their actual physical values depend also on post-inflationary
dynamics. The late motion of the apeiron induces a time-dependence of the
scanned SM parameters and masses. The time-variation of the vacuum energy
provides the strongest constraint and also the most universal, since it is
independent of the specific model implementation.

SOL predicts that the FPV distribution of $\phi$ is sharply peaked around a
given value ${\bar{\phi}}$ at the end of a long period of inflation. After
reheating, the apeiron will slow roll down the shallow potential according to
its equation of motion

${\dot{\phi}}\approx-\frac{V^{\prime}({\bar{\phi}})}{3H}~{}~{}~{},~{}V^{\prime}({\bar{\phi}})\approx\frac{\hbar\,H_{0}^{4}}{\pi^{2}M_{P}\sqrt{\alpha^{2}\beta}}~{},$
(3.24)

where $H$ is the Hubble rate of the radiation-dominated Universe and, for
simplicity, we set $\xi=1$. The apeiron potential can be approximated as

$V(\phi)\approx\Lambda^{4}+V^{\prime}({\bar{\phi}})(\phi-{\bar{\phi}})~{},$
(3.25)

where we have tuned the vacuum energy such that it is equal to the observed
value of the cosmological constant ($\Lambda=2.46\times 10^{-3}$ eV) in the
proximity of the configuration ${\bar{\phi}}$.

The dark-energy behaviour of $\phi$ is conveniently described in terms of the
ratio between the pressure $P_{\phi}$ and energy density $\rho_{\phi}$ of the
apeiron

$w=\frac{P_{\phi}}{\rho_{\phi}}=\frac{{\dot{\phi}}^{2}/2-V(\phi)}{{\dot{\phi}}^{2}/2+V(\phi)}\approx-1+\frac{{\dot{\phi}}^{2}}{V({\bar{\phi}})}~{}.$
(3.26)

Since present observations require that dark energy resemble a cosmological
constant with a discrepancy from $w=-1$ of less then percent, we must impose

$\alpha^{2}\beta>\left(\frac{\hbar\,H_{0}^{4}}{M_{P}H_{\rm
now}\Lambda^{2}}\right)^{2}=\left(\frac{H_{0}}{2\times 10^{-3}~{}{\rm
eV}}\right)^{8}~{},$ (3.27)

where $H_{\rm now}=1.37\times 10^{-42}$ GeV is the Hubble rate today.

It is possible to circumvent the bound in eq. (3.27) at the price of
introducing new dynamics into the model, preventing the field $\phi$ to evolve
in the thermal environment. One could imagine a trapping mechanism based on
the existence of a hidden-sector interaction whose non-perturbative effects
break the global shift symmetry, generating a periodic potential that stops
$\phi$ from further evolution during the thermal history of the Universe. An
even more economic solution is to interpret $\phi$ as the familiar axion and
use the non-perturbative QCD interactions to trap the field $\phi$.

Assuming that the trapping mechanism occurs when the temperature of the
Universe is $T_{T}$, the post-inflationary slow roll gives a relative shift of
the field $\phi$ from its initial location ${\bar{\phi}}$

$|\delta_{\phi}|\approx\frac{|V^{\prime}({\bar{\phi}})|\,M_{P}^{2}}{{\bar{\phi}}\,T_{T}^{4}}\approx\frac{H_{0}^{4}}{\alpha\beta\,{\bar{\varphi}}\,T_{T}^{4}}~{}.$
(3.28)

The quantity $\delta_{\phi}$ measures also the relative shift of the physical
parameter scanned by the apeiron. As long as
$|\delta_{\phi}|\\!\stackrel{{\scriptstyle<}}{{{}_{\sim}}}\\!1$, the post-
inflationary evolution does not modify significantly the original SOL
prediction. This implies a lower bound on the trapping temperature $T_{T}$.

###  3.6 EFT Parameters and SOL

It may be useful to summarise here some conditions on EFT parameters that
could be of relevance for SOL. We can express the conditions in terms of the
following three physical parameters: the inflationary background Hubble rate
$H_{0}$, the SM energy cutoff $M$ where new physics is expected to take place,
and the apeiron field excursion $f$ needed to scan the SM parameters in their
full range. Equivalently, we can use the parameters $H_{0}$, $\alpha$ and
$\beta$.

Two conditions we may wish to consider in the EFT framework are the request
that (i) the $\phi$ modulation of the energy density is only a perturbative
correction to the background value, see eq. (2.17), and (ii) the dark-energy
equation of state is not significantly modified by the apeiron slow-roll after
inflation, see eq. (3.27):

$\\!\\!\\!\left\\{\begin{array}[]{c}{\displaystyle\alpha>\left(\frac{H_{0}}{2\times
10^{19}~{}{\rm GeV}}\right)^{2}}\\\
{\displaystyle\alpha^{2}\beta>\left(\frac{H_{0}}{2\times 10^{-3}~{}{\rm
eV}}\right)^{8}}\end{array}\right.~{}~{}{\rm
or}~{}~{}\left\\{\begin{array}[]{cc}{\displaystyle\frac{H_{0}}{\rm
GeV}>\left(\frac{M}{10^{8}~{}{\rm
GeV}}\right)^{2}10^{-3}}&\\!\\!\Big{(}\\!\\!\begin{array}[]{c}{\rm
perturbative}\\\ {\rm domain}\end{array}\\!\\!\Big{)}\\!\\!\vspace{0.1cm}\\\
{\displaystyle\frac{f}{M_{P}}>\left(\frac{M}{10^{8}~{}{\rm
GeV}}\right)^{4}10^{79}}&\\!\\!\Big{(}\\!\\!\begin{array}[]{c}{\rm
dark~{}energy}\\\ {\rm EoS}\end{array}\\!\\!\Big{)}\\!\\!\end{array}\right.$
(3.29)

where we have taken $g_{*}={\mathcal{O}}(1)$. Both conditions are not strictly
necessary for SOL. The first condition can be evaded by going beyond the
perturbative expansion and considering a wide field range in which the energy
variation due to the $\phi$ excursion is large. The second condition does not
apply in presence of a post-inflation trapping mechanism. Nevertheless, these
conditions give an indication for the parameter range of simple SOL models.

When the two constraints in eq. (3.29) are taken together, we find that SOL in
the perturbative region and without trapping (i) can be in the C regime only
if $M\stackrel{{\scriptstyle<}}{{{}_{\sim}}}10$ MeV and444The C regime must
also satisfy $H_{0}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}(M/10~{}{\rm
MeV})^{2}\,10^{-14}$ eV, but this bound at best saturates the stronger
requirement that the reheating temperature must be large enough to allow for
nucleosynthesis which imposes
$H_{0}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}10^{-14}$ eV.
$H_{0}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}(10~{}{\rm MeV}/M)\,10^{-3}$ eV;
(ii) can be in the QV regime only if
$M\stackrel{{\scriptstyle<}}{{{}_{\sim}}}$ TeV and
$H_{0}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}10^{-3}$ eV; (iii) must be in the
Q2V regime if $M\stackrel{{\scriptstyle>}}{{{}_{\sim}}}$ TeV.

The value of $H_{0}$ is related to the reheating temperature after inflation,
which is

$T_{\rm
RH}=\left(\frac{90\,H_{0}^{2}\,M_{P}^{2}}{\pi^{2}\,g_{R}}\right)^{1/4}=\sqrt{\frac{H_{0}}{10^{8}~{}{\rm
GeV}}}~{}9\times 10^{12}~{}{\rm GeV}~{},$ (3.30)

where $g_{R}$ is the number of relativistic degrees of freedom in the thermal
bath. Moreover, $H_{0}$ is directly related to the ratio $r$ of tensor-to-
scalar primordial perturbations, according to $H_{0}=\sqrt{r}\,2.58\times
10^{14}$ GeV. Hence, the non-observation of primordial tensor modes implies
$M\stackrel{{\scriptstyle<}}{{{}_{\sim}}}10^{16}$ GeV for the perturbative
expansion to be applicable.

If the system is in the Q2V regime, the number of inflationary $e$-folds
necessary to reach the asymptotic state is

$N\sim\frac{\pi g_{*}fM_{P}}{\sqrt{\hbar}\,M^{2}}>\left(\frac{M}{10^{8}~{}{\rm
GeV}}\right)^{2}10^{100}~{}.$ (3.31)

The colossal super-Planckianity of $f$, see eq. (3.29), and the order-googol
number of required $e$-folds, see eq. (3.31), manifestly display the issues
about living in the swampland and about the validity of the semi-classical
approach discussed in sect. 2.2.

##  4 Near-Criticality of the Higgs Self-Coupling from SOL

The discovery of the Higgs boson [79, 80] has revealed the surprising
coincidence that the SM parameters (most notably the Higgs quartic and top
Yukawa couplings) lie critically at the edge of a metastability region, where
the SM vacuum is close to a phase transition into a high-field vacuum [81, 82,
83, 84]. Such an intriguing feature of near-criticality could be a hint that
the SOL mechanism is operating in the early Universe.

###  4.1 Phase Diagram

Following the general approach described in sect. 3.4, we consider a setup in
which all SM couplings are $\phi$-dependent scanning parameters although, to
simplify the problem, we fix the weak scale and the cosmological constant at
their observed values. We will describe how one could address these additional
questions in sects. 5 and 6.

The EFT potential is a general function of $\varphi\equiv\phi/f$ and
$g_{*}^{2}h^{2}/M^{2}$, where $h=({2H_{h}^{\dagger}H_{h}})^{1/2}$ is the real
and positive field describing the Higgs boson in unitary gauge, contained in
the Higgs doublet $H_{h}$. For our purposes, the potential can be written in
the form of eq. (3.21) as555It may seem that our parametrisation is unnatural
since we are assuming a precise correlation between the way the quartic and
quadratic Higgs terms scan with $\varphi$. In reality, this choice is made
only for the sake of simplifying the presentation and it has no impact on our
results, since the relevant energy scales are much larger than $v$ and we
could equally well set $v=0$. Also, as discussed in sect. 3.5 the strongest
bound on $f$ comes from the time variation of the vacuum energy and not from
time variations of the Fermi constant or fermion masses which, unlike the
vacuum energy, are sensitive to the scanning procedure of the Higgs
parameters.

$V(\varphi,h)=\frac{M^{4}}{g_{*}^{2}}\,\omega(\varphi)+\frac{\lambda(\varphi,h)}{4}\left(h^{2}-v^{2}\right)^{2}~{},$
(4.1)

where $v=246$ GeV. The effective quartic coupling $\lambda$ is scanned by the
apeiron variation in the range $|\varphi|\leq 1$ and, with no loss of
generality, we can make a field redefinition such that

$\lambda(\varphi,M/g_{*})=-g_{*}^{2}\,\varphi~{},~{}~{}~{}\frac{d\,\lambda(\varphi,h)}{d\ln
h^{2}}=\beta_{\lambda}(h)~{},$ (4.2)

where $\beta_{\lambda}$ is the SM beta-function describing the RG evolution
for $h<M/g_{*}$. In practice, each value of $\varphi$ identifies one RG
trajectory of the quartic coupling. The dependence on all other SM couplings
(which, in turn, are also $\varphi$-dependent) is encoded in the beta-
function.

To simplify the discussion, let us focus on the case in which
$\beta_{\lambda}$ is negative at all scales, as $\varphi$ scans the SM
couplings. Then, the potential in eq. (4.1) has two possible Higgs minima. For
values of $\varphi$ such that $\lambda(v)$ is positive, we find the usual SM
vacuum $\langle h\rangle=v$. For $\varphi>0$, the Higgs potential develops an
unstable direction at large field values. We assume that the UV completion
gives a positive contribution to $\beta_{\lambda}$, sufficient to stabilise
the potential at $\langle h\rangle=c_{{\rm UV}}M$, where $c_{{\rm UV}}$ is a
coefficient of order unity. When $\lambda(h)$ changes sign at an intermediate
scale, it is possible to have coexistence of the two vacua for the same value
of $\varphi$ with a potential barrier separating the two.

Expanding for small $\varphi$ and integrating out the Higgs, the two branches
of the $\varphi$ potential in the IR and UV phases around the critical point
are

$\frac{V(\varphi,\langle
h\rangle)}{M^{4}}=\left\\{\begin{array}[]{rll}\kappa_{{\rm
IR}}\varphi+\dots&~{}{\rm
for}~{}\varphi<\varphi_{+}{}&~{}${{(}IR~{}phase:}$~{}\langle
h\rangle=v)\vspace{0.2cm}\\\ -\kappa_{{\rm UV}}\varphi+\dots&~{}{\rm
for}~{}\varphi>0{}&~{}${{(}UV~{}phase:}$~{}\langle h\rangle=c_{{\rm
UV}}M)\end{array}\right.$ (4.3) $\kappa_{{\rm
IR}}=\frac{\omega^{\prime}(0)}{g_{*}^{2}}~{},~{}~{}~{}~{}\kappa_{{\rm
UV}}=\frac{g_{*}^{2}c_{{\rm UV}}^{4}}{4}-\kappa_{{\rm IR}}~{},$ (4.4)

where the parameters $\kappa_{{\rm IR,UV}}$ are generically of order one and