# Asymptotic Observables and the Swampland

Tom Rudelius<EMAIL_ADDRESS>Physics Department, University of
California, Berkeley CA 94720 USA

###### Abstract

We show that constraints on scalar field potentials and towers of light
massive states in asymptotic limits of scalar field space (as posited by the
de Sitter Conjecture and the Swampland Distance Conjecture, respectively) are
correlated with the prospects for defining asymptotic observables in expanding
FRW cosmologies. The observations of a “census taker” in an eternally
inflating cosmology are further related to the question of whether certain
domain walls satisfy a version of the Weak Gravity Conjecture. This suggests
that answers to fundamental questions about asymptotic observables in
cosmology could help shed light on the Swampland program, and vice versa.

###### pacs:

##  I Introduction

The past few years have seen significant advances in our understanding of the
physics of black holes. Recent computations of the Page curve [1, 2, 3] have
shed light on the black hole information paradox [4] and led to a concrete
realization of black hole complementarity [5, 6, 7]. Meanwhile, strong
evidence has been provided for the absence of global symmetries in quantum
gravity [8, 9, 10, 11, 12, 13, 14] and the Weak Gravity Conjecture (WGC) [15,
16, 17, 18, 19]—two conjectures motivated by demanding consistent black hole
decay—as well as the Swampland Distance Conjecture (SDC) [20, 21, 22, 23, 24,
25, 26]. The consequences of these conjectures have been studied extensively.

At the same time, there has been a revival of interest in questions about de
Sitter space and inflation, though as of yet these questions have not been
clearly answered. Various works have considered bounds on scalar field
potentials in quantum gravity [27, 28, 29, 30, 31, 32], called into question
the existence of de Sitter vacua in quantum gravity [33, 27, 34], and
advocated for alternative models of dark energy [35, 36, 37, 38]. Yet at the
same time, other works have made progress in putting claimed de Sitter
constructions in string theory on more solid footing [39, 40, 41, 42, 43, 44].
Similarly, various works have proposed constraints on inflation in quantum
gravity [45, 46, 47, 48, 49, 35], while other works have attempted to evade
these constraints [50, 51, 52].

The goal of this paper is to draw parallels between the physics of black holes
and the physics of de Sitter cosmologies, which ideally may allow us to
translate some of the recent lessons learned in the former area into progress
in the latter. At the same time, we will also highlight connections between
semiclassical analyses of black holes and cosmology and recent developments in
the “Swampland program.” In particular, we will argue that older studies on
the difficulties of defining asymptotic observables in an expanding universe
may shed light on recent studies of scalar field potentials in string theory,
and in turn, recent progress on the Weak Gravity Conjecture and black hole
complementarity may shed light on asymptotic observables in de Sitter space.

The remainder of this paper is structured as follows. In Section II, we review
the difficulties of defining asymptotic observables in expanding spacetimes.
In Section III, we review the difficulties of defining asymptotic observables,
and making predictions, in an eternally inflating cosmology. In Section IV, we
connect the problem of defining asymptotic observables to the de Sitter
Conjecture [27]. We advocate for a particular “strong form” of the de Sitter
Conjecture, recently proposed by the author in [53], and we explain why this
Strong de Sitter Conjecture may be thought of as a sort of Weak Gravity
Conjecture [15]. In Section V, we connect the thermal fluctuations that occur
in quintessence models to the Swampland Distance Conjecture [20]. In Section
VI, we emphasize that the experiences of observers in an eternally inflating
cosmology are dictated by whether or not domain walls satisfy a version of the
Weak Gravity Conjecture, which suggests a parallel between black hole decay
(which involves the ordinary Weak Gravity Conjecture for charged particles)
and de Sitter vacuum decay (which involves this version of the Weak Gravity
Conjecture for domain walls). We conclude with a discussion of our results and
remaining questions in Section VII.

##  II Asymptotic Observables in an Expanding Universe

To date, all precise formulations of quantum gravity involve either Anti-de
Sitter (AdS) spacetimes or asymptotically flat spacetimes. This is related to
the fact that such spacetimes allow us to define asymptotic observables: in
the former, such observables are represented by correlation functions of a
conformal field theory living at the boundary of AdS, though the AdS/CFT
correspondence. In the latter, observables are represented by S-matrix
elements.

In cosmology, on the other hand, it is not nearly so simple to define
asymptotic observables. In many cosmologies, the presence of an initial
singularity precludes the existence of an S-matrix, though this issue may
perhaps be sidestepped by assuming a unique initial state and writing down an
S-vector that describes only the final state amplitudes [54, 55]. However, an
even larger issue looms: as emphasized in e.g. [56, 57], FRW cosmologies do
not have a property known as “asymptotic coldness”: the energy density of
these spacetimes does not tend to zero at spatial infinity on a fixed Cauchy
slice, and relatedly fluctuations of the geometry extend indefinitely. Any
observer looking into the past at finite time can only access a finite portion
of such a universe, whereas the unobserved region of space contains an
infinite amount of energy and (perhaps) an infinite amount of information.
This prevents the observer from accessing the global state of the universe no
matter how long they wait, and it precludes an S-matrix or S-vector
description.

It is possible that exact asymptotic observables simply do not exist in
cosmology, and trying to define them is nothing but a fool’s errand. But on
the other hand, it is possible that the above issues could be partially
circumvented, and some sort of asymptotic observables could be defined even in
the absence of asymptotic coldness. The prospects for this depend strongly on
the type of cosmology under consideration. Following [56], we review three
such possibilities: de Sitter space (with equation of state parameter $w=-1$),
Q-space ($-1<w<-1/3$), and a decelerating universe ($w>-1/3$).

For future reference, let us write down the metric of a (spatially flat) 4d
FRW cosmology:

$ds^{2}=-dt^{2}+a(t)^{2}\left(dr^{2}+r^{2}d\Omega_{2}^{2}\right)\,,$ (1)

where $a(t)$ is the scale factor, which we assume evolves according to

$a(t)=\left\\{\begin{array}[]{cc}t^{\frac{2}{3(w+1)}}&w>-1\\\
e^{Ht}&w=-1\end{array}\right..$ (2)

Here we define the Hubble parameter $H:=\dot{a}/a$, which is constant for
$w=-1$ but vanishes in the $t\rightarrow\infty$ limit for $w>-1$. The energy
density is then given by $\rho=H^{2}/3$, and the pressure is given by
$p=w\rho$. Throughout this paper we set $8\pi G=1$.

###  II.1 De Sitter

De Sitter space (dS) has equation of state parameter $w=-1$: the scale factor
$a(t)$ expands exponentially at a constant rate for all time. The maximal
extension of de Sitter has the metric

$ds^{2}=\frac{1}{H^{2}\sin^{2}\eta}\left(-d\eta^{2}+d\chi^{2}+\sin^{2}\chi
d\Omega_{2}^{2}\right)\,.$ (3)

and its Penrose diagram is shown in Figure 1. The spatial sections at constant
$\eta$ are 3-spheres, so there is no notion of spatial infinity or null
infinity: the only asymptotic regions are $I^{\pm}$. A comoving observer has
both past and future horizons of radius $H^{-1}$. The region limited by these
respective horizons is called the causal diamond.

Figure 1: The Penrose diagram of de Sitter space.

Several factors preclude the existence of asymptotic observables in de Sitter
space. To begin, de Sitter space has a future horizon: there are regions of
spacetime that are forever out of contact of a comoving observer, so no
observer can ever witness the final state of the entire universe. Second, de
Sitter has a finite entropy,

$S_{dS}=\frac{8\pi^{2}}{H^{2}}\,,$ (4)

and the Hilbert space of quantum gravity in de Sitter is (likely) finite-
dimensional, which puts a limit on the complexity of any sort of measurement
apparatus that could exist in de Sitter space [54]. Finally, as noted in [56],
de Sitter has a constant temperature

$T_{\text{dS}}=\frac{H}{2\pi}\,,$ (5)

and the resulting thermal fluctuations, if sufficiently energetic, may destroy
any observers. Such high-energy thermal fluctuations are Boltzmann-suppressed
by a factor of $\exp(-E/T_{\text{dS}})$, but if one waits long enough such
fluctuations are bound to occur, and thus no observers can exist eternally in
de Sitter space.

For practical purposes in our own universe, these issues are not so important:
the de Sitter entropy is enormous, of order $10^{123}$, and the temperature is
minuscule, of order $10^{-61}$, so observables can be defined to very good
approximation. In addition, one might also entertain the possibility of “meta-
observables” [54], which are exact quantities that could be determined by a
“meta-observer” who has access to the entirety of $I^{+}$ and $I^{-}$ (see
Figure 1). However, neither of these approaches quite addresses the conceptual
problem at hand: no observer living in de Sitter space can perform a
measurement with arbitrary precision, so exact observables do not exist in de
Sitter.

###  II.2 Q-space

Figure 2: The Penrose diagram of Q-space.

Q-space ($-1<w<-1/3$) describes a quintessence-dominated universe, which
accelerates indefinitely into the future but with decreasing Hubble parameter
$H=\dot{a}/a$. Its associated Penrose diagram is shown in Figure 2. An
observer’s causal diamond is bounded by a light-like big bang singularity and
a future event horizon of radius

$R_{E}=-\frac{3(w+1)}{3w+1}t=-\frac{2}{3w+1}H^{-1}\,.$ (6)

There is also an apparent horizon of radius

$R_{A}=H^{-1}=\frac{3}{2}(w+1)t\,.$ (7)

Note that the size of these horizons grows without bound in the limit
$t\rightarrow\infty$, and correspondingly there is no bound on the amount of
entropy that can exist within a comoving observer’s causal diamond. Indeed,
such an observer will experience a thermal heat bath with temperature

$T_{Q}=\frac{H}{2\pi}=\frac{3}{\pi(w+1)t}\,,$ (8)

and although this temperature decreases indefinitely with time, arbitrarily
high-entropy fluctuations will occur even at late times. However, unlike in de
Sitter, the Boltzmann suppression of high-energy fluctuations increases with
time, and shortly after the Hubble scale $H$ drops below a given energy $E$,
the probability of ever again observing a thermal fluctuation of energy $E$
drops quickly to zero [56].

Thus, life is not quite as bad for an observer in Q-space as it is for an
observer in de Sitter: although there is still a future cosmic horizon, there
is no limit on the entropy contained within such a horizon, and
correspondingly there is no inherent limit on the precision of a measurement.
Furthermore, observers in Q-space are rarely destroyed by high-energy thermal
fluctuations, whereas in de Sitter this will eventually occur with probability
1.

Nonetheless, an important problem faces any observer in Q-space who wants to
measure some observable with ever-greater precision: although the maximal
entropy allowed within the horizon may increase indefinitely, the only source
of such entropy is the thermal fluctuations of massless (or nearly
massless—see Section V) particles from the horizon. Even in a more realistic
cosmology with a period of matter/radiation domination preceding accelerated
expansion, the entropy of conventional matter/radiation that enters the causal
diamond of the observer by classical evolution is finite: eventually, a pair
of comoving objects will exit one another’s horizons and cease to interact.
Thus, constructing an experimental apparatus of arbitrarily large complexity
would require the observer to somehow harness the entropy of thermal radiation
from the horizon, and it is not clear that this is possible [56].

In summary, while the prospects for defining asymptotic observables are more
promising in Q-space than in de Sitter, there are nonetheless some important
difficulties that must be overcome. We will now see that some of these
difficulties are avoided in a decelerating universe.

###  II.3 Decelerating Universes

Figure 3: The Penrose diagram of a decelerating FRW cosmology.

The Penrose diagram of a decelerating universe $w>-1/3$ is shown in Figure 3:
essentially, it is obtained by turning the Penrose diagram of Q-space upside
down. A decelerating universe does not have a future horizon: eventually, any
two coming objects will enter one another’s past light cones, and
correspondingly the entropy accessible to a given observer diverges in the
asymptotic future. There is therefore no limitation on the complexity of a
possible experimental apparatus, and there is no limit on the size of space
visible to a late-time observer. The theory at late times approaches that of
particles interacting in flat space: in short, the conditions for observers
are as ideal as they could be in a universe without asymptotic coldness.

##  III Asymptotic Observables in Eternal Inflation

Our universe is experiencing accelerated expansion, and thus far dark energy
has no observed departure from $w=-1$. This is difficult to square with the
aforementioned difficulties facing de Sitter space (among other difficulties
[58, 59, 60]), but string theory may point us in the right direction: in
string theory, de Sitter vacua are generally expected to be metastable,
decaying via bubble nucleation to other vacua in the string Landscape [61].
These vacua may in turn decay via bubble nucleation, and this process
continues until a terminal vacuum is reached [62, 63]: presumably either a
$\Lambda<0$ AdS vacuum (which promptly crunches) or a $\Lambda=0$ vacuum. In
the latter case, the vacuum in question may lie in an asymptotic limit of
scalar field space, in which case bubble nucleation may produce either an
accelerating or decelerating universe, depending on the shape of the
potential—we will elaborate on the details of this point in the next section.

The upshot of this is that the difficult task of defining asymptotic
observables in de Sitter space has been exchanged for the difficult task of
defining asymptotic observables in an eternally inflating cosmology.111A
number of works have questioned the plausibility of eternal inflation [64, 65,
66, 67, 33, 31, 68, 69], though at present there is no sharp no-go result
which forbids it. In such a cosmology, anything that _can_ happen _does_
happen an infinite number of times, raising the difficult question of how to
define a measure on the Landscape of all possible universes. For a nice review
of the measure problem in eternal inflation, see [70].

One approach to the measure problem, notably advocated by Susskind [57], is to
introduce a comoving observer who looks back into the past and collects data.
The expected observations of this observer then define a measure on the
Landscape: namely, the probability $p_{i}$ of outcome $i$ is given in terms of
the number of measurements $N(t)$ in the past light cone of the observer at
time $t$ and the number of times $N_{i}(t)$ the observer measures $i$ by [71]

$p_{i}:=\lim_{t\rightarrow\infty}\frac{N_{i}(t)}{N(t)}\,.$ (9)

Such an observer is called a “census taker.”

Of course, there are different possible census takers one could choose, i.e.,
different locations where the “census bureau” could be located. Some census
takers will end their lives in an AdS vacuum, which ends with a singular
crunch. Such a census taker only has access to a finite amount of entropy
before meeting their demise, so they cannot measure any observable to
arbitrary precision. Other census takers may end up in a bubble of Q-space, in
which the acceleration of the universe asymptotes to zero. This is a much more
promising location for a census bureau, but the challenges of defining
asymptotic observables here are just as thorny here as they were in Q-space:
in particular, a pair of comoving objects will eventually exit one another’s
horizons, and no observer can see all the way to spatial infinity.

The most promising location for a census bureau is in a “hat region”: a bubble
containing a decelerating universe, so named because of the hat-like feature
it introduces to the Penrose diagram (see Figure 4). Such a region has many of
the nice features of a decelerating universe: a census taker in the hat region
region will observe an infinite amount of entropy if she waits long enough,
any two comoving objects in the hat region will eventually come into causal
contact, and physics at late times can be described by an effective field
theory in flat space.

Figure 4: A hat region of an eternally inflating cosmology.

Some preliminary but noteworthy progress has been made in understanding the
physics in such regions. In particular, [72] proposed that the physics in a
hat region has a dual description in terms of a CFT coupled to Liouville
gravity living at spatial infinity $\Sigma$ (see Figure 4). In [73], Harlow
and Susskind further conjectured that the maximum precision of a dual
description of a cosmological geometry is determined by the maximal entropy
bound on the past light cone of a census taker, and they argued that this
description can be exact, i.e., ultraviolet complete, only if there exists a
census taker with an infinite entropy bound (also known as a “maximal” census
taker). In principle, a census taker in a bubble of Q-space fits this
description as well, since they too have an infinite entropy bound, but to
date there has been little to no work in understanding the dual descriptions
of such universes.

One issue facing a census taker in hat region, which is absent in the case of
a decelerating FRW universe, is the existence of a cosmic horizon: as shown in
Figure 4, such a census taker can see all the way to spatial infinity $\Sigma$
in her own bubble, but there are other parts of spacetime hidden beyond the
cosmic horizon. This issue, however, is probably not devastating from the
perspective of defining asymptotic observables: indeed, black hole horizons
exist in asymptotically AdS spacetimes. Here, a crucial role is played by the
notion of black hole complementarity [5, 6, 7]: the physics of the black hole
interior is encoded in the radiation emitted by the black hole horizon. It is
plausible that a similar notion of cosmic horizon complementarity is relevant
here, so that the global structure of the eternally inflating spacetime is
encoded in the radiation emitted from the cosmic horizon [57, 74], ostensibly
allowing the census taker to define a global measure on the spacetime beyond
her horizon [71]. Similarities between de Sitter cosmic horizons and black
hole horizons have recently been investigated in [75, 76, 77].

##  IV Asymptotic Observables and the de Sitter Conjecture

While the difficulties of constructing de Sitter vacua in string theory are
not new, they have drawn renewed interest since the publication of what is now
referred to as the “de Sitter Conjecture (dSC)” [27]. In this work, the
authors proposed a bound on the gradient of scalar field potentials $V$ in
quantum gravity in $d$ spacetime dimensions of the form

$|\nabla V|\geq c_{d}V\,,$ (10)

where $c_{d}$ is an $O(1)$ constant in $d$-dimensional Planck units to be
determined later. This bound is almost surely violated at the maximum of the
standard model Higgs potential, so it is very unlikely that it applies
universally in quantum gravity [78]. However, it is quite plausible that a
bound of this form applies in asymptotic regions of scalar field space [79,
29], and indeed many examples in string theory satisfy this criterion [80, 27,
81]. In the following section, we will review one general argument for the
validity of this bound in asymptotic regions of scalar field space, originally
given in [82].

In order to determine the domain of validity of the dSC bound (10), it is
crucial to fix the precise value of the $O(1)$ constant $c_{d}$, and relatedly
to understand the physical principle underlying this bound. As an analogy, the
precise $O(1)$ coefficient $\gamma_{d}$ appearing in the WGC bound
$|q|/m\geq\gamma_{d}$ is fixed by the physical principle that non-
supersymmetric black holes must be able to decay. We would like a similar
statement here to fix the value of $c_{d}$.

####  IV.0.1 Review: Sharpening the dSC

In [53], it was noted that the dSC is exactly preserved under dimensional
reduction if we set

$c_{d}=c_{d}^{\textrm{strong}}:=\frac{2}{\sqrt{d-2}}\,.$ (11)

In other words, a theory which saturates the bound (10) with this value of
$c_{d}$ will still saturate the bound after dimensional reduction. Of course,
it is not obvious that _every_ valid Swampland conjecture must be exactly
preserved under dimensional reduction, but it is worth noting that some of the
most well-supported Swampland conjectures are, including the Weak Gravity
Conjecture and the absence of global symmetries [83, 84].

The value $c_{d}=c_{d}^{\textrm{strong}}$ is also special from the perspective
of the late-time expansion of the universe: a scalar field rolling in a
potential of the form

$V\propto\exp(-\lambda\phi)$ (12)

will drive expansion of the universe with equation of state

$w=-1+\frac{1}{2}\frac{d-2}{d-1}\lambda^{2}\,,$ (13)

which for $\lambda>c_{d}^{\textrm{strong}}$ implies

$w\geq-\frac{d-3}{d-1}\,.$ (14)

This is precisely the strong energy condition in $d$ dimensions, which is
precisely the condition that forbids accelerated expansion of the universe.
Consequently, we define the “Strong de Sitter Conjecture” as the statement
that the strong energy condition should be satisfied at late times222The “late
times” condition is necessary to exclude the possibility that a scalar field
in an asymptotic region is given a kick of kinetic energy so that it violates
the strong energy condition for a brief time. We thank Thomas Van Riet for
pointing out this possibility to us. in asymptotic limits of scalar field
space, which is equivalent to the de Sitter bound (10) with
$c_{d}=c_{d}^{\textrm{strong}}:=2/\sqrt{d-2}$ if we suppose that the energy
density is dominated by the scalar field condensate. Note that we do not
expect this condition to hold outside of such asymptotic regions of scalar
field space: indeed, the strong energy condition is violated even in our own
universe.

####  IV.0.2 Supersymmetry and the Strong dSC

String vacua with vanishing vacuum energy $V=0$ are expected to be
supersymmetric, except in certain asymptotic limits of scalar field space [73]
such as the weak coupling limit of non-supersymmetric, $SO(16)\times SO(16)$
heterotic string theory [85].333We thank Irene Valenzuela for explaining this
example to us. Many string vacua in asymptotic limits of scalar field space
_are_ supersymmetric, however, and in this context compelling evidence for the
Strong dSC was given in [55]: the authors of that paper argued that within an
accelerating cosmology in four dimensions, a scalar field cannot asymptote to
a zero-energy supersymmetric minimum, and their argument can be extended
trivially to general spacetime dimensions. In particular, the potential around
a stable supersymmetric minimum in $d$ dimensions must take the form [86, 87]:

$V(\phi^{i})=2(d-2)\left((d-2)(\nabla W)^{2}-(d-1)W^{2}\right)\,,$ (15)

where $W$ is the superpotential. If we then assume that the potential takes
the asymptotic form $V\propto\exp(-\lambda\phi)$ as $\phi\rightarrow\infty$,
we must impose $W\approx W_{0}\exp(-\lambda\phi/2)$, which leads to

$V(\phi)\approx
2(d-2)W_{0}^{2}e^{-\lambda\phi}\left((d-2)\frac{\lambda^{2}}{4}-(d-1)\right)\,.$
(16)

This is positive only if

$\lambda>2\sqrt{\frac{d-1}{d-2}}\,,$ (17)

which immediately implies $\lambda>2/\sqrt{d-2}$, thereby satisfying the
Strong dSC.

####  IV.0.3 Asymptotic Observables and the Strong dSC

From the discussion in the previous section, we see that the Strong dSC may
also be motivated by a more physical principle: the existence of asymptotic
hat regions, which may be necessary for the existence of well-defined
observables in spacetimes with positive vacuum energy, requires not only that
$V\rightarrow 0$ limits exist in scalar field space, but also that these
limits occur in regions with decelerating expansion. Conversely, the
difficulty of defining exact observables in asymptotic Q-space may signal an
inconsistency with such spacetimes, implying that a scalar field potential
which violates the Strong dSC bound in asymptotic regions of scalar field
space must reside in the Swampland. Note that the existence of $V\rightarrow
0$ regions with decelerating expansion is uncontroversial: the existence of
$\mathcal{N}=2$ Minkowski vacua in string theory is beyond dispute. The
radical suggestion here is the converse: the nonexistence of asymptotic
Q-space in string theory would be surprising.

In a decelerating universe (in contrast to Q-space), physics at late times is
governed by an effective theory of interacting particles in flat space. In a
sense, the Strong dSC should therefore be thought of as a type of Weak Gravity
Conjecture: just as the ordinary WGC holds that gravity can be decoupled from
electromagnetism at low energies and the scalar WGC [88] holds that gravity
can be decoupled from interactions mediated by a scalar field at low energies,
the Strong dSC holds that gravitational expansion of the universe can be
decoupled from other interactions at late times.

####  IV.0.4 String Theory and the Strong dSC

Many examples in string theory satisfy the Strong dSC, such as the KKLT
scenario and the LVS scenario. More examples are discussed in [53].

Here, we instead focus on a pair of caveats: apparent counterexamples to the
Strong dSC in string theory. The first class of apparent counterexamples come
from string compactifications which saturate the “Transplanckian Censorship
Conjecture” bound [32]:

$|V^{\prime}|\geq\frac{2}{\sqrt{(d-1)(d-2)}}V\,.$ (18)

String compactifications saturating this bound were studied in e.g.[80, 81,
89]. Since the coefficient on the right-hand side of this inequality is
smaller than $c_{d}^{\textrm{strong}}$, these examples naively seem to violate
the Strong dSC.

However, as previously discussed in [53], this conclusion is premature: the
expression $V^{\prime}$ appearing in (18) is the derivative of the potential
with respect to the geodesic distance along certain geodesics in field space.
The expression $\nabla V$ appearing in the Strong dSC is instead the gradient
of the potential, which receives contributions from all the scalar fields in
the theory. Once these additional contributions are included, it is
conceivable that the Strong dSC may be satisfied even while (18) is saturated.
Indeed, one can check that this happens in e.g. heterotic string theory
compactified to four dimensions [53].

Finally, let us note a second class of important examples, which arise from
compactifications of supercritical string theory.444We are grateful to Miguel
Montero and Irene Valenzuela, as well as Eva Silverstein, for independently
bringing this example to our attention. Supercritical string theories actually
saturate the Strong dSC bound [90], but certain compactifications of
supercritical string theories violate the bound and give rise to accelerated
expansion [91, 92]. However, this accelerated expansion lasts for only a
finite time before perturbative control is lost, so these cosmologies do not
contradict the Strong dSC, which by definition forbids accelerated expansion
only at arbitrarily late times in asymptotic limits of scalar field space.

##  V Thermal Fluctuations and the Swampland Distance Conjecture

We have argued that the Strong dSC value
$c_{d}^{\textrm{strong}}=2/\sqrt{d-2}$ is a particularly natural choice for
the $O(1)$ constant $c_{d}$ appearing in the dSC bound, motivated both by
dimensional reduction and by the improved prospects for defining asymptotic
observables in a decelerating universe. Nonetheless, it is worthwhile to
consider the alternative possibility of an observer who ends their life in
Q-space.

Recall from our discussion above that the prospects for asymptotic observables
in Q-space are not completely hopeless: perhaps the greatest difference
between a census taker in Q-space and a census taker in a hat region is that
the Q-space observer cannot access arbitrarily large amounts of information
through classical evolution: the only way to build a measuring device of
arbitrary complexity is through low-energy, high-entropy thermal fluctuations
coming from the cosmic horizon. To assess the possibility of asymptotic
observables in Q-space, we must therefore investigate the thermal fluctuations
that occur in such cosmologies.

At late times, the temperature of the future horizon in Q-space tends to zero,
so the thermal fluctuations become softer and softer. Eventually, one expects
that only massless modes will be emitted _unless_ there are states whose
energy decreases appropriately with time. Indeed, the existence of such states
is a stipulation of another well-tested Swampland conjecture called the
Swampland Distance Conjecture (SDC) [20],555Strictly speaking, the Swampland
Distance Conjecture applies only to massless scalar field moduli in
supersymmetric theories, whereas we are concerned here with a “refined
version” of the conjecture, which holds that this conjecture should apply in
asymptotic limits of scalar field space in non-supersymmetric theories as well
[93]. Compelling, general arguments for both the SDC and its refinement have
been given in [21, 22], and we will not distinguish the two conjectures in
this work. which holds that in any asymptotic limit $\phi\rightarrow\infty$ in
scalar field space, there must exist a tower of particles, each labeled by a
positive integer $n$, whose masses scale with $\phi$ and $n$ as

$m_{n}\sim nm_{0}e^{-\alpha\phi}\,,$ (19)

for $m_{0}$ an arbitrary mass scale and $\alpha$ some $O(1)$ constant in
Planck units. Like the value of $c_{d}$ appearing in the dSC bound (10), the
question of what values of $\alpha$ are allowed in quantum gravity is a topic
of ongoing research.

Motivated by the SDC, let us now consider thermal emission from the Q-space
horizon in four dimensions. As discussed in [56], the probability of thermal
emission per unit time of a state of energy $E$ is given roughly by

$P_{E}\sim\frac{1}{R_{A}}\exp\left[S(E)-2\pi ER_{A}\right]\,,$ (20)

where $R_{A}=H^{-1}$, $\exp(S(E))$ is the number of states of energy $E$, and
we are assuming that there are only $O(1)$ states of energy below the Q-space
temperature $T_{Q}=H/(2\pi)$. According to Bousso’s D-bound [94], which
generalizes the Bekenstein bound [95, 96] to de Sitter space, the term in the
exponent in (20) is necessarily negative.

Integrating this result over all time $t>t_{0}$, we find the probability of a
thermal fluctuation of energy $E$ after time $t_{0}$:

$\displaystyle\mathcal{P}(E)$ $\displaystyle=\int_{t_{0}}^{\infty}dtP_{E}(t)$
$\displaystyle=\int_{t_{0}}^{\infty}dt\frac{1}{R_{A}(t)}\exp\left[S(E)-2\pi
ER_{A}\right]\,.$ (21)

For constant $E\gg R_{A}^{-1}(t_{0})$, the Boltzmann suppression $\exp(-2\pi
ER_{A}(t_{0}))$ dominates, and the total probability $\mathcal{P}(E)$ is
small. However, in a theory satisfying the SDC, the energy of a single-
particle state in the tower itself changes with time, $E=E(t)$. In particular,
if we have

$m_{n}\sim nm_{0}e^{-\alpha\phi}\,,~{}~{}~{}V(\phi)\sim
V_{0}e^{-\lambda\phi}\,,$ (22)

for $\phi$ a canonically normalized scalar field, then the energy of such a
state behaves as $E\sim E(t_{0})\exp(-\alpha\phi)$, whereas the radius $R_{A}$
scales as $R_{A}\sim R_{A}(t_{0})\exp(\lambda\phi/2)$. This means that if

$\alpha=\frac{\lambda}{2}\,,$ (23)

then the Boltzmann suppression factor $\exp(-2\pi ER_{A})$ remains roughly
constant over time, and the total probability is bounded from below by

$\displaystyle\mathcal{P}(E)$
$\displaystyle\geq\int_{t_{0}}^{\infty}dt\frac{1}{R_{A}(t)}\exp\left[-2\pi
E(t)R_{A}(t)\right]$ $\displaystyle\gtrsim\frac{2}{3(w+1)}\exp\left[-2\pi
E(t_{0})R_{A}(t_{0})\right]\int_{R_{A}(t_{0})}^{\infty}\frac{dR_{A}}{R_{A}}\,,$
(24)

where we have used the fact that $R_{A}=\frac{3(w+1)}{2}t$. This integral
diverges logarithmically, so we see that thermal emission of particles in the
SDC tower continues indefinitely when $\alpha\geq\lambda/2$.

Let us make a few remarks on how this result fits with existing literature.
Reference [89] proposed precisely the relationship $\alpha\geq c_{d}/2$
between the exponent of the SDC tower $\alpha$ and the coefficient of the dSC
bound (10), based on the fact that many examples in string theory seem to
satisfy or saturate this bound [89, 21] if one sets

$c_{d}=c_{d}^{\textrm{TCC}}:={2\over\sqrt{(d-1)(d-2)}}\,,$ (25)

which is the value of $c_{d}$ specified by the “Transplanckian Censorship
Conjecture (TCC)” [32]. We see here that the bound $\alpha\geq\lambda/2$
follows from the requirement that states in the SDC tower are thermally
produced at late times, which together with the dSC bound $\lambda\geq c_{d}$
implies $\alpha\geq c_{d}/2$. Whether or not such a requirement should
hold—and whether or not asymptotic Q-space can even occur in quantum
gravity—is a topic for future research.

Under the assumption of the SDC, [82] presented a compelling argument for the
validity of the dSC in asymptotic regions of scalar field space: in the limit
$\phi\rightarrow\infty$, the SDC implies the existence of a large number of
light species $N$, which in turn lead to a UV cutoff on effective field theory
that is parametrically below the 4d Planck scale [97, 98, 99, 100]:

$\Lambda_{\text{UV}}\sim\frac{M_{\text{Pl}}}{\sqrt{N(\Lambda_{\text{UV}})}}\,.$
(26)

According to the SDC, the number of light species is given by

$N(\Lambda_{\text{UV}})\gtrsim\frac{\Lambda_{\text{UV}}}{m_{0}e^{-\alpha\phi}}\,,$
(27)

so

$\Lambda_{\text{UV}}\sim e^{-\alpha\phi/3}\,.$ (28)

On the other hand, the Hubble scale $H$ acts as an IR cutoff on effective
field theory, and this scales with $\phi$ as

$H\sim V^{1/2}\sim e^{-\lambda\phi/2}\,.$ (29)

Requiring that the UV cutoff $\Lambda_{\text{UV}}$ is larger than the IR
cutoff $H$ as $\phi\rightarrow\infty$ therefore implies

$\alpha\leq\frac{3}{2}\lambda\,,$ (30)

which is consistent with the condition $\alpha\geq\lambda/2$.

It is worth emphasizing that the relationship $\alpha\geq\lambda/2$ we have
found here assumes that we are in Q-space, so the universe is accelerating at
late times, i.e., $0<\lambda<2/\sqrt{d-2}$. Our calculation tells us nothing
about the coefficient $\alpha$ of an SDC tower in a decelerating universe
($\lambda>2/\sqrt{d-2}$) or a supersymmetric theory in flat space ($V\equiv
0$), so in particular the bound $\alpha\geq c_{d}/2$ is not applicable if one
assumes the Strong dSC.

##  VI Vacuum Decay and the Weak Gravity Conjecture

So far, we have seen how the experiences of different observers in an
expanding universe point towards the Strong de Sitter Conjecture and perhaps
the Swampland Distance Conjecture: census takers have helped shed light on the
Swampland. We now present an instance of the opposite: the Weak Gravity
Conjecture (WGC) may help shed light on the observations of census takers in a
hat region.

Consider a maximal census taker, i.e., a census taker who begins her life in a
de Sitter phase and ends her life in a hat region. Such a census taker is
maximal in the sense that she will eventually have access to infinite entropy.
However, in the standard description of eternal inflation, her past light cone
will not contain the entirety of the expanding universe (see Figure 4): her
observations are limited by a cosmic horizon. As discussed in Section III
above, this issue could be circumvented by a suitable notion of cosmic horizon
complementarity, in which the region behind the observer’s horizon is somehow
encoded in the radiation she receives from this horizon.666Some support of
this idea may come from recent work [101], which found islands in a dS2
spacetime with a hat region, indicating that an observer in the hat region has
access to information about the inflating region, analogous to the way islands
in black hole spacetimes indicate that an asymptotic observer has access to
information about the black hole interior [1, 2, 3]. It is unclear at present
if this story can be extended to de Sitter spacetimes in more than two
dimensions, however.

Nonetheless, there is a somewhat surprising aspect of this picture: not all
decays from de Sitter space into the $\Lambda=0$ vacuum can occur in the past
light cone of a maximal census taker [70]. This is due to the counterintuitive
fact that the domain wall separating a $\Lambda_{i}>0$ vacuum from a
$\Lambda_{f}=0$ vacuum may appear to recede from the perspective of _both_ the
inside observer and the outside observer. (This fact can be visualized by
analogy with stereographic projection of the 2-sphere, as shown in Figure 5.)
The radius of curvature of concentric shells inside/outside a spherical bubble
at constant time decreases away from the bubble wall on both sides. In this
case, observers on both sides of the bubble believe that they are actually on
the inside of an expanding bubble, and the $\Lambda_{f}=0$ bubble never grows
to encompass the observer in the $\Lambda_{i}>0$ region. As a result, the
observer in the $\Lambda_{i}>0$ region never enters the past horizon of the
$\Lambda_{f}=0$ census taker, so he will never be counted in her census.

Figure 5: Stereographic projection of the 2-sphere. Latitudes in the southern
hemisphere map to circles in the interior of the unit disk, and their radius
of curvature on the 2-sphere grows with increasing size in the stereographic
plane. Latitudes in the northern hemisphere, on the other hand, map to circles
outside the unit disk, and the radius of curvature of these latitudes on the
2-sphere actually _decreases_ while the size of the associated circles in the
plane _increases_.

The question of whether the observer in the $\Lambda_{i}>0$ region sees the
domain wall approaching or receding depends on the relative sizes of
$\Lambda_{i}$ and the tension $T$ of the domain wall. In particular, the
qualification requirement for the census is given in 4d reduced Planck units
by [102]

$T\leq\frac{2}{\sqrt{3}}\left(\sqrt{\Lambda_{i}}-\sqrt{\Lambda_{f}}\right)\,,$
(31)

where in the case at hand, $\Lambda_{f}=0$. If this bound is satisfied, the
$\Lambda_{i}>0$ observer will eventually enter the past light cone of the
maximal census taker. If it is violated, the $\Lambda_{i}>0$ observer will
remain forever beyond her horizon, and information about the $\Lambda_{i}>0$
observer can reach the maximal census taker only in a highly scrambled form
through the radiation she receives from her horizon [71].

This bound is familiar from the classic paper on vacuum decay by Coleman and
de Luccia [61]: when (31) is satisfied comfortably, gravitational effects are
weak and can be largely neglected from the computation of the decay rate. When
it is violated, gravitational effects are strong. Indeed, for a decay from a
$\Lambda_{i}=0$ vacuum to a $\Lambda_{f}<0$ vacuum, these gravitational
effects become so strong that bubble nucleation shuts off entirely. BPS domain
walls in the supersymmetric $\Lambda_{i}=0$ vacuum saturate this bound [103],
and their decay rate vanishes, so the supersymmetric vacuum is (marginally)
stable.

An upper bound on the tension of a brane which is saturated by a BPS brane
smells an awful lot like a Weak Gravity Conjecture, and indeed this connection
was discussed at length by Freivogel and Kleban in [104], who conjectured that
for every vacuum in the Landscape with vacuum energy $\Lambda_{i}$, there
should exist another vacuum of energy $\Lambda_{f}<\Lambda_{i}$ and a domain
wall between them whose tension satisfies (31). If true, this conjecture would
imply that the lifetime of any de Sitter vacuum must be shorter than its
Poincaré recurrence time, and it would imply that any non-supersymmetric Anti-
de Sitter vacuum will decay in finite time (see also [105]).

The relationship between this conjecture and the usual Weak Gravity Conjecture
for $p$-form gauge fields can be made more concrete in the case of domain
walls separating vacua distinguished by different values of flux [104, 105].
In particular, consider the 4d description of axion monodromy inflation [106,
107] pioneered in [108, 109], which has a Lagrangian of the form

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}|F_{4}|^{2}+g\phi
F_{4}\,,$ (32)

with $F_{4}=dC_{3}$. The 3-form field has no propagating degrees of freedom in
4d, so it can be integrated out to produce a quadratic, multi-branched
potential for the axion $\phi$,

$V=\frac{1}{2}(nf_{0}+g\phi)^{2}\,.$ (33)

Here, $n\in\mathbb{Z}$, and $f_{0}$ is the coupling constant of $C_{3}$. The
potential is preserved under shifts $\phi\rightarrow\phi+2\pi f$,
$n\rightarrow n-1$ after imposing the consistency condition $2\pi fg=f_{0}$.
The difference of vacuum energies between a pair of neighboring flux vacua is
given roughly by $nf_{0}^{2}$, so for $n$ of order 1, (31) becomes

$T\lesssim f_{0}\,.$ (34)

Up to an unspecified $O(1)$ prefactor, this is simply the WGC bound for a
domain wall charged under a 3-form gauge field.

Why is this significant? In past discussions, the fact that the observations
of a census taker in a hat region could depend on “arcane details” such as the
tension of a particular domain wall was viewed as an unwelcome surprise, and
potentially even as an indication that such a census taker is not well-suited
to making predictions in eternal inflation [70]. However, recent work on the
WGC suggests a very different perspective: we have seen in many contexts that
the violation/observation of a WGC bound is _not_ an arcane detail, but rather
a crucial consistency condition of quantum gravity. In a theory which violates
the ordinary WGC bound $q/m\geq\gamma_{d}$, non-supersymmetric extremal black
holes cannot decay. Analogously, we see here that a $\Lambda_{i}>0$ vacuum
decay to a $\Lambda_{f}=0$ vacuum cannot occur in the past light cone of a
census taker in the latter vacuum if no domain wall between them satisfies the
WGC-like bound (31).

Perhaps, like the ordinary WGC, this should be interpreted as a consistency
condition on the effective field theory of the $\Lambda_{i}>0$ vacuum: given a
$\Lambda_{i}>0$ vacuum, there necessarily exists a $\Lambda_{f}=0$ vacuum and
a domain wall separating the two vacua whose tension satisfies (31). Such a
condition would represent a strengthening of the conjecture of [104], which
demands the existence of a domain wall with tension satisfying (31) for every
$\Lambda_{i}>0$, but does not demand $\Lambda_{f}=0$. This is also related to
the conjecture of McNamara and Vafa that there should be no nontrivial
cobordisms in quantum gravity [110], which implies that there must exist a
domain wall between any two vacua but does not impose a constraint on the
tension of this domain wall.

On the other hand, it is quite hard to imagine that such a decay channel
should exist for any $\Lambda_{i}>0$ vacuum: it seems very unlikely, for
instance, that our standard model vacuum can decay to a $\Lambda_{f}=0$ vacuum
via a domain wall whose tension satisfies (31), though it is difficult to rule
out this possibility without a better understanding of the quantum gravity
Landscape. In addition, it is worth noting that that the condition (31) is not
satisfied by the domain wall interpolating between the $\Lambda_{i}>0$ vacuum
and the asymptotic region of scalar field space in the KKLT proposal [111]
(though the Strong dSC is satisfied in the asymptotic regime of scalar field
space in that example). Whether this is due to an inconsistency in the KKLT
proposal, indicates the existence of some domain wall not considered in the
original KKLT paper, or represents a counterexample to the supposal of the
preceding paragraph remains to be seen.

##  VII Discussion

In this paper, we have established a connection between various Swampland
conjectures and the difficulty of defining asymptotic observables in expanding
and eternally inflating cosmologies. We have seen that a strong version of the
dSC in asymptotic regions of scalar field space—previously distinguished by
dimensional reduction—would offer the best prospects for defining asymptotic
observables in eternal inflation, as it implies the existence of hat regions
(i.e., decelerating bubble universes) and rules out the possibility of
terminal quintessence vacua in asymptotic regions of scalar field space. We
have seen that the observations of a census taker in a hat region depends on
whether or not a WGC-like bound is satisfied, and we have considered the
possibility that this may be a consistency condition on dS vacuum decay in
quantum gravity, similar to how the ordinary WGC is a consistency condition on
black hole decay. We also saw, however, that a recently-proposed, sharpened
version of the SDC implies that a tower of light states are thermally produced
at late times in Q-space. This may be related to the definition of precise
asymptotic observables in such spacetimes, which necessarily depend on thermal
fluctuations, since these light massive particles may perhaps be better
candidates than massless modes for the construction of an arbitrarily precise
measuring device.

On the other hand, we have not tackled the most important questions: do
asymptotic observables exist in expanding spacetimes? If so, what are they,
and how are they computed? Can census takers in an eternally inflating
universe be used to define a measure on the string Landscape, and if so, what
is it? It is very possible that these questions will ultimately lead us in
directions orthogonal to the the ones we have pursued in this paper: as
discussed, even a decelerating cosmology suffers from asymptotic warmness, and
in light of this it is not clear that even the Strong dSC is sufficient to
guarantee the existence of well-defined observables. Another possibility is
that our semiclassical description of expanding cosmologies must be modified
significantly in the full quantum gravity, so that distant regions of space
contain little new information, and asymptotic coldness is restored. This
possibility is worth exploring further, especially in light of the recent
realization of black hole complementarity in terms of islands [1, 2, 3].
Ultimately, our current cartoon picture of de Sitter and/or eternal inflation
may be subject to serious revision.

Clearly, the correct picture of quantum gravity in cosmology is still coming
into focus, and the ground rules are not yet clear. Nonetheless, it is
heartening to think that progress in the Swampland program may shed light on
the problem of asymptotic observables in cosmology, and vice versa. Certainly,
these are beautiful times to ponder the mysteries of quantum gravity.

Acknowledgments. We thank Lars Aalsma, David Andriot, Raphael Bousso, Juan
Carlos Carrasco Martínez, Hugo Marrochio, Liam McAllister, Jacob McNamara,
Miguel Montero, Matthew Reece, Gary Shiu, Eva Silverstein, Cumrun Vafa, Irene
Valenzuela, and Thomas Van Riet for useful discussions. We further thank Lars
Aalsma, Raphael Bousso, Hugo Marrochio, Liam McAllister, Matthew Reece, and
Gary Shiu for comments on a draft. We thank the String Phenomenology 2021
conference, where some of these useful discussions took place. The work of TR
was supported by NSF grant PHY1820912, the Simons Foundation, and the Berkeley
Center for Theoretical Physics.

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