aainstitutetext: Institute for Theoretical Physics, KU Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgiumbbinstitutetext:  Department of
Physics and Astronomy, University of British Columbia,
6224 Agricultural Road, Vancouver, BC, V6T 1Z1ccinstitutetext: Center for
Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USAddinstitutetext: Dipartimento di Fisica e Astronomia,
Universitá di Firenze,
Via G. Sansone 1, I-50019, Sesto Fiorentino, Firenze, Italy

# Islands in Multiverse Models

Sergio E. Aguilar-Gutierrez a,b Aidan Chatwin-Davies a Thomas Hertog c,d
Natalia Pinzani-Fokeeva a and Brandon Robinson
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider multiverse models in two-dimensional linear dilaton-gravity
theories as toy models of false vacuum eternal inflation. Coupling conformal
matter we calculate the Von Neumann entropy of subregions. When these are
sufficiently large we find that an island develops covering most of the rest
of the multiverse, leading to a Page-like transition. This resonates with a
description of multiverse models in semiclassical quantum cosmology, where a
measure for local predictions is given by saddle point geometries which
coarse-grain over any structure associated with eternal inflation beyond one’s
patch.

##  1 Introduction

The multiverse suffers from an information loss problem akin to that of black
holes: the so-called “measure problem” Linde:1993xx . This arises in
cosmological models that assume a classical near de Sitter (dS) background, in
which quantum fluctuations produce physically distinct patches where inflation
locally ends and a more interesting cosmological evolution can ensue. The
archetypal example is false vacuum-driven eternal inflation with multiple
decay channels. As in the case of black holes, the “problem” concerns a
breakdown of predictivity. The formation of a mosaic of bubbles or patches
with different physical properties, say different statistical features of the
Cosmic Microwave Background (CMB), means the theory fails to predict what we
should observe.

Semiclassical quantum cosmology (QC) in low-energy gravity offers a very
different description of the multiverse that is seemingly at odds with the
view of a cosmic patchwork of bubbles. The global fine-grained mosaic of
bubbles in the description above is replaced in semiclassical QC with a small
number of distinct saddle point geometries. The latter are associated with
coarse-grained descriptions of the universe.111This is terminology from
decoherent histories quantum mechanics. In this context, by “fine-graining” we
mean retaining information on the largest scales whereas “coarse-graining”
does not. Indeed the specific coarse-grained saddle geometries that will be of
interest to us later will contain fine-grained information in a local region.
Specifically, each individual saddle geometry contains information about a
limited cosmic patch or bubble only, while coarse-graining, or averaging, over
any putative mosaic structure on much larger scales. It has been argued that
this semiclassical description resolves the information loss problem
associated with multiverse cosmology Hartle:2010dq . The semiclassical theory
encodes any “global” information that is relevant to the prediction of local
observations of a given observer not as a mosaic structure, but as distinct
past (saddle point) histories of a given observer, with the relative weighting
of saddle points specifying a measure.

The semiclassical recovery of information in multiverse cosmology bears
striking similarities to the recent low-energy gravity description of black
hole evaporation as a unitary process Penington:2019npb ; Almheiri:2019psf ;
Penington:2019kki ; Almheiri:2019qdq ; Marolf:2020xie ; Almheiri:2020cfm ;
Marolf:2020rpm . In both cases, the semiclassical low-energy framework appears
to capture the essential quantum physics without an explicit knowledge of the
microscopic quantum state. Equally striking, semiclassical reasoning appears
to cast doubt on the assumption that a definite spacetime background with
independent degrees of freedom exists well beyond horizons, let alone
indefinitely, in a manner that is independent of the observable of interest.
Instead, an additional saddle appears when a given observer aims to perform
some of the extraordinarily complicated measurements needed to recover a
significant amount of information. For example, the semiclassical calculation
of the fine-grained Von Neumann entropy of Hawking radiation which reproduces
the “Page curve” Page:1993wv ; Page:2013dx , long regarded as a key signature
of unitary evolution, involves additional saddles: replica wormholes
Penington:2019kki ; Almheiri:2019qdq ; Marolf:2020xie ; Almheiri:2020cfm ;
Marolf:2020rpm .

A complementary and calculationally tractable description of the semiclassical
purification process is provided by the “island rule.” According to this, the
Von Neumann entropy of Hawking radiation collected in a region $R$ can be
obtained by extremizing the generalized entropy over possible configurations
$R\cup I$, where $I$ is an additional island region, and then taking the
resulting global minimum,

$S(\rho_{R})=\min\underset{I}{\mathrm{ext}}~{}S_{\mathrm{gen}}(R\cup I),$ (1)

where

$S_{\mathrm{gen}}(X)=S_{\mathrm{semi-cl}}(X)+\frac{{\rm Area}(\partial
X)}{4G_{N}}.$ (2)

Here, $S_{\mathrm{semi-cl}}(X)$ is the Von Neumann entropy of quantum fields
of region $X$ of a classical background geometry, and ${\rm Area}(\partial X)$
is the gravitational area term of the boundary $\partial X$. While originally
motivated on the basis of considerations of holographic entanglement entropy
Ryu_2006 ; Hubeny:2007xt ; Faulkner:2013ana ; Engelhardt:2014gca , the island
rule Eq. (1) in the context of black holes was later found to be consonant
with an analysis based on the semiclassical gravitational path integral
Penington:2019kki ; Almheiri:2019qdq .

Moving back to cosmology, the island prescription opens up a new semiclassical
angle to study the multiverse. This is interesting, for an oft-voiced
criticism against the semiclassical quantum cosmology resolution of the
measure problem has been that the saddle point approximation of the
wavefunction of the universe simply misses relevant information in the global
fine-grained patchwork that eternal inflation supposedly generates. If,
however, large islands were to develop in multiverse configurations whenever
one calculates sufficiently refined observables, then this would suggest that
the coarse-graining inherent in the semiclassical theory is not a bug but a
feature, and an interesting one indeed. The goal of this paper is to explore
precisely this possibility. We do so in two-dimensional toy model multiverse
cosmologies where explicit computations of the Von Neumann entropy of matter
fields are possible, and we then relate our findings in these models to the
more general semiclassical QC description of eternal inflation.

We pursue our analysis in the Jackiw-Teitelboim (JT) theory of two-dimensional
linear dilaton gravity Teitelboim:1983ux ; Jackiw:1984je . JT gravity has seen
a recent resurgence in interest as a simple solvable theory of quantum gravity
Almheiri:2014cka ; Maldacena:2016upp  and given its implications in low
dimensional holography (see e.g. Saad:2019lba ). For our purposes, we shall be
primarily interested in the de Sitter version of JT gravity Maldacena:2019cbz
; Cotler:2019nbi , which has featured in earlier studies of islands in low-
dimensional cosmological toy models Chen:2020tes ; Hartman:2020khs ;
Balasubramanian:2020xqf ; Sybesma:2020fxg ; Aalsma:2021bit ; Manu:2020tty . We
construct a first toy model multiverse by analytic continuation of the dS2
solution. Then, inspired by Hartman:2020khs , we generalize the model by
allowing for regions to be excised and replaced with bubbles of zero- or
negative-curvature spacetime, and we couple conformal matter to the background
metric. The result is a low-dimensional model for the global mosaic spacetimes
featuring in traditional (classical) studies of eternal inflation. We use the
value of the dilaton to characterize regions of spacetime with different
physical properties, identifying regions of weak gravity and of strong gravity
along the way. We then consider interval subregions $R$ located in weakly
gravitating regions of the background. Using the island formula (1), we
compute the Von Neumann entropy associated to $R$ and study its dependence on
properties of $R$ and properties of the global spacetime.

In all cases that we analyze, we are able to show that for a sufficiently
large region $R$, and at sufficiently late times, an island develops.
Consequently, while initially the Von Neumann entropy of the region grows with
its size, a Page-like transition occurs at a critical point beyond which a
configuration with a non-trivial island minimizes the generalized entropy (2).

Further, we find rather universally that islands, when they exist, cover
nearly all of the multiverse structure outside $R$. This agrees with the
results of a recent work Langhoff:2021uct  which considers the formation of
islands in a higher dimensional multiverse setting using the “island finder”
prescription Bousso:2021sji . Both sets of results lend support to the
intuition emanating from the semiclassical QC description of the multiverse
that distant regions may not carry independent degrees of freedom, and thus
that the huge coarse-graining which the semiclassical theory encodes may be
appropriate to derive well-defined predictions for local observations.

The precise point at which the Page-like transition occurs depends on the
details of the multiverse configuration. Nonetheless, reading the
semiclassical QC description the other way around, we are led to conjecture
that, quite generally, islands should form at the threshold of the regime of
eternal inflation that surrounds the weakly or non-gravitating patch
containing $R$, provided of course one considers an appropriate observable.
The picture that arises is that of an “inside out” version of black holes, in
which the definite classical spacetime around us corresponds to an oasis
surrounded by (quantum) uncertainty Hartle:2010dq ; Hartle:2016tpo .

The organization of this paper is as follows. We begin in Sec. 2.1 with a
brief review of de Sitter, flat, and anti de Sitter versions of JT gravity,
and a discussion of how to glue these solutions together to construct JT
multiverse models. We also review in Sec. 2.2 results for the generalized
entropy of an interval in a probe conformal field theory (CFT) with large
central charge coupled to JT gravity. In Sec. 3, we analyze the generalized
entropy of an interval region in a single de Sitter or flat patch of the JT
multiverse, and demonstrate the late time, large interval entropy preferring
the formation of a large single island. We comment on other configurations
including multiple small islands and intervals spanning several patches in
App. A. In Sec. 4, we develop the analogy between the qualitative general
lessons from our investigations of the JT multiverse and a semiclassical
quantum cosmology description of higher dimensional inflationary multiverses.
We conclude with an extensive discussion of our results in Sec. 5, and point
to some future directions.

###  1.1 Erratum

We did not properly account for contributions from the Weyl anomaly and the
Casimir energy in Secs. 2.2 and 3. They combine to give a source for the
metric equation of motion; see, e.g., Balasubramanian:2020xqf . These
contributions cancel when $n=1$.222We thank Edgar Shaghoulian for pointing
this out to us. As a consequence, only the calculations and results in Sec. 3
that operate in the $\phi_{r}/G_{N}\gg c$ limit—where we can neglect
backreaction with certain caveats—are reliable. We discuss further details in
the addendum in App. B. However, the following items which assumed
$\phi_{r}/G_{N}\ll c$ should not be considered reliable:

  * •

The paragraph containing Eqs.(42) and (43).

  * •

The numerical Page curves in Figs. 5-6, 8-9, 13-18, 20.

  * •

The sentence containing Eq. (52) and the one that immediately follows it.

##  2 Jackiw–Teitelboim “cosmology”

In this section, first we discuss our toy model of cosmology in Sec. 2.1 and
then, in Sec. 2.2, we prepare the formulae for the computation of the
generalized entropy of subregions. We borrow techniques developed in Refs.
Chen:2020tes ; Hartman:2020khs  and expand them to build two-dimensional de
Sitter solutions with multiple flat, crunching, and expanding bubbles.

###  2.1 Bubbles in de Sitter JT gravity

Our starting point is the de Sitter version of JT gravity, extensively studied
in Refs. Anninos:2017hhn ; Anninos:2018svg ; Maldacena:2019cbz ;
Cotler:2019nbi . It is a theory of two-dimensional spacetime with positive
curvature coupled to a dilaton field, $\phi$. The action is given by

$I_{dS\text{-}JT}[g_{\mu\nu},\phi]=\frac{\phi_{0}}{16\pi
G_{N}}\int_{\mathcal{M}}d^{2}x\sqrt{-g}\mathcal{R}+\frac{1}{16\pi
G_{N}}\int_{\mathcal{M}}d^{2}x\sqrt{-g}\phi(\mathcal{R}-2)+I_{GHY}[g_{\mu\nu},\phi],$
(3)

where $\mathcal{R}$ is the bulk scalar curvature, $I_{GHY}$ is the Gibbons-
Hawking-York counterterm,

$I_{GHY}[g_{\mu\nu},\phi]=\frac{\phi_{0}}{8\pi
G_{N}}\int_{\partial\mathcal{M}}K+\frac{1}{8\pi
G_{N}}\int_{\partial\mathcal{M}}\tilde{\phi}(K-1),$ (4)

$\tilde{\phi}$ is the boundary value of the dynamical dilaton, and $K$ is the
trace of the extrinsic curvature of the boundary $\partial\mathcal{M}$ of a
manifold $\mathcal{M}$. In writing Eq. (3), we have also included a
topological term proportional to $\phi_{0}$, a positive constant. Moreover, we
have set the length scale of the cosmological constant to one.

Varying $I_{dS\text{-}JT}$ with respect to $\phi$ enforces $\mathcal{R}=2$;
i.e., the spacetime is fixed to be locally $\mathrm{dS}_{2}$. Consider first
the exactly $\mathrm{dS}_{2}$ solution. We may write its line element in terms
of compact global coordinates $(\sigma,\varphi)$ as

$\mathrm{d}s^{2}=\sec^{2}\sigma\left(-\mathrm{d}\sigma^{2}+\mathrm{d}\varphi^{2}\right),$
(5)

where the timelike coordinate $\sigma$ takes values in $(-\pi/2,\pi/2)$ and
the spacelike coordinate $\varphi\in(-\pi,\pi)$ is periodically identified at
its endpoints. These coordinates cover the whole $\mathrm{dS}_{2}$ manifold
and are useful for depicting its conformal structure in a Penrose diagram, as
shown in Fig. 1. Varying $I_{dS\text{-}JT}$ with respect to $g_{\mu\nu}$ gives
the metric equation of motion

$(g_{\mu\nu}\nabla^{2}-\nabla_{\mu}\nabla_{\nu}+g_{\mu\nu})\phi=0,$ (6)

whose solution for the line element (5) is given by

$\phi(\sigma,\varphi)=\phi_{r}\frac{\cos\varphi}{\cos\sigma},$ (7)

where $\phi_{r}>0$, which satisfies $\tilde{\phi}=\phi_{r}/\varepsilon$ for a
small UV cutoff $\varepsilon$.

Figure 1: Penrose diagram for $\mathrm{dS}_{2}$. The horizontal line
$\sigma=\pi/2$ corresponds to $\mathcal{I}^{+}$. The expanding patch (the past
domain of dependence of the part of $\mathcal{I}^{+}$ where $\phi$ diverges to
$+\infty$) is shaded in yellow, and the crunching patch (the past domain of
dependence of the part of $\mathcal{I}^{+}$ where $\phi$ diverges to
$-\infty$) is shaded in green.

Although the spacetime metric has $\mathcal{R}=2$ everywhere, we can think of
this $\mathrm{dS}_{2}$ solution as a simple, low-dimensional model of a
cosmological spacetime that has two types of regions with different physical
properties. These two types of regions are distinguished by the dilaton’s
behavior in the asymptotic future Chen:2020tes ; Hartman:2020khs . Namely,
$\mathcal{I}^{+}$ can be partitioned into an interval where
$\phi\rightarrow+\infty$ and an interval where $\phi\rightarrow-\infty$. The
two types of region in question are then identified with these intervals’ past
domains of dependence. In previous literature, the past domain of dependence
of the part of $\mathcal{I}^{+}$ where $\phi\rightarrow+\infty$ has been
called an “expanding patch”, and the past domain of dependence of its
complement has been called a “crunching patch.” The intuition for this
termninology comes from viewing JT gravity as descending from a higher
dimensional theory, which we briefly review here for completeness; see, e.g.,
Maldacena:2019cbz ; Cotler:2019nbi  for more details. However, we emphasize
that we will always treat the two-dimensional de Sitter JT gravity theory as a
standalone toy model of cosmology.

Starting from the de Sitter-Schwarzschild black hole solution to four-
dimensional Einstein gravity,

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{2}^{2}\qquad\text{with}\qquad
f(r)=1-2M/r-r^{2}/L_{4d}^{2},$ (8)

one can obtain de Sitter JT gravity via dimensional reduction of the near
horizon geometry. The procedure follows by taking the limit where the zeros
($0<r_{-}<r_{+}$) of $f(r)$ degenerate ($r_{-}=r_{+}=\hat{r}$), i.e. the zero
temperature limit, which produces the Nariai dS${}_{2}\times\,\text{S}^{2}$
spacetime geometry:

$\mathrm{d}s^{2}=\hat{r}^{2}\mathrm{d}s_{\mathrm{dS}_{2}}^{2}+\hat{r}^{2}(1+\delta)^{2}\mathrm{d}\Omega_{2}^{2}\qquad\text{with}\qquad
1+\delta=r/\hat{r}.$ (9)

Expanding the four-dimensional theory perturbatively to zeroth order in
$\delta$ and dimensionally reducing on the transverse $\mathrm{S}^{2}$ gives
the topological terms in Eqs. (3) and (4) with
$\phi_{0}=L_{4d}^{2}\hat{r}^{2}/4G_{4d}$. By including the leading deformation
at $O(\delta)$, the same dimensional reduction yields the full JT gravity
action in Eq. (3) with dynamical dilaton $\phi=2\phi_{0}\delta$. Locations
where the dilaton becomes negative therefore correspond to the black hole
interior in the higher dimensional picture, and positive values of the dilaton
correspond to the black hole exterior, where spacetime expands eternally as
$\phi\rightarrow+\infty$. While $\phi$ cannot be less than $-1$ according to
Eq. (9), $\phi\rightarrow-\infty$ in the two-dimensional model is commonly
viewed as signalling the eventual black hole singularity in the higher
dimensional theory. This motivates the nomenclature “expanding patch” and
“crunching patch,” which we will continue to use throughout this work. More
relevantly, we call spacetime regions in which $\phi\rightarrow+\infty$
regions of weak gravity, and regions where $\phi\rightarrow-\infty$ regions of
strong gravity in our model. Much like the terms “expanding” and “crunching,”
this identification is inspired by the higher dimensional theory; the black
hole singularity is clearly a region of strong gravity, while the four-
dimensional Newton’s constant is small when the dilaton is large. However, we
take this identification to be intrinsic to the two-dimensional model itself
and independent of any specific choice of parameters.

We can push this low-dimensional cosmological model further by making two
additional observations. First, one can analytically extend the spacetime by
allowing the angular coordinate $\varphi$ to be $2\pi n$-periodic for natural
numbers $n\geq 1$. This results in a larger spacetime where the line element
is still given by Eq. (5) and on which the dilaton is still given by Eq. (7),
but now we allow $\varphi$ to take values in $(-n\pi,n\pi)$. The Penrose
diagram of such an extension consists of $n$ copies of the diagram in Fig. 1
that are glued together before being periodically identified along the
leftmost and rightmost sides, as illustrated in Fig. 2 with $n=3$. In terms of
an embedding of $\mathrm{dS}_{2}$ as a hyperboloid in $\mathbb{R}^{1,2}$, such
an extension covers the hyperboloid $n$ times. We will denote this $n$-fold
extension of $\mathrm{dS}_{2}$ by $\mathrm{dS}_{2}^{n}$, and the
decompactified limit is obtained by formally taking
$n\rightarrow+\infty$.333While an $n$-fold extension of $\mathrm{dS}_{2}$ is a
well-defined classical solution that can function as a background for a
quantum field theory, subtleties arise if one tries to define a quantum state
for the gravitational sector. We will discuss this point in Sec. 5.

Figure 2: Penrose diagram for extended $\mathrm{dS}_{2}^{n}$. The case with
$n=3$ is drawn here for illustration. The region $R$ lies in the expanding
patch that is centered about $\varphi=0$. We take as an ansatz an island, $I$,
whose endpoints $(\sigma_{I},\pm\varphi_{I})$ are in the crunching patches
that are adjacent to $R$.

Second, as was pointed out in Ref. Hartman:2020khs , one can excise an
expanding patch or a crunching patch and replace it with a patch of flat
spacetime. A flat version of JT gravity is obtained by replacing the integrand
of the second term in Eq. (3) with $\sqrt{-g}\left(\phi\mathcal{R}-2\right)$,
leading to444Notice that there are other possible choices in place of Eq. (10)
that would give flat spacetime solutions in two dimensions. We followed the
conventions of Hartman:2020khs , which result in the dilaton diverging to
$+\infty$ toward the future.

$I_{flat\text{-}JT}[g_{\mu\nu},\phi]=\frac{\phi_{0}}{16\pi
G_{N}}\int_{\mathcal{M}}d^{2}x\sqrt{-g}\mathcal{R}+\frac{1}{16\pi
G_{N}}\int_{\mathcal{M}}d^{2}x\sqrt{-g}(\phi\mathcal{R}-2)+I_{GHY}[g_{\mu\nu},\phi].$
(10)

The resulting dilaton equation of motion is $\mathcal{R}=0$, and the metric
equation of motion is

$(g_{\mu\nu}\nabla^{2}-\nabla_{\mu}\nabla_{\nu})\phi+g_{\mu\nu}=0.$ (11)

In terms of the usual planar coordinates $(t,x)$ for which
$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}x^{2}$, the general solution for
the dilaton is $\phi(t,x)=\frac{1}{2}(t^{2}-x^{2})+At+Bx+C$ for constants $A$,
$B$, and $C$. Let us instead choose coordinates $(\sigma,\varphi)$ by defining

$\displaystyle t$
$\displaystyle=\tan\left(\frac{\sigma+\varphi}{2}\right)+\tan\left(\frac{\sigma-\varphi}{2}\right),$
(12) $\displaystyle x$
$\displaystyle=\tan\left(\frac{\sigma+\varphi}{2}\right)-\tan\left(\frac{\sigma-\varphi}{2}\right).$

The range of these coordinates is $|\sigma\pm\varphi|<\pi$, and the line
element reads

$\mathrm{d}s^{2}=\frac{-\mathrm{d}\sigma^{2}+\mathrm{d}\varphi^{2}}{\tfrac{1}{4}(\cos\sigma+\cos\varphi)^{2}}=\frac{-\mathrm{d}\sigma^{2}+\mathrm{d}\varphi^{2}}{\cos^{2}(\frac{\sigma+\varphi}{2})\cos^{2}(\frac{\sigma-\varphi}{2})}.$
(13)

If we set the integration constants $A=B=0$ and $C=\phi_{r}$, the dilaton
reads

$\phi(\sigma,\varphi)=\phi_{r}+2\tan\left(\frac{\sigma+\varphi}{2}\right)\tan\left(\frac{\sigma-\varphi}{2}\right),$
(14)

and we can continuously join the flat solution in Eqs. (13) and (14) to the
$\mathrm{dS}_{2}$ solution in Eqs. (5) and (7) along the line segments
$\sigma=|\varphi|$; see Fig. 3. The dilaton’s first derivatives will be
discontinuous whenever $\phi_{r}\neq 1$, which signals that the interface must
carry some tension. We will return to this point in the next subsection. Then,
by extension, it follows that for the right choice of integration constants
(as well as an appropriate offset for $\varphi$), one can substitute a flat
patch as defined by Eqs. (13) and (14) for any expanding or crunching patch in
an extended $\mathrm{dS}_{2}^{n}$ manifold. In this way, we can build up a
model which we call a “JT multiverse” that consists of a pattern of expanding,
crunching, and flat patches that can be arbitrarily long.

Figure 3: (Left) Penrose diagram for $\mathbb{R}^{1,1}$. (Right) Penrose
diagram for $\mathrm{dS}_{2}$ where the expanding patch has been replaced with
a bubble of flat spacetime. The potion of full $\mathbb{R}^{1,1}$ that this
bubble corresponds to is shaded in the left diagram.

Figure 4: (Left) Penrose diagram for global $\mathrm{AdS}_{2}$. (Right)
Penrose diagram for $\mathrm{dS}_{2}$ where the crunching patch has been
replaced with a bubble of $\mathrm{AdS}_{2}$. The potion of the
$\mathrm{AdS}_{2}$ manifold that this bubble corresponds to is shaded in the
left diagram. Although not illustrated here, the diamond centred about
$(\sigma^{\prime},\varphi^{\prime})=(0,\pi/2)$ could be used to replace the
expanding patch of $\mathrm{dS}_{2}$.

It is also possible to patch in a portion of a two-dimensional anti de Sitter
($\mathrm{AdS}_{2}$) spacetime in lieu of an expanding or crunching
$\mathrm{dS}_{2}$ patch.555The extension of $\varphi$’s range and the
inclusion of $\mathrm{AdS}_{2}$ bubbles are both departures from the model
proposed in Ref. Hartman:2020khs . Upon replacing $(\mathcal{R}-2)$ with
$(\mathcal{R}+2)$ in Eq. (3), the usual story for $\mathcal{R}=-2$ follows
Maldacena:2016upp ; Sarosi:2017ykf : in terms of Poincaré coordinates $(t,z)$,
the $\mathrm{AdS}_{2}$ line element reads
$\mathrm{d}s^{2}=z^{-2}(-\mathrm{d}t^{2}+\mathrm{d}z^{2})$ and the general
solution for the dilaton is $\phi=(A+Bt+C(t^{2}+z^{2}))/z$. We will instead
work with global coordinates $(\sigma^{\prime},\varphi^{\prime})$, where $t\pm
z=\tan((\sigma^{\prime}\pm\varphi^{\prime})/2)$. The $\mathrm{AdS}_{2}$ line
element then reads

$\mathrm{d}s^{2}=\csc^{2}\varphi^{\prime}(-\mathrm{d}\sigma^{\prime}\,{}^{2}+\mathrm{d}\varphi^{\prime}\,{}^{2}),$
(15)

where $\sigma^{\prime}\in\mathbb{R}$ and $\varphi^{\prime}\in(0,\pi)$; see
Fig. 4 for a Penrose diagram. For the dilaton, we set $A=C=0$ and $B=\phi_{r}$
to obtain

$\phi=\phi_{r}\frac{\sin\sigma^{\prime}}{\sin\varphi^{\prime}},$ (16)

where $\phi_{r}$ is the same as in Eq. (7) so that we may perform a continuous
gluing. If we shift the global coordinates by defining
$\varphi=\varphi^{\prime}-\pi/2$ and $\sigma=\sigma^{\prime}-\pi/2$ (resp.
$\sigma=\sigma^{\prime}+\pi/2$), then we can glue a diamond with
$\sigma>|\varphi|$ into a crunching patch (resp. expanding patch). In terms of
these shifted coordinates, the $\mathrm{AdS}_{2}$ line element reads

$\mathrm{d}s^{2}=\sec^{2}\varphi(-\mathrm{d}\sigma^{2}+\mathrm{d}\varphi^{2}),$
(17)

and the dilaton is given by

$\phi=\pm\phi_{r}\frac{\cos\sigma}{\cos\varphi},$ (18)

where we must take the positive sign when substituting for an expanding patch
and the negative sign when substituting for a crunching patch.

A bubble of $\mathrm{AdS}_{2}$ behaves somewhat similarly to a crunching patch
regarding whether or not an island forms, and so we will not focus too much on
such bubbles. Nevertheless, it is interesting and satisfying that one can
construct toy JT multiverses that contain both flat and negative curvature
bubbles in an ambient positive curvature spacetime. As such, these JT
multiverses are low-dimensional models for the sorts of mosaic universes
predicted by traditional eternal inflation. In such universes, instantons can
nucleate bubbles that have different values of the cosmological constant and
different physical properties within an ambient, eternally inflating spacetime
that has a positive cosmological constant. In the JT multiverses considered
here, the background curvature and asymptotic behavior of the dilaton are
proxies for different, distinguishable cosmological properties.

###  2.2 CFTs and generalized entropy in JT gravity

Here we consider deforming the action of de Sitter and flat JT gravity in Eq.
(3) and Eq. (10), respectively, by coupling to the background metric
$g_{\mu\nu}$ a two-dimensional CFT with field content collectively denoted by
$\psi$, where $I_{CFT}[g_{\mu\nu},\psi]$ is the action of the Lorentzian CFT
describing the matter sector. Crucially, we assume as usual that the CFT does
not couple to the dilaton, and so there is no backreaction to take us away
from the background solution of the dilaton equation of motion. Further, we
assume that the central charge of the CFT is very large, $c\gg 1$, such that
we can consistently treat the gravitational sector in the presence of matter
at the semiclassical level and neglect fluctuations in the boundary mode of
the dynamical dilaton.666Given that our perspective on JT gravity coupled to a
probe CFT is purely two-dimensional, we may freely dial $\phi_{0}$ and
$\phi_{r}$ so long as $c\gg 1$. However, if we were to consider our model as
embedded in a higher dimensional theory, e.g. a feature necessary in the
analysis of Aalsma:2021bit , then the parameters of the lower dimensional
theory would have to lie in a hierarchy $1\ll
c\ll\phi_{r}/G_{N}\ll\phi_{0}/G_{N}$ in order to work in a semiclassical
regime where gravity is weak and the matter sector is a probe of the classical
background. Finally, we require that the CFT is in a global vacuum and
therefore that the stress tensor has vanishing one-point function $\langle
T_{\mu\nu}\rangle=0$ in the geometries that we consider below. With these
assumptions, the metric equations of motion in Eqs. (6) and (11) are left
unmodified.

While requiring $\langle T_{\mu\nu}\rangle=0$, we need to be careful about
trace anomaly contributions,

$\displaystyle\langle T^{\mu}{}_{\mu}\rangle=\frac{c}{12}\mathcal{R},$ (19)

arising in regions of our JT multiverse solutions with non vanishing
$\mathcal{R}$. Such a term can enter as a source for the dilaton as can be
seen, for example, by computing the trace of Eq. (6),

$\displaystyle(\nabla^{2}+2)\phi=8\pi G_{N}\langle T^{\mu}{}_{\mu}\rangle.$
(20)

However, including boundary contributions, the integrated trace anomaly of a
two-dimensional CFT takes the form

$\displaystyle\int d^{2}x\sqrt{-g}\langle
T^{\mu}{}_{\mu}\rangle=\frac{c}{24\pi}\int_{\mathcal{M}}d^{2}x\sqrt{-g}\mathcal{R}+\frac{c}{12\pi}\int_{\partial\mathcal{M}}K.$
(21)

Therefore, at the level of the action, we can redefine the constant value of
the dilaton $\phi_{0}\to\phi^{\prime}_{0}=\phi_{0}+2cG_{N}/3$ to remove the
source. Thus, without loss of generality, we will assume the dilaton obeys
source-free metric equations of motion in what follows.

Ultimately, we will be interested in computing the Von Neumann entropy
$S(\rho_{R})$ associated to a subregion $R$ in our JT multiverse plus CFT
model. According to the island formula Eq. (1), we will therefore need to
compute the generalized entropy for different configurations of $R$ and $I$.
From Eq. (2), we obtain

$\displaystyle S_{\rm gen}(R\cup I)=S_{\rm CFT}(R\cup
I)+\frac{\rm{Area}(\partial I)}{4G_{N}}-S_{\rm{ct}}(\partial I),$ (22)

where $S_{\rm CFT}$ is the semiclassical entropy of CFT fields—that is, the
entropy of the quantum fields on a fixed background geometry evaluated
according to the conventional techniques of quantum field theory in curved
spacetime. The second term in Eq. (22) is the gravitational contribution to
generalized entropy coming from the boundary of the island. In JT gravity, the
“area” of the boundary of the island is just $\phi_{0}+\phi$, evaluated at and
summed over all of the island’s endpoints. We omit an area term due to the
boundary of $R$; in principle we could include this contribution, but it would
not change any of our conclusions, as we will see shortly. In writing Eq.
(22), we have included $S_{\mathrm{ct}}$, a counterterm originating from the
gravitational contribution that renormalizes the UV divergence in $S_{\rm
CFT}$ coming from the boundary of $I$.

To compute the semiclassical entropy of fields, owing to the simplicity of our
model, we can use standard universal results of Von Neumann entropy of a two-
dimensional CFT in Minkowski vacuum Holzhey:1994we ; Calabrese:2004eu ;
Calabrese:2009qy . For a subregion taken to be a single interval of proper
length $\ell$, it is given by

$\displaystyle S_{\rm CFT}=\frac{c}{6}\log\frac{\ell^{2}}{\epsilon_{\rm
uv}^{2}}+s_{0},$ (23)

where $\epsilon_{\rm uv}\ll\ell$ is a UV regulator and $s_{0}$ is a scheme-
dependent constant. In our case, we are working under the assumption that all
CFT fields are in a vacuum state of a JT multiverse geometry written in
$(\sigma,\varphi)$ coordinates. Therefore, we need to translate our global
coordinates to those in which the CFT is in a Minkowski vacuum.

For all values of $\mathcal{R}$, we can put the background metric in the form

$\displaystyle\mathrm{d}s^{2}=\frac{1}{\omega^{2}(\sigma,\varphi)}(-\mathrm{d}\sigma^{2}+\mathrm{d}\varphi^{2}),$
(24)

where

$\omega(\sigma,\varphi)=\left\\{\begin{array}[]{ll}\cos\sigma&(\sigma,\varphi)~{}\text{in
a}~{}\mathcal{R}=2~{}\text{patch}\\\\[5.69054pt]
\tfrac{1}{2}(\cos\sigma+\cos\varphi)&(\sigma,\varphi)~{}\text{in
a}~{}\mathcal{R}=0~{}\text{patch}\\\\[5.69054pt]
\cos\varphi&(\sigma,\varphi)~{}\text{in
a}~{}\mathcal{R}=-2~{}\text{patch.}\end{array}\right.$ (25)

By rescaling $\sigma=n\tilde{\sigma}$ and $\varphi=n\tilde{\varphi}$, such
that the spatial coordinate takes values $\tilde{\varphi}\in(-\pi,\pi)$, we
may perform the coordinate transformation

$\displaystyle z=e^{-i(\tilde{\sigma}+\tilde{\varphi})},\hskip
35.56593pt\bar{z}=e^{-i(\tilde{\sigma}-\tilde{\varphi})},$ (26)

and the metric Eq. (24) becomes

$\displaystyle\mathrm{d}s^{2}=\frac{n^{2}\mathrm{d}z\mathrm{d}\bar{z}}{\omega^{2}(n\tilde{\sigma},n\tilde{\varphi})e^{-2i\tilde{\sigma}}}=:\frac{\mathrm{d}z\mathrm{d}\bar{z}}{\Omega^{2}(z,\bar{z})}.$
(27)

With respect to $(z,\bar{z})$ coordinates, the CFT is in a Minkowski vacuum up
to a Weyl rescaling.

Thus, given the entangling region be an interval with endpoints at
$(z_{1},\bar{z}_{1})$ and $(z_{2},\bar{z}_{2})$, using the metric Eq. (27) in
the universal formula Eq. (23), we find

$\displaystyle S_{\rm
CFT}=\frac{c}{6}\log\left[\frac{z_{12}\bar{z}_{12}}{\epsilon_{\rm
uv}^{2}\Omega_{1}\Omega_{2}}\right],$ (28)

where $z_{ij}:=z_{i}-z_{j}$ (resp. $\bar{z}_{ij}$) and
$\Omega_{i}:=\Omega(z_{i},\bar{z}_{i})$. Following from the coordinate
transformations above for a CFT on the background described in Eq. (24), we
find the following expression for the single interval Von Neumann entropy

$\displaystyle S_{\rm
CFT}=\frac{c}{6}\log\left[\frac{2n^{2}(\cos(\sigma_{ij}/n)-\cos(\varphi_{ij}/n))}{\epsilon_{\rm
uv}^{2}\omega_{1}\omega_{2}}\right]+s_{0},$ (29)

where we adopt the notation $\sigma_{ij},\,\varphi_{ij}$ from above for
$z_{ij}$ and $\omega_{i}:=\omega(\sigma_{i},\varphi_{i})$. Eq. (29) will prove
useful for comparisons in the following sections in our search for islands in
dS${}_{2}^{n}$.

Since the matter sector we are considering is a two-dimensional CFT, the
regularizing term $S_{\rm ct}(\partial I)$ takes a simple form. That is, if we
consider an island configuration of a system of disjoint intervals
$I=\bigsqcup_{j}I_{j}$ with $k$ endpoints, then

$\displaystyle S_{\rm ct}=k\frac{c}{6}\log\frac{\epsilon_{\rm
rg}}{\epsilon_{\rm uv}},$ (30)

where $\epsilon_{\rm rg}\gg\epsilon_{\rm uv}$ is an arbitrary renormalization
scale. The appearance of this scale can be thought of as due to contact terms
in the non-minimally coupled CFT which contributes to the RG flow of $1/G_{N}$
Bousso:2015mna .777This new scale, $\epsilon_{\rm rg}$, can be absorbed into
$G_{N}$, but we will keep it explicit throughout the following sections. As we
will use in the subsequent section for single island configurations, and for
multiple disjoint islands in the appendix, the net effect of $S_{\rm ct}$ on
the rest of the non-geometrical part of the generalized entropy, i.e. $S_{\rm
CFT}$, will be to renormalize $\epsilon_{\rm uv}^{2}\to\epsilon_{\rm
rg}\epsilon_{\rm uv}$.

Before moving on, there are a few remaining subtleties that we must address.
The above review of generalized entropy for large $c$ CFTs in JT gravity
implicitly assumed a smooth gluing of the interfaces between different
patches. However, since we consider configurations with patches of different
$\mathcal{R}$ glued together below, we should address the possible
shortcomings of our approach.

First, we will assume below that it is sufficient to consider transparent
boundary conditions for the CFT matter at the interface between patches. That
is, from the perspective of the CFT, the interface is trivial. However, in the
case that the interface carries some non-trivial tension due to $\phi_{r}\neq
1$, it is not clear a priori that this assumption holds insofar as the
presence of such an interface could break conformal symmetry by interface
couplings between the bulk conformal matter and interface-localized degrees of
freedom. For the following analysis, we can either assume that no interface
couplings appear, such that the probe CFT is completely decoupled, or that the
conformal symmetry enjoyed by the probe CFT is manifest, at least
approximately, in regions far from a non-trivial interface. With either of
these assumptions, it is possible to apply the above results for the Von
Neumann entropy of the CFT (reliably in regions far from a non-trivial
interface) in all cases.

This brings us to the last point that we need to address regarding the
configurations of the entangling region $R$ and the islands $I$. It is well
known that in two-dimensional CFTs on a background with a non-empty boundary
the Von Neumann entropy for a region $R$ that has non-trivial intersection
with the boundary is not simply given by Eq. (23) but rather picks up an
additional universal $\log(g)$ term Affleck:1991tk ; Calabrese:2009qy . The
same $\log(g)$ could ostensibly appear in the generalized entropy if there
exists a non-empty intersection between a non-trivial interface between
patches in dS${}_{2}^{n}$ and $R\cup I$. However, since the $g$-function is
not extensive in the size of the region, neglecting its effects will not
change the results of our analysis in any meaningful way.

##  3 Islands in JT multiverses

In this section, we use the island formula (1) to compute the Von Neumann
entropy associated to a spacelike interval $R$ in the JT multiverses coupled
to a CFT described above. In particular, we consider regions $R$ that are
confined to a single patch, and we look for islands $I$ that are supported
outside of $R$’s patch. We first consider the case of $\mathrm{dS}_{2}^{n}$,
followed by the case where we include flat and negatively curved bubbles.

###  3.1 Extended dS2

Consider an $n$-fold extension of $\mathrm{dS}_{2}$ with a line element and
dilaton given by Eqs. (5) and (7) respectively, and where the coordinate
$\varphi$ runs from $-n\pi$ to $n\pi$. Let $R$ be a spacelike interval with
endpoints $(\sigma_{R},\varphi_{R})$ and $(\sigma_{R},-\varphi_{R})$,888More
accurately, the endpoints of $R$ define a causal diamond to which the entropy
of $R$ is associated. where we take $0<\varphi_{R}\leq\sigma_{R}$ so that $R$
is contained within a single expanding patch, as shown in Fig. 2. For this
configuration, let us compute the von Neumann entropy of the reduced state on
$R$, per the island formula. We must therefore look for extrema of the
generalized entropy $S_{\mathrm{gen}}(R\cup I)$ with respect to the inclusion
of island regions, $I$, and identify the extremum that gives the smallest
generalized entropy.

One extremum is of course the trivial island, $I=\varnothing$. In this case,
the entropy of $R$ reduces to

$S_{\mathrm{gen}}(R)\equiv
S_{\mathrm{CFT}}(R)=\frac{c}{3}\log\left[{\frac{2n\sin(\varphi_{R}/n)}{\epsilon_{\mathrm{uv}}\cos\sigma_{R}}}\right],$
(31)

where we have used Eq. (29) with $\omega=\cos\sigma$, and here and henceforth
we drop the non-universal constant $s_{0}$. Following Hartman:2020khs , we
neglect the (gravitational) area term contribution to $S_{\mathrm{gen}}(R)$
coming from the boundary of $R$ because we will choose the latter to lie near
$\mathcal{I}^{+}$ where $\phi\rightarrow+\infty$, which is our proxy for a
non-gravitating region in any parametric regime. Including this contribution
would just shift $S_{\mathrm{gen}}(R)$ by $(\phi+\phi_{0})/4G_{N}$ evaluated
at the endpoints of $R$. For any nontrivial island, $S_{\mathrm{gen}}(R\cup
I)$ would shift by the same amount, therefore an area term due to $\partial R$
would not affect the competition among extrema.

Motivated by the results of Hartman:2020khs , next we search for a nontrivial
island contained in the causal complement of $R$, whose endpoints are
$(\sigma_{I},-\varphi_{I})$ and $(\sigma_{I},\varphi_{I})$; see Fig. 2. Since
the CFT is in a pure vacuum state, we have that $S_{\mathrm{gen}}(R\cup I)$ is
equal to the generalized entropy evaluated for the complement,
$S_{\mathrm{gen}}((R\cup I)^{c})$, where $(R\cup I)^{c}$ denotes the
complement of $R\cup I$ on any Cauchy slice that contains $R\cup I$. $(R\cup
I)^{c}$ is therefore a symmetric pair of intervals whose endpoints are
$(\pm\varphi_{I},\sigma_{I})$ and $(\pm\varphi_{R},\sigma_{R})$, respectively.
In the operator product expansion (OPE) limit, the disconnected components of
$(R\cup I)^{c}$ are each small and spaced far apart, and so, the reduced state
approximately factorizes across them. Thus, in the OPE limit,
$S_{\mathrm{gen}}((R\cup I)^{c})$ is determined by the sum of the entropies of
its two constituent intervals. Using Eq. (29), we get

$S_{\text{gen}}((R\cup
I)^{c})=\frac{c}{3}\log\left[\frac{2n^{2}\left(\cos(\frac{\sigma_{I}-\sigma_{R}}{n})-\cos(\frac{\varphi_{I}-\varphi_{R}}{n})\right)}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\cos\sigma_{I}\cos\sigma_{R}}\right]+2\phi_{r}\frac{\cos\varphi_{I}}{\cos\sigma_{I}}+2\phi_{0}.$
(32)

Note that we again omit any area term contribution from $R$, but we include
the area term due to $\partial I$. The latter has also the effect of
renormalizing $\epsilon_{\mathrm{uv}}$, as discussed in Sec. 2.2. The OPE
limit approximation is checked in App. A.2. Here and henceforth we set
$4G_{N}=1$.

In order for $I$ to be an entanglement island, the boundary of $I$ must be a
quantum extremal surface. In other words, $S_{\mathrm{gen}}(R\cup I)$ (or
equivalently, $S_{\mathrm{gen}}((R\cup I)^{c})$) must be stationary with
respect to variations of the endpoint coordinates $\sigma_{I}$ and
$\varphi_{I}$. The system of equations

$\displaystyle\frac{\partial}{\partial\sigma_{I}}S_{\text{gen}}((R\cup
I)^{c})$ $\displaystyle=0$ (33)
$\displaystyle\frac{\partial}{\partial\varphi_{I}}S_{\text{gen}}((R\cup
I)^{c})$ $\displaystyle=0$

has no general closed-form solution that we could discern, but it can be
solved in the limits $\phi_{r}\gg c$ and $\phi_{r}\ll c$, as well as
numerically in other parametric regimes. In all cases, we find a critical
point, $(\sigma_{I*},\varphi_{I*})$, located in the upper left corner of the
crunching patch that is adjacent to $R$’s patch, as illustrated in Fig. 2. We
remark that this critical point is actually a maximum with respect to
variations of both $\sigma_{I}$ and $\varphi_{I}$, but evaluation of the
Hessian reveals that this point is still a saddle of $S_{\text{gen}}((R\cup
I)^{c})$ as a function of $\sigma_{I}$ and $\varphi_{I}$.999Using the local
hyperbolic coordinates $X$ and $T$ introduced in Eq. (7.5) of Ref.
Hartman:2020khs  instead, it is possible to show that the saddle that we found
is a maximum in $T$ and a minimum in $X$. The critical point that we identify
here coincides with that found in Ref. Hartman:2020khs  when we set $n=1$.
Further note that the result of App. B of Ref. Hartman:2020khs  only
guarantees that the critical point of $S_{\mathrm{gen}}$ is a timelike maximum
and makes no statement about the spacelike direction. We can then evaluate the
generalized entropy Eq. (32) at this critical point to obtain
$S_{\mathrm{island}}(R)$, which we denote as such to distinguish it from the
(non-extremized) ansatz (32).

Having in mind that $\sigma_{R}$, $\varphi_{R}$, $\sigma_{I}$, and
$\varphi_{I}$ are all close to the corners of their respective patches, let us
write

$\displaystyle\sigma_{R}$
$\displaystyle=\frac{\pi}{2}-\delta\sigma_{R}\qquad\,\sigma_{I}=\frac{\pi}{2}-\delta\sigma_{I}$
(34) $\displaystyle\varphi_{R}$
$\displaystyle=\frac{\pi}{2}-\delta\varphi_{R}\qquad\varphi_{I}=\frac{\pi}{2}+\delta\varphi_{I},$

where $\delta\sigma_{R}$, $\delta\varphi_{R}$, $\delta\sigma_{I}$, and
$\delta\varphi_{I}$ are all positive and small. Making these substitutions in
Eq. (32), we get

$S_{\text{gen}}((R\cup
I)^{c})\approx\frac{c}{3}\log\left[\frac{2n^{2}\left(\cos(\frac{\delta\sigma_{I}-\delta\sigma_{R}}{n})-\cos(\frac{\delta\varphi_{I}+\delta\varphi_{R}}{n})\right)}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\delta\sigma_{I}\delta\sigma_{R}}\right]-2\phi_{r}\frac{\delta\varphi_{I}}{\delta\sigma_{I}}+2\phi_{0}.$
(35)

Next, let us also assume that the sum $\delta\varphi_{I}+\delta\varphi_{R}$
and the difference $\delta\sigma_{I}-\delta\sigma_{R}$ are small, giving

$S_{\text{gen}}((R\cup
I)^{c})\approx\frac{c}{3}\log\left[\frac{(\delta\varphi_{I}+\delta\varphi_{R})^{2}-(\delta\sigma_{I}-\delta\sigma_{R})^{2}}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\delta\sigma_{I}\delta\sigma_{R}}\right]-2\phi_{r}\frac{\delta\varphi_{I}}{\delta\sigma_{I}}+2\phi_{0}.$
(36)

Notice that the generalized entropy is independent of $n$ to leading order.
Let us further assume that $\delta\sigma_{I}\gg\delta\sigma_{R}$, which we can
justify later. With that assumption, so that
$(\delta\sigma_{I}-\delta\sigma_{R})\approx\delta\sigma_{I}$ in the numerator
above, the system of equations
$\partial_{\delta\sigma_{I}}S_{\mathrm{gen}}=0$,
$\partial_{\delta\varphi_{I}}S_{\mathrm{gen}}=0$ has a very simple solution:

$\delta\sigma_{I}=\frac{6\phi_{r}}{c}\delta\varphi_{R},\qquad\delta\varphi_{I}=\sqrt{1+\frac{36\phi_{r}^{2}}{c^{2}}}\delta\varphi_{R}.$
(37)

Note that $\delta\varphi_{I}>\delta\sigma_{I}$, and so the endpoint of $I$ is
in the crunching patch, as we initially required. Plugging this solution back
into Eq. (36), we get

$S_{\text{island}}(R)\approx\frac{c}{3}\log\left[\frac{c}{3\phi_{r}\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}}\left(1+\sqrt{1+\frac{36\phi_{r}^{2}}{c^{2}}}\right)\frac{\delta\varphi_{R}}{\delta\sigma_{R}}\right]-\frac{c}{3}\sqrt{1+\frac{36\phi_{r}^{2}}{c^{2}}}+2\phi_{0}.$
(38)

Now let us consider two separate parametric limits and choose the endpoint of
$R$ accordingly. First, suppose that $\phi_{r}\gg c$. In this case, choose the
endpoint of $R$ such that $\delta\sigma_{R}=\delta\varphi_{R}/N$, where $N$ is
at least $O((\phi_{r}/c)^{0})$. In other words, we suppose that as we drag the
right endpoint of $R$ toward the upper right corner of the expanding patch, we
keep the ratio $\delta\sigma_{R}/\delta\varphi_{R}$ fixed. It follows that
$\delta\varphi_{R}>\delta\sigma_{R}$, so that the endpoint of $R$ is indeed in
the expanding patch, and in this regime where $\phi_{r}\gg c$, the assumption
$\delta\sigma_{I}\gg\delta\sigma_{R}$ is justified. The endpoints of $R$ are
also parametrically close to $\mathcal{I}^{+}$. Making this choice and
dropping subdominant terms, we arrive at

$S_{\text{island}}(R)\approx\frac{c}{3}\log\left[\frac{2N}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}}\right]-2\phi_{r}+2\phi_{0}.$
(39)

If we further drop the logarithmic correction, we have that
$S_{\mathrm{island}}(R)\approx 2(\phi_{0}-\phi_{r})$, which is just twice the
value of the dilaton evaluated at the boundary of the crunching patch. Either
way, $S_{\mathrm{island}}(R)$ is approximately constant, while
$S_{\mathrm{CFT}}(R)$ diverges as the endpoints of $R$ approach
$(\pi/2,\pm\pi/2)$. There is therefore a “Page transition” beyond which the
nontrivial island entropy is the smaller extremum.

Let us compare $S_{\mathrm{island}}(R)$ to $S_{\mathrm{CFT}}(R)$ to determine
the location of the Page transition. Plugging Eq. (34) into Eq. (31), we get

$S_{\mathrm{CFT}}(R)\approx\frac{c}{3}\log\left[\frac{f_{n}}{\epsilon_{\mathrm{uv}}\delta\sigma_{R}}\right],$
(40)

where $f_{n}=2n\sin(\pi/2n)$. Equating $S_{\mathrm{CFT}}(R)$ and
$S_{\mathrm{island}}(R)$, we find that the Page transition occurs at

$\delta\sigma_{R}^{\rm
Page}=\frac{f_{n}\epsilon_{\mathrm{rg}}}{2N}e^{-\frac{6}{c}(\phi_{0}-\phi_{r})},\qquad\delta\varphi_{R}^{\rm
Page}=\frac{f_{n}\epsilon_{\mathrm{rg}}}{2}e^{-\frac{6}{c}(\phi_{0}-\phi_{r})}.$
(41)

We can also read off the mild dependence of the Page transition on $n$.
Because $f_{n}$ monotonically increases up to $\pi$ as $n\rightarrow+\infty$,
we see that the size of $R$ at which the Page transition occurs
correspondingly monotonically decreases to a finite size.

If we instead suppose that $\phi_{r}\ll c$, choose the endpoints of $R$ such
that $\delta\sigma_{R}=(6\phi_{r}/Nc)\delta\varphi_{R}$ for the same
consistency reasons as above, where $N$ is at least $O((c/\phi_{r})^{0})$. It
then follows that

$S_{\text{island}}(R)\approx\frac{c}{3}\left(\log\left[\frac{Nc^{2}}{9\phi_{r}^{2}\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}}\right]-1\right)+2\phi_{0},$
(42)

which again remains constant while $S_{\mathrm{CFT}}(R)$ diverges. In this
case, the Page transition occurs at

$\delta\sigma_{R}^{\rm
Page}=\frac{f_{n}\epsilon_{\mathrm{rg}}}{4N}\left(\frac{6\phi_{r}}{c}\right)^{2}e^{-\frac{6}{c}(\phi_{0}-1)},\qquad\delta\varphi_{R}^{\rm
Page}=\frac{f_{n}\epsilon_{\mathrm{rg}}}{4}\left(\frac{6\phi_{r}}{c}\right)e^{-\frac{6}{c}(\phi_{0}-1)}.$
(43)

For other parametric regimes, we must turn to numerics; see, for example,
Figs. 5 and 6.101010A Mathematica notebook that reproduces the plots in this
manuscript is included as supplementary material. In all cases, however, we
see the same basic physics at play: $S_{\mathrm{CFT}}(R)$ gives the lesser
entropy for small $R$, but there is a Page transition after which
$S_{\mathrm{island}}(R)$ is the lesser entropy for sufficiently large $R$. The
crossover point monotonically decreases as a function of $n$, and in numerical
analyses, we can study the limiting behavior by taking the
$n\rightarrow+\infty$ limit of Eq. (32), which gives

$\lim_{n\rightarrow\infty}S_{\text{gen}}((R\cup
I)^{c})=\frac{c}{3}\log\left[\frac{(\varphi_{I}-\varphi_{R})^{2}-(\sigma_{I}-\sigma_{R})^{2}}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\cos\sigma_{I}\cos\sigma_{R}}\right]+2\phi_{r}\frac{\cos\varphi_{I}}{\cos\sigma_{I}}+2\phi_{0}.$
(44)

Figure 5: $S_{\mathrm{CFT}}(R)$ (brown, blue, dark green) versus $S_{\rm
island}(R)$ (orange, red, black) in dS${}^{n}_{2}$ for $n=1,\;2,\;+\infty$,
respectively, with $\sigma_{R}=\pi/2-10^{-5}$ held fixed. The size of $R$
beyond which the island contribution to generalized entropy becomes dominant
slightly decreases as $n$ increases. Because $\sigma_{R}$ is held fixed in
this plot, taking $\varphi_{R}$ all the way to $\pi/2$ moves the endpoint of
$R$ outside of the expanding patch. In this limit, the endpoint of $I$ also
moves outside of the crunching patch and $R\cup I$ tends to a full Cauchy
slice on which the state is pure, resulting in vanishing entropy. The
parameter values used for this plot are $c=600$, $\phi_{r}=10$, $\phi_{0}=0$,
$\epsilon_{\mathrm{uv}}=1$, $\epsilon_{\mathrm{rg}}=1$.

Fig. 5 Shows the competition between $S_{\mathrm{CFT}}(R)$ and
$S_{\mathrm{island}}(R)$ for different $n$ as $\varphi_{R}$ varies with
$\sigma_{R}$ held fixed. These curves reproduce the same qualitative features
that followed from the approximate analysis above. We can also examine the
competition between $S_{\mathrm{CFT}}(R)$ and $S_{\mathrm{island}}(R)$ as
$\sigma_{R}$ is varied, as shown in Fig. 6 for $n=1$. A Page transition still
occurs as $\sigma_{R}$ is decreased, and the value of $\varphi_{R}$ at which
the transition occurs also decreases. We also find that moving the subregion
$R$ back in time pushes the island forward in time toward ${\cal I}^{+}$.
Below a limiting value $\sigma_{R}^{\star}$, the island is formally pushed
beyond ${\cal I}^{+}$, outside the allowed range of the parameters of the
crunching patch. A similar behavior persists for all values of $n$.

Figure 6: $S_{\rm island}(R)$ (orange, red, black) versus
$S_{\mathrm{CFT}}(R)$ (dashed, same color scheme) in dS2 (i.e. $n=1$) for
$\sigma_{R}=\pi/2-0.001$, $\pi/2-0.02$, and $\pi/2-0.051$, respectively. As we
decrease $\sigma_{R}$, the size of $R$ at which the Page transition occurs
also decreases. The parameter values used for this plot are $c=10000$,
$\phi_{r}=100$, $\phi_{0}=0$, $\epsilon_{\mathrm{uv}}=1$,
$\epsilon_{\mathrm{rg}}=1$. These exaggerated choices of parameters were made
to clearly illustrate the shifts in entropy.

While we have found an island that extremizes $S_{\mathrm{gen}}(R\cup I)$, one
should ask whether there are other island configurations consisting of
multiple disjoint components that give smaller values of
$S_{\mathrm{gen}}(R\cup I)$. Heuristically, such islands are disfavored by the
island formula. One would expect that a single large island, such that $R\cup
I$ covers as much of a Cauchy surface as possible, would be more efficient at
purifying the state of $R$ compared to several smaller disjoint components,
thus lowering the CFT entropy cost. Moreover, the area of the boundary of
every disjoint piece of an island contributes to the total generalized
entropy. Therefore (at least when $\phi_{0}\gg\phi_{r}$) the geometric cost to
form an island is larger for a greater number of disconnected components.
While we cannot prove that the single large island is the minimal extremum, we
were able to verify that the extrema for which $I$ consists of two
disconnected components result in a larger generalized entropy for a theory of
$c\gg 1$ free Dirac fermions. The details of our numerical analysis are
elaborated in App. A.3. In particular, a plot of $S_{\mathrm{gen}}(R\cup I)$
for these non-minimal extrema $I$ as a function of the size of $R$ is shown in
Fig. 20. This constitutes evidence that the single large island is indeed
likely the minimal extremum.

We can also consider the case where $R$ is in a crunching patch. However, if
we look for an island whose endpoints lie in the surrounding expanding
patches, we find that the extremality conditions (33) cannot be
satisfied.111111This is consistent with the fact that the necessary conditions
for island formation presented in Ref. Hartman:2020khs  are not satisfied in
the expanding patch. In other words, there are no quantum extremal surfaces,
and so no islands of this type form.

###  3.2 Extended dS2 with bubbles

We now consider an $n$-fold extension of $\mathrm{dS}_{2}$ where the expanding
patch centred about $\varphi=0$ has been replaced with a flat bubble with the
line element Eq. (13) and on which the dilaton is given by Eq. (14); see Fig.
7. Let $R$ have endpoints $(\sigma_{R},\pm\varphi_{R})$ contained within this
flat bubble; we will examine the entropy of $R$ as its size increases while
keeping its endpoints close to $\mathcal{I}^{+}$ (i.e.
$\varphi_{R}+\sigma_{R}\approx\pi$).

We again compute $S(\rho_{R})$ using the island formula. In this case, the
trivial island gives

$S_{\text{gen}}(R)\equiv
S_{\mathrm{CFT}}(R)=\frac{c}{3}\log\left[\frac{2n\sin(\varphi_{R}/n)}{\epsilon_{\mathrm{uv}}\cos(\tfrac{1}{2}(\sigma_{R}-\varphi_{R}))\cos(\tfrac{1}{2}(\sigma_{R}+\varphi_{R}))}\right],$
(45)

where we used Eq. (29) with the flat Weyl factors for our chosen coordinates.
Next, we look for an island with endpoints $(\sigma_{I},\pm\varphi_{I})$ that
lie in the crunching patches that are adjacent to the flat bubble, as depicted
in Fig. 7. The generalized entropy for such an island is

$S_{\text{gen}}((R\cup
I)^{c})=\frac{c}{3}\log\left[\frac{2n^{2}\left(\cos(\frac{\sigma_{I}-\sigma_{R}}{n})-\cos(\frac{\varphi_{I}-\varphi_{R}}{n})\right)}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\cos\sigma_{I}\cos(\tfrac{1}{2}(\sigma_{R}-\varphi_{R}))\cos(\tfrac{1}{2}(\sigma_{R}+\varphi_{R}))}\right]\\!+2\phi_{r}\frac{\cos\varphi_{I}}{\cos\sigma_{I}}+2\phi_{0}$
(46)

where we again compute the entropy of the complement and have invoked an OPE
limit approximation.

Figure 7: A single-component island $I$ in an extended JT multiverse for $R$
in a flat bubble.

As before, we can explicitly solve the extremality conditions (33) in the
limits $\phi_{r}\gg c$ and $\phi_{r}\ll c$, as well as numerically in other
regimes. In our current configuration, we set

$\displaystyle\sigma_{R}$
$\displaystyle=\frac{\pi}{2}+\delta\sigma_{R},\qquad\,\sigma_{I}=\frac{\pi}{2}-\delta\sigma_{I},$
(47) $\displaystyle\varphi_{R}$
$\displaystyle=\frac{\pi}{2}-\delta\varphi_{R},\qquad\varphi_{I}=\frac{\pi}{2}+\delta\varphi_{I},$

so that the endpoints of $R$ and $I$ are near $\mathcal{I}^{+}$. With these
definitions, Eq. (46) approximately reduces to

$S_{\text{gen}}((R\cup
I)^{c})\approx\frac{c}{3}\log\left[\frac{(\delta\varphi_{I}+\delta\varphi_{R})^{2}-(\delta\sigma_{I}+\delta\sigma_{R})^{2}}{\epsilon_{\mathrm{uv}}\epsilon_{\mathrm{rg}}\delta\sigma_{I}(\delta\varphi_{R}-\delta\sigma_{R})}\right]-2\phi_{r}\frac{\delta\varphi_{I}}{\delta\sigma_{I}}+2\phi_{0}.$
(48)

Again assuming that $\delta\sigma_{I}\gg\delta\sigma_{R}$, upon extremizing
Eq. (48) with respect to $\delta\sigma_{I}$ and $\delta\varphi_{I}$, we find
the same critical point as Eq. (37). This gives

$S_{\text{island}}(R)\approx\frac{c}{3}\log\left[\frac{c}{3\phi_{r}\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}}\left(1+\sqrt{1+\frac{36\phi_{r}^{2}}{c^{2}}}\right)\frac{\delta\varphi_{R}}{\delta\varphi_{R}-\delta\sigma_{R}}\right]-\frac{c}{3}\sqrt{1+\frac{36\phi_{r}^{2}}{c^{2}}}+2\phi_{0}$
(49)

for the generalized entropy corresponding to $R\cup I$. Similarly, the no-
island entropy is

$S_{\mathrm{CFT}}(R)\approx\frac{c}{3}\log\left[\frac{f_{n}}{\epsilon_{\mathrm{uv}}(\delta\varphi_{R}-\delta\sigma_{R})}\right]\approx\frac{c}{3}\log\left[\frac{f_{n}}{\epsilon_{\mathrm{uv}}\delta\varphi_{R}}\right],$
(50)

where the second step follows because we will always choose $\delta\sigma_{R}$
to be much less than $\delta\varphi_{R}$.

In the limit where $\phi_{r}\gg c$, we again choose
$\delta\sigma_{R}=\delta\varphi_{R}/N$. This gives

$S_{\mathrm{island}}(R)\approx\frac{c}{3}\log\left[\frac{2}{(1-N^{-1})\epsilon_{\mathrm{uv}}\epsilon_{\mathrm{rg}}}\right]-2\phi_{r}+2\phi_{0},$
(51)

and results in a Page transition at the same location as in Eq. (41). In the
limit where $\phi_{r}\ll c$, we choose
$\delta\sigma_{R}=(6\phi_{r}/Nc)\delta\varphi_{R}$ as in the previous section,
which gives

$S_{\mathrm{island}}(R)\approx\frac{c}{3}\left(\log\left[\frac{2c}{3\phi_{r}\epsilon_{\mathrm{uv}}\epsilon_{\mathrm{rg}}}\right]-1\right)+2\phi_{0}.$
(52)

The Page transition in this case happens as in Eq. (43).

A plot of $S_{\mathrm{CFT}}(R)$ and $S_{\mathrm{island}}(R)$ outside of the
regimes discussed above for different values of $n$ is shown in Fig. 8. The
$n\rightarrow+\infty$ limit is computed in the same way as Eq. (44), but with
$\cos\sigma_{R}$ replaced with the Weyl factor for flat spacetime,
$\tfrac{1}{2}(\cos\sigma_{R}+\cos\varphi_{R})$. The endpoints of $R$ are
initially chosen such that $\sigma_{R}>\pi/2$ and are dragged toward
$\sigma_{R}=\varphi_{R}=\pi/2$ to increase the size of $R$. For these
parameter choices, the Page transition is visually very clear.

Figure 8: $S(\rho_{R})$ evaluated for no islands (brown, blue, dark green
ascending lines) and a single large island for $n=1,2,+\infty$ (orange, red,
black curves, respectively, from bottom to top). For a given $\varphi_{R}$, we
set $\sigma_{R}=-(2/\pi)(\pi/2-10^{-3})\varphi_{R}+\pi-10^{-3}$ so that $R$
grows large while its endpoints remain near $\mathcal{I}^{+}$. Other parameter
values are $c=600$, $\phi_{r}=10$, $\phi_{0}=0$, $\epsilon_{\mathrm{uv}}=1$,
$\epsilon_{\mathrm{rg}}=1$.

As in our analysis of $\mathrm{dS}_{2}^{n}$, we can also consider the case
where $R$ is in a flat bubble that is surrounded by two expanding patches,
e.g., a flat bubble that replaces the crunching patch centred about
$\varphi=\pi$. Once again, we find that the extremality conditions (33) cannot
be satisfied for an island whose endpoints are in the adjacent expanding
patches.

Finally, we note that islands continue to develop when the crunching
$\mathrm{dS}_{2}$ patches are replaced with bubbles of $\mathrm{AdS}_{2}$. To
see this, we return to the case where $R$ has endpoints
$(\sigma_{R},\pm\varphi_{R})$ that lie in a central $\mathrm{dS}_{2}$
expanding patch, but we now replace the adjacent crunching patches with
$\mathrm{AdS}_{2}$ bubbles, in which the line element and dilaton are given by
Eqs. (17) and (18), respectively. For an island with endpoints
$(\sigma_{I},\pm\varphi_{I})$ with $\pi/2<\varphi_{I}<3\pi/2$, we now have

$S_{\text{gen}}((R\cup
I)^{c})=\frac{c}{3}\log\left[\frac{2n^{2}\left(\cos(\frac{\sigma_{I}-\sigma_{R}}{n})-\cos(\frac{\varphi_{I}-\varphi_{R}}{n})\right)}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\cos(\varphi_{I}-\pi)\cos\sigma_{R}}\right]-2\phi_{r}\frac{\cos\sigma_{I}}{\cos(\varphi_{I}-\pi)}+2\phi_{0}.$
(53)

The behavior of the dilaton near the corners of an AdS2 bubble is very
different compared to its behavior in dS2 crunching patches. In the latter
case, $\phi\rightarrow-\infty$ near a crunching patch’s corners at
$\mathcal{I}^{+}$, whereas $\phi\rightarrow 0$ near the corners of an AdS2
bubble. This ultimately locates the endpoints of $I$ away from the corners of
the AdS2 bubbles surrounding $R$, and so a perturbative expansion like Eq.
(47) is no longer useful. Nevertheless, we still find nontrivial islands
numerically and we see that an island produces the minimal generalized entropy
past a critical value of $\varphi_{R}$, which we show in Fig. 9.

Figure 9: $S(\rho_{R})$ evaluated for no islands (brown, blue, dark green
ascending lines) with $R$ located in an expanding patch and a single large
island for $n=1,2,+\infty$ (orange, red, black curves, respectively, from
bottom to top) whose endpoints lie in adjacent $\mathrm{AdS}_{2}$ bubbles.
$\sigma_{R}$ is fixed to $\pi/2-10^{-5}$. Other parameter values are $c=600$,
$\phi_{r}=10$, $\phi_{0}=0$, $\epsilon_{\mathrm{uv}}=1$,
$\epsilon_{\mathrm{rg}}=1$.

###  3.3 Features of island formation

To briefly summarize the last two subsections, we applied the island formula
to compute the fine-grained entropy associated to a region $R$ that is
confined to either an expanding patch of $\mathrm{dS}_{2}^{n}$, or a flat
bubble embedded in $\mathrm{dS}_{2}^{n}$. In both cases, once $R$’s endpoints
are close enough to $\mathcal{I}^{+}$, once $R$ exceeds a certain size, and
provided that $R$’s patch is surrounded by crunching patches or
$\mathrm{AdS}_{2}$ bubbles on either side, an island forms outside of $R$’s
patch that covers most of the external universe. Exactly where this transition
happens depends mildly on $n$ (i.e., the size of the universe), but it
monotonically decreases toward a limiting value $\varphi^{\rm
Page}_{R}(n\rightarrow\infty)<\pi/2$. Moreover, changing position of $R$ in
time, we observe that an island may appear only for sufficiently large
$\sigma_{R}$. As $\sigma_{R}$ approaches $\mathcal{I}^{+}$, the islands’
endpoints move back in time toward a limiting location.

There is a clear interpretation for the island entropy, at least when
$\phi_{0}\gg\phi_{r}$ and $\phi_{0}\gg c$. Examining Eqs. (39) and (42) or
Eqs. (51) and (52), we see that $S_{\mathrm{island}}(R)$ is equal to
$2\phi_{0}$ up to $O(\phi_{r})$ or $O(c\log c)$ corrections depending on
whether $\phi_{r}\gg c$ or $\phi_{r}\ll c$, respectively. That is, in the
regime of parametrically large $\phi_{0}$, $S_{\mathrm{island}}(R)\approx
2\phi_{0}$ is the two-dimensional de Sitter horizon entropy. Therefore, the
Page transition in this cosmological setting occurs when $R$ grows so large
that its matter entropy would exceed the de Sitter entropy. If we view a
maximum entropy as a bound on Hilbert space dimension, it is tempting to
speculate that the appearance of islands is a semiclassical signal that the
dimension of the Hilbert space for putative fine-grained degrees of freedom
associated with $R$ is bounded by the de Sitter entropy. Moreover, this bound
persists regardless of how much spacetime lies outside of $R$’s patch.

In particular, one might have thought that the island entropy would depend on
the pattern of patches and bubbles outside of $R$’s patch, but this is not the
case. The value of $S_{\mathrm{CFT}}(R)$ depends only on the size of $R$ and
on $n$ and, to leading order, $S_{\mathrm{island}}(R)$ depends only on the
former. In other words, $S_{\mathrm{island}}(R)$ is the same irrespective of
the spacetime that lies beyond $R$’s patch.

The fact that $S_{\mathrm{island}}(R)$ depends only on the endpoints of $R$
has further consequences for when an island forms and gives the minimal
extremum of $S_{\mathrm{gen}}(R\cup I)$. As long as the endpoints of $R$ are
sufficiently close to the corners of expanding patches or flat bubbles, an
island will form with endpoints lying in the adjacent patches to those
containing $R$’s endpoints provided they are crunching patches or
$\mathrm{AdS}_{2}$ bubbles. Examples of such configurations are illustrated in
Figs. 18(a) and 18(b). Under these conditions, $(R\cup I)^{c}$ is locally
identical to the cases that we examined in the previous two subsections, and
so a Page transition occurs for sufficiently large $R$.

Much as crunching patches or $\mathrm{AdS}_{2}$ regions are necessary to form
islands, we also observed that islands do not form when $R$ is surrounded by
expanding patches, or by flat bubbles; the extremality conditions (33) cannot
be satisfied in expanding $\mathrm{dS}_{2}$ patches, or flat bubbles. One
might wonder, then, whether islands whose endpoints lie in the nearest
crunching or $\mathrm{AdS}_{2}$ regions can form and whether they can give a
lower generalized entropy than the absence of islands. Examples of such
configurations are illustrated in Figs. 16(a), 16(b), and 16(c). Though these
configurations lie outside of the OPE limit, we can compute their associated
entropies for a specific choice of CFT. Taking the CFT to be a theory of $c\gg
1$ free Dirac fermions, we find that islands do indeed form, in the sense that
extrema of $S_{\mathrm{gen}}(R\cup I)$ with nontrivial $I$ exist, but that the
configurations shown in Figs. 16(a) and 16(b) always result in an entropy that
is larger than $S_{\mathrm{CFT}}(R)$. Therefore, it appears that surrounding
$R$ with expanding patches can “screen” the rest of the universe from $R$ to a
certain extent. However, this is not a hard and fast rule since, for example,
the configuration shown in Fig. 16(c) still exhibits a Page transition for
sufficiently large $R$. Computational details and the associated Page curves
are elaborated in App. A.2.

##  4 False vacuum inflation in quantum cosmology

In the previous section, we have seen islands appear in the calculation of the
Von Neumann entropy associated with a spacelike interval $R$ confined to a
patch of the global spacetime in various two-dimensional toy-model multiverses
(provided $R$ is taken sufficiently large). The formation of an island
suggests that, if we had been working in the framework of semiclassical
quantum cosmology, an additional saddle point geometry of the gravitational
path integral would have come into play in the calculation of the Von Neumann
entropy, and perhaps also of “observables” with a sufficiently rich
information content.

In this section, we elaborate on this point by examining a similar, but more
conventional toy-model multiverse in four dimensions, one that is often
associated with the decay of an inflating false vacuum through bubble
nucleation. To be precise, we consider the Hartle-Hawking quantum state for
universes that contain a scalar field whose potential possesses false and true
vacua. We will compare the calculation of Von Neumann entropy in the two-
dimensional model to the calculation of probabilistic predictions for local
cosmological observables using the Hartle-Hawking state in this model, and we
will discuss how each model informs the other.

In the current setting, different gravitational saddles contribute to the
calculation of probabilities depending on the level of detail of the local
observation in question. This leads us to draw an analogy between the
appearance of new saddles here, when the observational question is made
sufficiently precise, and the appearance of a large island in the two-
dimensional model, when $R$ is sufficiently large. These saddles are saddle
point geometries of the Hartle-Hawking wavefunction, and in particular, they
involve an enormous coarse-graining over the external (with respect to the
local observation) fine-grained multiverse structure. This further resonates
with the fact that the formation of an island is insensitive to almost all of
the multiverse structure external to $R$.

Altogether, the comparisons drawn here are meant to exemplify how
semiclassical QC appears to incorporate the huge reduction of degrees of
freedom suggested by the islands program in cosmology, while retaining some
information in terms of a multiplicity of pasts. Our discussion in this
section closely follows part of Hartle:2016tpo  albeit with a somewhat
different emphasis.

###  4.1 Multiverse Model

We consider four-dimensional Einstein gravity coupled to a single scalar field
$\chi$ moving in a positive potential. We take the potential to have a false
vacuum $F$ with two quantum decay channels to two vacua $A$ and $B$ where the
potential vanishes. Fig. 10 gives an example. Classically, this theory has an
eternally inflating de Sitter solution with an effective cosmological constant
given by the value of the potential in the false vacuum. Quantum mechanically,
this solution decays through the nucleation of bubbles of true vacuum. The
geometry inside these bubbles is that of an open universe which expands in the
de Sitter background.

Figure 10: A potential with one false vacuum and two true vacua $A$ and $B$.
The false vacuum is assumed to be separated from both true vacua by a barrier
followed by a relatively flat patch where the slow roll conditions for
inflation hold. The different shapes of the two barriers and of the potential
in the two slow roll regimes leading on to the true vacua gives different
false vacuum decay rates and different predictions for CMB related observables
in universes ending up in either $A$ or $B$.

We allow for different decay rates of the false vacuum to $A$ and $B$. We
further assume that the potential toward the vacua has flat patches where the
slow roll conditions hold so that while the scalar slowly rolls down, the open
universes121212Whether the local geometry inside is open remains a matter of
debate Cespedes:2020xpn . inside the bubbles undergo a period of inflation
before the bubble universe reheats and standard cosmological evolution ensues.
Finally, we assume the potential is such that detailed CMB-related
observables, say the spectral tilt or the tensor to scalar ratio, enable
observers inside one of the bubbles to determine whether they live in $A$ or
$B$.

The quantum mechanical nucleation of bubbles of type $A$ or $B$ in the false
vacuum background is thought to give rise to a toy-model multiverse. These
bubbles are the analog of the expanding or flat bubbles we patched in, in the
two-dimensional toy-model multiverses in the preceding sections. The
“crunching,” or strong gravity patches in the two-dimensional models
correspond to the false vacuum background here. The CMB observables
discriminating between $A$ and $B$ are the analog of the different dilaton
behaviors in the patches containing $R$ in the two-dimensional models.

Figure 11: A conformal representation that evokes a fine-grained configuration
of (possibly infinitely many) bubble universes in a false vacuum de Sitter
background. The false vacuum is indicated in blue, regions inside bubbles of
type A are in yellow, and regions inside bubbles of type B are in green. A
quantum state of the universe $\Psi$ does not describe one specific such
configuration, but an ensemble of possible ones.

A particular eternally inflating history consisting of a specific
configuration of bubble universes in a false vacuum de Sitter background is
illustrated in Fig. 11. In an essentially classical approach to eternal
inflation, such a particular fine-grained global configuration is taken as a
starting point for the calculation of predictions for local observations. In
the absence of a quantum state, these predictions are derived by counting the
number of Hubble volumes (or bubbles) in a global configuration where
observables take different values. This requires an extraneous notion of
typicality in sync with a prescription for regulating infinities because a
fine-grained history typically follows an infinite number of bubbles, each of
which is itself infinite. This prescription, which specifies a measure,
supplements the theory. It consists, e.g., of specifying a spacelike three-
surface beyond which one no longer counts instances of observations. It is
well known, and hardly surprising, that the resulting predictions are highly
regulator-dependent. This is known as the measure problem of eternal
inflation. It is essentially a problem of information; the theory is
underdetermined and that gives rise to a breakdown of predictivity.

In the previous section, we have taken the analog of such a particular fine-
grained configuration as a starting point for a semiclassical calculation of
the Von Neumann entropy of subregions. It is tempting to interpret the
formation of an island in that context as a signature that the idea of a
normal definite global spacetime may be questionable. Here we take the
complementary viewpoint and follow up on this reasoning with a conventional
semiclassical quantum cosmology treatment of this false vacuum model. In the
next Section, we substantiate the resonances between both analyses.

###  4.2 Local predictions from coarse-grained saddle point geometries

We consider the Hartle-Hawking no-boundary wavefunction (NBWF) in the model
above on a closed spacelike three-surface $\Sigma$. Schematically, we have

$\Psi_{HH}=\Psi[h(\vec{x}),\chi(\vec{x}),\zeta(\vec{x})],$

where $\zeta$ represents linear scalar perturbations around the background
saddles, $h(\vec{x})$ is the induced metric on $\Sigma$ and $\chi$ is the
scalar field.

In the semiclassical approximation, the NBWF is given by a sum of saddle
points, each contributing a term of the form Hartle:1983ai

$\Psi[h,\chi,\zeta]\sim\exp(-I/\hbar)=\exp\\{(-I_{R}[h,\chi,\zeta]+iS[h,\chi,\zeta])/\hbar\\}.$
(54)

Here, $I_{R}[h,\chi,\zeta]$ and $-S[h,\chi,\zeta]$ are the real and imaginary
parts of the Euclidean action $I$, evaluated on a saddle point solution of the
field equations that matches $(h,\chi,\zeta)$ on its only boundary $\Sigma$
and is otherwise regular. In regions of superspace $\\{h,\chi,\zeta\\}$ where
$S$ varies sufficiently rapidly, the semiclassical wavefunction (54) describes
a family of locally classical Lorentzian cosmologies that are the integral
curves of $S$ and have amplitudes to leading order in $\hbar$ that are
proportional to $\exp[-I_{R}(h,\chi,\zeta)]/\hbar]$, which is constant along
the integral curve Hartle:2008ng .

In the model we consider, the wavefunction comprises two distinct sets of
cosmological backgrounds. First, there is a one-parameter family of saddle
points, labeled by the absolute value $\chi_{0}$ of the scalar field at their
“South Pole”, with $\chi_{0}$ somewhere on the slow roll slope near one of the
true vacua $A$ or $B$. Each of these saddles corresponds to a closed
inflationary Friedmann-Lemaître-Robertson-Walker (FLRW) background, without
eternal inflation, ending up in either $A$ or $B$ Hartle:2008ng . Second,
there are two isolated saddle geometries describing the expansion of a bubble,
either of type $A$ or $B$, embedded in the false vacuum background at
$\chi=0$. These saddles are no-boundary versions of the well-known Coleman-De
Luccia (CDL) or Hawking-Moss (HM) instantons (which one dominates depends on
the shape of the barrier). As regular compact solutions of the Euclidean field
equations, these instantons are valid saddle points of the NBWF. In this
interpretation, CDL instantons are associated with histories following a
single bubble evolving toward a true vacuum that expands in a false vacuum
background. The nucleation point of the bubble lies at the throat of the de
Sitter background. Indeed the saddle describes the creation of both bubble and
background. Crucially, the saddle does not keep track of other bubbles which
may or may not be nucleating at various other locations in the false vacuum
background. Instead it averages over everything happening outside one bubble
Hartle:2016tpo . This is illustrated in Fig. 12.

In the language of decoherent histories quantum mechanics, one says that CDL
instantons, as NBWF saddle points, correspond to coarse-grained histories
Hartle:1992as . Quantum mechanical coarse-graining amounts to some sort of
averaging whereby one bundles together detailed histories in coarser grained
sets retaining less information. Taking the semiclassical wavefunction at face
value, therefore, we see that this includes a huge amount of coarse-graining
over possible multi-bubble configurations. Nevertheless, the semiclassical
theory repackages some information contained in that putative fine-grained
structure in terms of a limited set of distinct coarse-grained saddle
geometries. While these saddles are consistent with there being no independent
degrees of freedom in those far-flung regions at all, they do not necessarily
imply this, although the dynamics of eternal inflation does suggest that at
the very least, the wavefunction is very much spread out over a wide range of
configurations on the largest scales.

Figure 12: The semiclassical prediction of local observations in the false
vacuum eternal inflation model of Fig 10 involves two distinct saddle point
geometries. Each of these corresponds to a coarse-grained history describing a
different possible past of a given “observer” $D$, in which she evolves either
toward the true vacuum $A$ or $B$. The saddle geometries average over any
multi-bubble configuration outside—represented by gray—but enter as a
superposition of pasts of $D$.

Decoherent histories quantum cosmology speaks of different descriptions of
quantum systems at different levels of coarse-graining. Which level is the
appropriate one depends on the correlation of interest Hartle:1992as . The
crux of previous semiclassical quantum cosmology calculations in models of
this kind Hartle:2016tpo ; Hartle:2010dq ; Hertog:2013mra  is that the coarse-
graining inherent in the semiclassical description is the relevant and
appropriate one for the calculation of probabilities for local observations.

To adopt an intrinsic perspective of a local observer, it is convenient to
specify an observational situation in terms of data $D$ treated as part of the
quantum system. Consider thus the correlation $p({\cal F}|D^{\geq 1})$ between
a feature ${\cal F}$ of the local universe, e.g., a statistical feature of the
CMB that we seek to predict, and at least one instance of a set of local data
$D$ (denoted by $D^{\geq 1}$), which we take also to select a surface of
homogeneity of a given density. One can think of $D$ as a specific local
configuration of matter fields and geometry inside one Hubble volume that has
ultimately evolved from the primordial perturbations $\zeta$ and whose
probability to occur in any Hubble volume can therefore in principle be
computed from the Gaussian fluctuation wavefunctions. All one knows from local
observations is that there is at least one instance of $D$. For any kind of
realistic data, this probability is of course exceedingly small.

The condition on $D^{\geq 1}$ in the correlation $p({\cal F}|D^{\geq 1})$
suppresses the first class of no-boundary saddle points labeled by $\chi_{0}$,
i.e. those not associated with the false vacuum and the nucleation of a bubble
of kind $A$ or $B$. The reason is that the surfaces of homogeneity in the
former class are relatively small, rendering $p(D^{\geq 1})\ll 1$. On the
other hand, bubbles have extremely large or even infinite surfaces of
homogeneity, rendering the condition on $D$ trivial, i.e. $p(D^{\geq
1})\approx 1$ Hartle:2016tpo . This selection of bubble saddles by taking $D$
sufficiently precise is the analog of taking $R$ sufficiently large in the
island calculations above. Adding more bubbles does not change anything, for
the condition on $D^{\geq 1}$ obviously remains moot. Assuming the probability
of bubble collisions is negligible, a coarse-graining that follows what
happens inside one bubble and ignores what goes on outside thus appears to be
adequate for evaluating the quantum mechanical probabilities $p({\cal
F}|D^{\geq 1})$. If the feature ${\cal F}$ distinguishes between a bubble of
type $A$ or $B$, then the relevant coarse-grained set of histories consists of
the two remaining saddle point geometries, each following only one bubble
(ours) but distinguished by whether this is of type $A$ or $B$. The picture
one might have in mind here is one that is familiar from holography—that these
two saddle geometries corresponds to two distinct ways of filling in the past
(say, the bulk) leading to a given set of data $D$ on a homogeneous boundary
surface of given density and subject to a no-boundary condition deep in the
interior.

Evidently the relative probabilities for the outcome ${\cal F}_{A}$ or ${\cal
F}_{B}$ will be specified by the action $I$ of the dominant saddle mediating
the decay of the false vacuum toward resp. $A$ or $B$. Thus for the relative
probabilities that we, systems characterized by data $D$, observe the physical
properties of bubble $A$ or $B$, we get

$\frac{p({\cal F}_{A}|D^{\geq 1})}{p({\cal F}_{B}|D^{\geq
1})}=e^{2I_{R}^{B}-2I_{R}^{A}}$ (55)

where $I_{R}^{A}$ and $I_{R}^{B}$ are the real parts of the Euclidean no-
boundary actions of the (CDL or HM) saddle point geometries. This includes a
weighting of the false vacuum background because no-boundary saddles describe
the nucleation of both a bubble and the background, in contrast with the use
of CDL or HM instantons in tunneling transitions where one assumes a pre-
existing vacuum state.

For a broad barrier where the Hawking-Moss saddle gives the dominant decay
channel, the no-boundary weighting in Eq. (55) is given by

$-I_{R}=24\pi^{2}\left(\frac{1}{V(\chi_{max})}-\frac{1}{V(0)}\right)+\frac{24\pi^{2}}{V(0)}\
,$ (56)

where $\chi_{max}$ denotes the value of $\chi$ at the maximum of $V$ and we
have added and subtracted the false vacuum weighting. The terms inside the
brackets are simply equal to the difference in entropy of both de Sitter
backgrounds and combine to form the bubble nucleation rate, and the third term
represents the no-boundary weighting of the false vacuum background.

For a narrow barrier, there is a CDL saddle that straddles the maximum. CDL
instantons are slightly more complicated saddle points in which the field
$\chi$ varies from an initial value $\chi_{i}$ near the false vacuum to a
final value $\chi_{f}$ on the other side of the barrier. In the limit
$V_{,\chi\chi}/H^{2}(\chi_{max})\rightarrow-4$, the CDL solution tends to HM
and so does its action. By contrast, in the limit where the barrier is narrow
and sharp and hence $|\chi_{f}-\chi_{0}|\ll 1$, the decay rate implied by the
CDL action tends to the well-known thin-wall result, giving for the no-
boundary weighting in Eq. (55),

$-I_{R}=\left(\frac{27\pi^{2}T^{4}}{2V(0)^{3}}-\frac{24\pi^{2}}{V(0)}\right)+\frac{24\pi^{2}}{V(0)}\
,$ (57)

where $T$ is the tension of the narrow barrier separating $F$ from the true
vacuum,

$T=\int_{\chi_{i}}^{\chi_{f}}d\chi\sqrt{2V(\chi)}.$ (58)

The third term in Eq. (56) and Eq. (57) is just the de Sitter entropy of the
false vacuum. Thus the no-boundary weighting of the saddles has formally a
very similar structure to the island entropy we obtained above, given by the
dS entropy plus a small correction. In relative probabilities, the weighting
of the background cancels, of course, so we end up in Eq. (55) with a
difference between two relatively small corrections to the entropy of the
false vacuum.

To summarize, correlations of the kind $p({\cal F}|D^{\geq 1})$ that capture
predictions for local cosmological observations in the Hartle-Hawking state in
false vacuum models are specified in terms of a superposition of saddle point
geometries, each representing a highly coarse-grained configuration.

###  4.3 Comparative summary of the models

In Sec. 2, we built toy model multiverses out of solutions of two-dimensional
JT gravity theories, and we used the islands prescription to compute the Von
Neumann entropies associated to regions $R$. In this section, we examined a
more familiar four-dimensional multiverse model from quantum cosmology, and we
explained how to compute probabilities for cosmological observations using
saddles of the Hartle-Hawking wavefunction. We end this section with a short
recap of the similarities, differences, and resonances between the two models
and the two calculations.

The two models are similar in the sense that they aim to model the physics of
an inflationary multiverse with multiple vacua. Roughly speaking, bubbles of
the terminal $A$ and $B$ vacua in the QC model correspond to the weakly-
gravitating patches of a JT multiverse where we situated $R$, and the false
vacuum $F$ corresponds to strongly-gravitating regions.

When following the islands prescription, the appearance of a nontrivial island
generally signals the appearance of a new saddle in an underlying Euclidean
gravitational path integral. While we currently lack a Euclidean description
of the JT multiverses, the selection of particular gravitational saddles is
clearly exhibited in the calculation of probabilities in the QC model.
Furthermore, this calculation is a true quantum gravitational calculation,
albeit in the semiclassical approximation, in the sense that it explicitly
tracks the state of the gravitational sector. The QC calculation is of course
not a calculation of Von Neumann entropy. Nevertheless, the selection of
saddles in the QC model and the formation of islands in the JT multiverse
model are analogous in that both are a consequence of having asked a
sufficiently detailed question: Islands form when calculating the fine-grained
entropy of regions that nearly span an entire cosmological patch in a JT
multiverse, and sufficiently detailed observational conditions select saddles
associated with false vacuum decay in the Hartle-Hawking wavefunction.

Finally, both models suggest a huge redundancy in the global picture of an
eternally inflating spacetime. The saddles in the QC model explicitly coarse-
grain over scales larger than those of the observational condition in
question. In the JT multiverse models, an island always forms for sufficiently
large $R$, independent of the global structure beyond $R$ and despite the fact
that $R$ is locally the same for any such structure. Furthermore, the fine-
grained degrees of freedom of $R$ are supposed to encode the island itself, at
least at a semiclassical level. (We will further comment on this point in the
Discussion.) Given that the geometry of the JT multiverses reflect a more
traditional view of the spacetime produced by eternal inflation, it is
tempting to speculate that the formation of these islands is a semiclassical
hint that the global view of an operationally well-defined, eternally
inflating spacetime eventually breaks down, as is manifest in the QC model. In
particular, the fragmentation of the global description into a number of
disconnected, separate saddle points for each type of bubble in the QC model
suggests that something similar will happen in a quantum gravitational
analysis of the two-dimensional model, at least when the latter is considered
in (perhaps some appropriate generalization of) the Hartle-Hawking state.

##  5 Discussion

We have considered toy multiverse models inspired by false vacuum eternal
inflation. We found that in the semiclassical calculation of the Von Neumann
entropy associated with a sufficiently large spacelike interval $R$ in two-
dimensional models, an island $I$ develops covering most of the rest of the
multiverse. This further substantiates the quantum cosmology treatment of
models of this kind in which predictions for local cosmological observables
are specified by saddle points that discriminate between different pasts of
$R$ but otherwise coarse-grain, or “average,” over any large-scale multiverse
structure outside one bubble.

The two-dimensional multiverse geometries we considered contain bubbles of
zero or negative curvature within an analytic extension of $\mathrm{dS}_{2}$
that is a solution of the de Sitter version of JT gravity. Within these
geometries, we used the behavior of the dilaton to label regions of weak
gravity and of strong gravity. Coupling a CFT to these geometries as a matter
model, we then calculated the generalized entropy $S_{\mathrm{gen}}(R\cup I)$
associated to a spacelike interval $R$ and additional putative islands $I$.
This let us implement the islands program to compute the Von Neumann entropy
$S(\rho_{R})$: given a spacelike interval $R$ and the generalized entropy for
different island configurations $S_{\mathrm{gen}}(R\cup I)$, we looked for
islands $I$ that extremize $S_{\mathrm{gen}}(R\cup I)$, and then we took the
minimum of these extrema. For sufficiently large subregions $R$ in regions of
weak gravity, an island forms with endpoints in surrounding regions of strong
gravity. The island covers most of the multiverse to the exterior of $R$,
suggesting that the global spacetime does not capture fundamental degrees of
freedom independent from those in $R$.

In our analysis of false vacuum eternal inflation, we considered observables
localized within bubbles that have exited from eternal inflation where gravity
can be said to be relatively weak. The averaging entering in the semiclassical
QC setting amounts to a coarse-graining over the regime of eternal inflation
surrounding the observer’s patch. Stretching the analogy with our two-
dimensional toy models as far as we can, we view the “crunching” patches, the
proxies for a strong gravity regime, as the toy-model analog of the regime of
eternal inflation. In establishing this connection, our analysis has
implicitly assumed the validity of the islands program in a setting where $R$
and $I$ are part of the same spacetime.131313When $R$ and $I$ are (parts of)
disjoint spacetimes associated to a manifestly bipartite Hilbert space, such
as in the study of entanglement between disjoint closed universes
Balasubramanian:2020coy ; Balasubramanian:2020xqf ; Balasubramanian:2021wgd ;
Fallows:2021sge , the development of entanglement islands can be thought of as
a consequence of monogamy of entanglement. In such situations, one considers
entangling a collection of non-gravitating degrees of freedom, $A$, with a
disjoint collection of gravitating degrees of freedom, $B$. The latter are
entangled with degrees of freedom in the gravitational sector due to
gravitational interactions. A consequence of monogamy of entanglement is that
the structure of entanglement between $B$ and the gravitational sector cannot
be perfectly preserved as one increases the entanglement between $A$ and $B$.
Therefore, the Von Neumann entropy of $A$ eventually becomes sensitive to
geometric effects in $B$, which is manifested by the formation of entanglement
islands. Regardless, $R$ is always entangled with its complement, and taking
$R$ larger thus increases the amount of this entanglement. It is thus
plausible that the fine-grained Von Neumann entropy $S(\rho_{R})$ becomes
sensitive to geometric effects in the form of entanglement islands, and that
is what we found.

We now comment on some loose ends and open questions in our two-dimensional
multiverse models. In addition to the configurations that we considered,
another possibility would have been to locate the subregion $R$ within one
static patch of our JT multiverse models, in the spirit of Ref.
Sybesma:2020fxg . In that work, given a region $R$ in a static patch of dS3,
an island develops in the opposite wedge, which leads to a Page transition.
Yet another avenue would be to consider island configurations which are
timelike separated from the region $R$, as explored in Ref. Chen:2020tes  for
the simple case of a ${\rm dS}_{2}$ solution of JT gravity with positive
cosmological constant. Similarly to the analysis of Ref.
Balasubramanian:2020xqf , it should also be possible to entangle our
multiverse configurations with a disjoint non-gravitating system, with the
auxiliary system playing the role of $R$.

While islands and saddles are manifestly nonpertubative objects, an
interesting question to ask is whether the perturbative dependence of $I$ on
$R$ could be understood using the theory of bulk operator reconstruction. In
the AdS/CFT correspondence, bulk reconstruction gives a prescription for how
to represent bulk operators that lie in the entanglement wedge of a given
boundary subregion as CFT operators supported on that subregion
Almheiri:2014lwa ; Dong:2016eik ; Cotler:2017erl ; Chen:2019gbt . In the case
of an AdS black hole coupled to an external reservoir in which an island
develops in the black hole interior, bulk operators supported on the island
are then represented through this prescription as operators in the reservoir
Penington:2019kki ; Almheiri:2019qdq . In both cases, the bulk operator is
“represented” in the sense that both it and its reconstruction’s expectation
values agree on a restricted set of perturbatively close states known as the
code subspace. In the present cosmological setting, it would be interesting to
investigate whether operators supported on $I$ can be represented as operators
supported on $R$ for an appropriate code subspace and given a mapping between
effective degrees of freedom on $R\cup I$ and fine-grained degrees of freedom
on $R$.

A feature of the two-dimensional models that we considered is that they offer
precise quantum control over the CFT matter model and its contribution to
generalized entropy. We did not focus on quantum aspects of the spacetime in
these models; nevertheless, a Euclidean construction of JT multiverses and an
analysis based on the gravitational path integral are interesting avenues for
future inquiry. These are not straightforward tasks, however, as there are
subtleties involved in defining a quantum state for the gravitational sector.
Suppose, for example, that we wished to define a Hartle-Hawking-like state for
dS${}_{2}^{n}$ (see Fig. 2) by continuing the manifold into the Euclidean past
at the $\sigma=0$ slice. The resulting manifold possesses a conical excess,
and so it cannot be a solution of the JT theory, which has $R=+2$ everywhere.
In principle, one would therefore have to modify the theory at the level of
its action so that it could support a conical excess.141414Similarly, as
discussed in Sec. 2, additional degrees of freedom are required to support the
discontinuities in the dilaton’s first derivative in the bubble spacetimes.
That said, it appears that a sensible gravitational path integral can still be
defined, at least for pure dS${}_{2}^{n}$. Ref. Cotler:2019nbi  constructs a
gravitational path integral that prepares a state at $\mathcal{I}^{+}$ via a
double analytic continuation in time and of the Hubble length. The problem is
then mapped onto a path integral for Euclidean AdS2, which may be a possible
starting point for a Euclidean calculation of Von Neumann entropies in JT
multiverses.

In the language of semiclassical gravitational path integrals, the formation
of entanglement islands signals that one or more new saddles comes into play.
While we did not pursue a path integral analysis of our two-dimensional toy
multiverses in de Sitter JT gravity, we pointed out that this observation is
very much in line with existing results in quantum cosmology in models of this
kind in four dimensions. As an illustration, we gave a brief discussion of the
probabilistic predictions for local cosmological observables such as, say, the
CMB temperature anisotropies, in a false vacuum model of eternal inflation
with two distinct decay channels and in the Hartle-Hawking state. In these and
other models of inflation, probabilities for observations in quantum cosmology
typically involve a superposition of saddle point geometries that include an
averaging over any multiverse structure on the largest scales. This built-in
coarse-graining was one of the key elements behind the semiclassical
resolution of the measure problem.

The upshot of the quantum cosmology analysis of these models appears to be
that the global spacetime breaks up into a sum of a small number of distinct
saddle point geometries, each of which involving a huge coarse-graining over
much, if not all, of the bubble exterior. The fundamentally classical picture
of a global spacetime is thus basically replaced in semiclassical quantum
cosmology by a multiplicity of a few past histories of $R$, combined with much
“uncertainty” on super-bubble scales. That is, contrary to appearances, the
saddle geometries would not specify a global classical state, but rather
delineate the limitations of classical spacetime, a point much emphasized by
Hartle et al.Hartle:2010dq ; Hartle:2016tpo . This may not be entirely
inconsistent with the ideas behind bulk reconstruction that one can represent
operators on $I$ as operators on $R$. Imagine one were interested in
constructing, or better still, measuring some heavy operator in $R$ that
contains a significant amount of information about a distant patch. One
expects that such extraordinary and complex measurements would result in a
backreaction on the spacetime in $R$ to the extent that the measurement
amounts to selecting the saddle point corresponding to the patch in question.

The picture that emerges from the confluence of these analyses is one in
which, fundamentally, a definite spacetime geometry in cosmology comes about
in an “inside out” way. In a sense, the entire multiverse would be reduced to
an oasis consisting of a patch of classical spacetime around us surrounded on
all sides by quantum fuzziness. That was the essence of the “top-down”
approach to (quantum) cosmology advocated by Hawking Hartle:2010dq ;
Hawking:2006ur .

Acknowledgments

We thank Ning Bao, Arjun Kar, Jason Pollack, Jacopo Sisti, James Sully, and
Mark Van Raamsdonk for helpful discussions during the preparation of this
manuscript, as well as Kristan Jensen, Edgar Shaghoulian, and Edward Witten
for useful comments. We also thank the anonymous referee for their
contributions to the review and publication process. S.E.A.G., T.H. and B.R.
are supported in part by the KU Leuven research grant C16/16/005. S.E.A.G.
also thanks Uppsala University for its hospitality during part of this work.
A.C.D. was supported for a portion of this work as a postdoctoral fellow
(Fundamental Research) of the National Research Foundation – Flanders (FWO),
Belgium. A.C.D. acknowledges the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC), [funding reference number
PDF-545750-2020]. / La contribution d’A.C.D. à cette recherche a été financée
en partie par le Conseil de recherches en sciences naturelles et en génie du
Canada (CRSNG), [numéro de référence PDF-545750-2020]. N.P.F. is supported by
the European Commission through the Marie Skłodowska-Curie Action UniCHydro
(grant agreement ID: 886540).

##  Appendix A Exact island entropies in a theory of free fermions

The different multiverse configurations in Sec. 3 involve a region $(R\cup
I)^{c}$ which consists of two disjoint intervals. When the size of these
intervals is small compared to their separation, the two-interval entropy is
approximately given by the sum of the single interval components. This regime
is sometimes called the OPE limit.

The single interval entropy is a universal quantity, i.e. valid for any CFT
with a given central charge, up to a scheme-dependent constant. However, the
Von Neumann entropy of the reduced state of a CFT on disconnected intervals is
in general not universal. In this appendix, we compute exact multi-interval
entropies for free massless Dirac fermions, and we compare the exact result to
the OPE limit approximation in order to check the latter’s accuracy. We also
use the exact result to investigate island configurations in JT multiverses
for which the OPE limit is not valid. Finally, we compute
$S_{\mathrm{gen}}(R\cup I)$ directly for an island that consists of two
disjoint intervals in order to give evidence that an island consisting of a
single large interval is the extremum that gives the smallest value of
$S_{\mathrm{gen}}(R\cup I)$.

###  A.1 Multi-interval entanglement entropy

Consider $p$ disjoint intervals in $\mathbb{R}^{2}$ whose endpoints we label
by $(u_{i},v_{i})$, with $i=1,\dots,p$. The corresponding Von Neumann entropy
for a two-dimensional Euclidean CFT consisting of $c$ free massless Dirac
fermions was computed in Ref. Casini:2005rm  and is given by

$S_{\rm
CFT}^{(p)}=\frac{c}{3}\left(\sum_{i,j}\log|u_{i}-v_{j}|-\sum_{i<j}\log|u_{i}-u_{j}|-\sum_{i<j}\log|v_{i}-v_{j}|-p\,\log\epsilon_{\rm
uv}\right),$ (59)

where $\epsilon_{\rm uv}$ is a small UV regulator.

Let us first consider the case where $p=2$, and suppose that we place the
theory on a manifold with the line element
$ds^{2}=dzd\bar{z}/\Omega(z,\bar{z})^{2}$. Labelling the two intervals’
endpoints by $(\mathbf{z}_{1},\mathbf{z}_{2})$ and
$(\mathbf{z}_{3},\mathbf{z}_{4})$, where
$\mathbf{z}_{i}\equiv(z_{i},\bar{z}_{i})$, the two-interval entropy is

$S_{\rm CFT}^{(2)}=\frac{c}{6}{\rm
log}\bigg{[}\frac{|z_{12}|^{2}|z_{23}|^{2}|z_{34}|^{2}|z_{14}|^{2}}{\epsilon_{\rm
uv}^{4}|z_{13}|^{2}|z_{24}|^{2}\Omega_{1}\Omega_{2}\Omega_{3}\Omega_{4}}\bigg{]}\,,$
(60)

where $z_{ij}:=z_{i}-z_{j}$ and $\Omega_{i}=\Omega(z_{i},\bar{z}_{i})$. For
simplicity, in writing Eq. (60), we have dropped a scheme-dependent constant.

To calculate $S_{\rm CFT}^{(2)}$ using the replica trick is equivalent to
inserting twist operators at the endpoints of the intervals Calabrese:2009qy ,
meaning that with four endpoints, the computation reduces to evaluating a
four-point function. In the limit where the cross ratios

$z_{13}z_{24}/(z_{23}z_{14})\rightarrow
1\qquad\text{and}\qquad\bar{z}_{13}\bar{z}_{24}/(\bar{z}_{23}\bar{z}_{14})\rightarrow
1\,,$ (61)

the two contributions to $S_{\rm CFT}^{(2)}$ decouple, and the two-interval
entropy written in Eq. (60) reduces to the sum of the single interval
entropies of the form in Eq. (28). The limit Eq. (61) thus corresponds to
evaluating this four-point function in the OPE limit.

For $p=3$, the three-interval entropy is given by

$S_{\rm CFT}^{(3)}=\frac{c}{6}{\rm
log}\bigg{[}\frac{|z_{12}|^{2}|z_{14}|^{2}|z_{16}|^{2}|z_{23}|^{2}|z_{34}|^{2}|z_{36}|^{2}|z_{25}|^{2}|z_{45}|^{2}|z_{56}|^{2}}{\epsilon_{\rm
uv}^{6}|z_{13}|^{2}|z_{24}|^{2}|z_{15}|^{2}|z_{35}|^{2}|z_{26}|^{2}|z_{46}|^{2}\Omega_{1}\Omega_{2}\Omega_{3}\Omega_{4}\Omega_{5}\Omega_{6}}\bigg{]},$
(62)

where the three intervals’ endpoints are $(\mathbf{z}_{1},\mathbf{z}_{2})$,
$(\mathbf{z}_{3},\mathbf{z}_{4})$, and $(\mathbf{z}_{5},\mathbf{z}_{6})$. We
will utilize Eqs. (60) and (62) in the rest of this appendix.

###  A.2 Single component island

We start by focusing on configurations with a single island $I$ extending
throughout the multiverse to be able to examine several cases of interest.
First, we reproduce our results in the main text for dS${}^{n}_{2}$ and for
the case of flat bubbles in dS${}^{n}_{2}$ with the exact free fermion
entropy. Subsequently, with the aim of testing how close the island must be to
the radiation region in order to produce a Page transition, we consider
additional JT multiverse configurations where bubbles are inserted
asymmetrically and where the crunching regions are not adjacent to the
expanding patch. In all the following cases we set
$\varphi_{1}=-\varphi_{4}=\varphi_{I}$, $\sigma_{1}=\sigma_{4}=\sigma_{I}$,
$\varphi_{2}=-\varphi_{3}=\varphi_{R}$, and
$\sigma_{2}=\sigma_{3}=\sigma_{R}$.

For the models studied in this work, namely pure dS${}_{2}^{n}$ and
dS${}_{2}^{n}$ with bubbles, the two-interval entropy of free fermions can be
expressed in global coordinates via Eq. (26) as

$\displaystyle S_{\rm CFT}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{c}{6}{\rm
log}\bigg{[}4n^{4}\left(\cos\left(\frac{\sigma_{12}}{n}\right)-\cos\left(\frac{\varphi_{12}}{n}\right)\right)\left(\cos\left(\frac{\sigma_{23}}{n}\right)-\cos\left(\frac{\varphi_{23}}{n}\right)\right)\times$
$\displaystyle\qquad\qquad\times\left(\cos\left(\frac{\sigma_{14}}{n}\right)-\cos\left(\frac{\varphi_{14}}{n}\right)\right)\left(\cos\left(\frac{\sigma_{34}}{n}\right)-\cos\left(\frac{\varphi_{34}}{n}\right)\right)\bigg{]}$
$\displaystyle-\frac{c}{6}{\rm
log}\bigg{[}\left(\cos\left(\frac{\sigma_{31}}{n}\right)-\cos\left(\frac{\varphi_{31}}{n}\right)\right)\left(\cos\left(\frac{\sigma_{42}}{n}\right)-\cos\left(\frac{\varphi_{42}}{n}\right)\right)\epsilon_{\rm
uv}^{4}\prod_{i=1}^{4}{\omega_{i}}\bigg{]},$

with $\sigma_{ij}=\sigma_{i}-\sigma_{j}$,
$\varphi_{ij}=\varphi_{i}-\varphi_{j}$ and
$\omega_{i}=\omega(\sigma_{i},\varphi_{i})$. We will use this expression
shortly to evaluate the generalized entropy of interest.

####  A.2.1 Extended dS2

Consider the symmetric configuration $R\cup I$ shown in Fig. 2. The conformal
factors at the endpoints for this disjoint interval are given as
$\omega_{1}=\omega_{4}=\cos{\sigma_{I}}$ and
$\omega_{2}=\omega_{3}=\cos{\sigma_{R}}$. Plugging these factors into Eq.
(A.2), we compute the corresponding generalized entropy Eq. (22) and extremize
with respect to $\varphi_{I}$ and $\sigma_{I}$. We find numerically the
entropy plots shown in Fig. 13. Notice that, as guaranteed by subadditivity of
Von Neumann entropy, the OPE approximation that we used in the main text
provides an upper bound to the exact generalized entropy with a non-trivial
island. While the qualitative behavior of the Page curve is unchanged, the
precise value of $\varphi_{R}$ at which the Page transition occurs depends on
whether we consider the free fermion model or the OPE limit of the twist
operators. Nevertheless, as explained in the introduction of this Appendix,
both entropies agree for large enough $\varphi_{R}$.

Figure 13: $S_{\text{gen}}(R\cup I)$ as derived using an exact two-interval
formula for the generalized entropy in the context of free fermions (in red),
the OPE limit (black), and the corresponding no-island entropy (blue) for a
choice of parameters $n=10$, $\sigma_{R}=\frac{\pi}{2}-10^{-5}$, $c=100$,
$\phi_{r}=10$, $\phi_{0}=0$, and $\epsilon_{\rm uv}=1$, $\epsilon_{\rm rg}=1$.
The results coincide only for large spatial extension of the region $R$. The
comparison shows that the matter entropy evaluated in the OPE limit gives a
good approximation when the separation between the disjoint intervals is much
greater than their proper lengths, which might occur before or after the Page
transition depending on the parameters of the theory.

####  A.2.2 Extended dS2 with flat bubbles

Next, we compare the exact generalized entropy for a theory of free fermions
to the results in Sec. 3.2 for the configuration illustrated in Fig. 7. The
conformal factors at the endpoints of $R\cup I$ are
$\omega_{1}=\omega_{4}=\frac{1}{2}\left(\cos{\sigma_{R}}+\cos{\varphi_{R}}\right)$
and $\omega_{2}=\omega_{3}=\cos{\sigma_{I}}$. The plot showing this comparison
is displayed in Fig. 14. Again, for sufficiently large $R$, we find good
agreement between the exact and the approximated generalized entropy.

Figure 14: Comparison between generalized entropy with non-trivial island for
a free fermion theory (red), the OPE limit of twist operators (black), and the
no-island entropy (blue), illustrated when the endpoints of R satisfy the
relation
$\sigma_{R}=-\frac{2}{\pi}\left(\frac{\pi}{2}-10^{-3}\right)\varphi_{R}+\pi-10^{-3}$.
The parameters of the theory are chosen as $n=10$, $c=600$, $\phi_{r}=10$,
$\phi_{0}=0$, and $\epsilon_{\rm uv}=1$, $\epsilon_{\rm rg}=1$. The free
fermion and the OPE generalized entropies coincide once $R$ is large enough.

####  A.2.3 Alternative island configurations

Here, let us consider an alternative set of island configurations as displayed
in Figs. 16 and 18. We are interested in investigating under what
circumstances islands arise when the endpoints of $I$ lie in crunching regions
that surround $R$.151515There have been settings in which islands appear in
expanding patches, e.g. Aalsma:2021bit , albeit with a different assumption
imposed on the CFT state.

In particular, in Fig. 16(a)-16(c), we consider configurations where the
region $R$ is confined to one flat or expanding patch, which has at most one
adjacent crunching patch. In Figs. 16(a)-16(b), we observe that islands may
form, but they are somewhat screened by the intermediate additional flat and
expanding bubbles, in the sense that they never have lower generalized entropy
than the configuration with no island. In contrast, we find that the
configuration depicted in Fig. 16(c) exhibits a Page transition, albeit for a
very large value of $\varphi_{R}$, when R is close enough of
$\mathcal{I}^{+}$.

On the other hand, if we extend $R$ so that it spans multiple patches and so
that its endpoints lie very close to crunching patches, we see that a Page
transition occurs, as illustrated in Figs. 18(a)-18(b). These numerics are
fully consistent with the analytic arguments from Sec. 3.3. Intuitively, the
entropy of $R\cup I$ is once again almost that of a pure state in these
configurations.

(a)

(b)

(c)

Figure 16: (a)-(c) Penrose diagrams and the corresponding generalized entropy
for different $R\cup I$ configurations. The red curves indicate the
generalized entropy including islands, while the blue ones indicate the no-
island entropy. The endpoints of the non-trivial islands are inside crunching
patches and they do not have to be located adjacent to the patch where $R$
resides. We observe, however, that the generalized entropy for the case (a)
and (b) is minimal for the no-island configuration. The endpoints of $R$ are
held at
$\sigma_{R}=-\frac{2}{\pi}\left(\frac{\pi}{2}-10^{-3}\right)\varphi_{R}+\pi-10^{-3}$
for configurations (a)-(b), and $\sigma_{R}=\frac{\pi}{2}-10^{-5}$ for (c).
Other parameters used in the plots are $n=10$, $c=600$, $\phi_{r}=10$,
$\phi_{0}=0$, $\epsilon_{\rm rg}=1$, and $\epsilon_{\rm uv}=1$.

(a)

(b)

Figure 18: (a)-(b) Penrose diagrams and the corresponding Page curves, in
which we relocate region $R$ to be as close to $I$ as possible to produce
higher purification of the quantum state with respect to Figs. 16(a)-16(b).
Red and blue denote the entropy with or without islands, and the constants are
chosen as $n=10$, $c=600$, $\phi_{r}=10$, $\phi_{0}=0$, $\epsilon_{\rm rg}=1$,
and $\epsilon_{\rm uv}=1$. Both endpoints of $R$ are located at
$\sigma_{R}=\frac{\pi}{2}-10^{-5}$ in (a), while for (b) one endpoint is at
$\sigma_{R1}=-\frac{2}{\pi}\left(\frac{\pi}{2}-10^{-3}\right)\varphi_{R}+\pi-10^{-3}$
and the other one at $\sigma_{R2}=\frac{\pi}{2}-10^{-5}$.

###  A.3 Two-component islands

In this section, we consider the possibility of an island in ${\rm
dS}_{2}^{n}$ that consists of two disconnected components, as depicted in Fig.
19. To compute the generalized entropy $S_{\rm gen}(R\cup I)$, we once again
consider the complement $(R\cup I)^{c}$, which is now comprised of three
intervals. As already emphasized, such an entropy depends on the particular
model under consideration. Here, we use the three-interval formula Eq. (62),
valid for the case of $c$ free massless Dirac fermions, and we adapt it to the
global coordinates defined in Eqs. (5) and (7) through Eq. (26).

The result of our numerical computation for $n=2$ is shown in Fig. 20. We find
that a two-component island configuration only appears for sufficiently large
$\varphi_{R}$. The two components are symmetric about $R$, and the endpoints
of the right component lie in the range $(\pi/2,\pi)$ within the crunching
region to the right of $R$. For larger values of $n$, the value of
$\varphi_{R}$ beyond which these island configurations appear increases. In
all circumstances, the corresponding generalized entropy is always greater
than the case of no islands or a single component island, and so a two-
component island never dominates.

Figure 19: An island with two disconnected components in dS${}_{2}^{2}$.
Figure 20: Two-component island (green curve) versus single-component island
(red curve) versus no island (blue curve) for $n=2$. We plot the regime of
$\varphi_{R}$ for which a two-island configuration exists as an extremum of
$S_{\mathrm{gen}}(R\cup I)$ and we observe that such a configuration is never
dominant. We choose the endpoints of $R$ to be fixed at
$\sigma_{R}=\frac{\pi}{2}-10^{-5}$, and the parameters of the theory are
chosen as $n=2$, $c=600$, $\phi_{r}=10$, $\phi_{0}=0$, $\epsilon_{\rm rg}=1$,
and $\epsilon_{\rm uv}=1$.

##  Appendix B Addendum

In Sec. 2.1, we considered vacuum solutions of Jackiw-Teitelboim (JT) gravity
with positive, zero, and negative cosmological constant, and in Sec. 2.2, we
deformed these theories by coupling the background metric to a two-dimensional
CFT with field content collectively denoted by $\psi$.

We can describe each of these three deformed theories in a unified way with
the action

$\displaystyle I[g_{\mu\nu},\phi,\psi]$ $\displaystyle=\frac{\phi_{0}}{16\pi
G_{N}}\int\mathrm{d}^{2}x\sqrt{-g}\mathcal{R}-\frac{1}{16\pi
G_{N}}\int\mathrm{d}^{2}x\sqrt{-g}(\phi\mathcal{R}-V(\phi))$ (64)
$\displaystyle+I_{GHY}[g_{\mu\nu},\phi]+I_{CFT}[g_{\mu\nu},\psi],$

where $V(\phi)=2\phi$ results in a de Sitter (dS) solution, $V(\phi)=2$
results in a flat solution, and $V(\phi)=-2\phi$ results in an anti-de Sitter
(AdS) solution. Varying this action with respect to the metric and the dilaton
field $\phi$ produces the following equations of motion in the semiclassical
limit:

$\displaystyle\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\nabla^{2}\phi-\frac{1}{2}g_{\mu\nu}V(\phi)$ $\displaystyle=-8\pi
G_{N}\expectationvalue{T_{\mu\nu}}\,,$ (65) $\displaystyle\mathcal{R}$
$\displaystyle=V^{\prime}(\phi)\,,$ (66)

where $\langle T_{\mu\nu}\rangle$ is the expectation value of the covariant
stress-energy tensor of the CFT. The vacuum solution corresponds to setting
$\expectationvalue{T_{\mu\nu}}=0$.

Upon including the coupling to the CFT, we showed that a contribution to
$\expectationvalue{T_{\mu\nu}}$ coming from a trace anomaly can be removed by
a suitable field redefinition, and we subsequently proceeded with the vacuum
solution for $\phi$ in Sec. 2.2 and beyond. However, we did not account for
contributions from the Weyl anomaly and the Casimir energy when we examined an
$n$-fold extension of dS2. These contributions cancel when $n=1$.

Begin with the line element defined in Eqs. (24) and (25) and let
$x^{\pm}=\sigma\pm\varphi$. As before, we can remove
$\expectationvalue{T_{+-}}$ by a suitable redefinition of the constant
$\phi_{0}$ (as it arises from the conformal anomaly). The Weyl anomaly and
Casimir energy Balasubramanian:2020xqf  combine to give

$\expectationvalue{T_{\pm\pm}}=\frac{c}{48\pi}\left(1-\frac{1}{n^{2}}\right),$
(67)

where we recall that the spatial coordinate $\varphi$ is $2\pi n$-periodic,
and we have chosen the state in the $z$, $\bar{z}$ coordinates of (26) to be
in vacuum. Therefore, the sourceless solutions for the dilaton will acquire a
supplementary additive term due to the source on the right-hand-side of Eq.
(65) when $n>1$.

Our starting point was the case $\mathcal{R}=2$, i.e. dS${}_{2}^{n}$. In this
case, the solution of Eq. (65) with $V(\phi)=2\phi$ and the source (67) is

$\phi=\phi_{r}\frac{\cos\varphi}{\cos\sigma}-\frac{cG_{N}}{3}\quantity(1-\frac{1}{n^{2}})(\sigma\tan\sigma+1),$
(68)

cf. Eq. (7). Notice, however, that when $\phi_{r}/G_{N}\gg c$, the vacuum
contribution dominates over the additive correction, and so we can safely
neglect the correction in this limit.

In particular, we can still attempt to build up a JT multiverse using vacuum
solutions as follows. Starting with pure dS${}_{2}^{n}$ with
$\phi_{r}/G_{N}\gg c$, we drop the correction due to a CFT stress-energy
source and work with the vacuum dS dilaton. If we want to include bubbles, the
only choice is to patch in bubbles in which the dilaton obeys vacuum equations
of motion; otherwise, the gluing would result in a dilaton that is not
continuous across bubble interfaces. However, the CFT then cannot be in the
Minkowski vacuum of (26), for which there would have to be nonzero $\langle
T_{\pm\pm}\rangle$ sources in flat and AdS regions per Eq. (67). Instead, the
CFT is in some state such that $\langle T_{\pm\pm}\rangle$ vanishes
everywhere. It is unclear whether such a CFT state is well defined and whether
its entropy is close to that of the Minkowski vacuum of (26), so that (29)
continues to hold. But, with these caveats, the existing analysis goes
forward.

Alternatively, if we do not neglect the source (67), we can solve for a
backreacted dilaton on the manifolds that we specified in Eqs. (24) and (25).
Let us start with dS${}_{2}^{n}$ in the absence of any bubbles. Compared to
the vacuum solution, the modified dilaton solution (68) causes the parts of
$\mathcal{I}^{+}$ on which $\phi\rightarrow+\infty$ (i.e., the future boundary
of the expanding patches) to shrink. By inspection, these are the parts of
$\mathcal{I}^{+}$ for which

$\cos\varphi>\frac{\pi cG_{N}}{6\phi_{r}}\left(1-\frac{1}{n^{2}}\right)\,.$
(69)

For such regions of $\mathcal{I}^{+}$ to exist, one must have that

$\frac{\phi_{r}}{G_{N}}>\frac{\pi c}{6}\left(1-\frac{1}{n^{2}}\right)\,,$ (70)

which we assume here. This is illustrated in Fig. 21.

Figure 21: Penrose diagram for dS${}_{2}^{n}$ with $n=2$. An expanding patch
is shaded in yellow, and it intersects $\mathcal{I}^{+}$ on the interval
$(-\varphi_{0},\varphi_{0})$ where $\varphi_{0}<\pi/2$.

Our analysis of quantum extremal islands then proceeds essentially verbatim.
We again consider a region $R$ whose endpoints lie near the corners of the
expanding patch centred about $\varphi=0$, and we posit an island whose
endpoints lie just beyond the patch’s corners. Because the locations of the
corners are now shifted relative to the vacuum case, instead of the ansatz
(34), we write

$\displaystyle\sigma_{R}$
$\displaystyle=\frac{\pi}{2}-\delta\sigma_{R}\,,\qquad\,\sigma_{I}=\frac{\pi}{2}-\delta\sigma_{I}\,,$
(71) $\displaystyle\varphi_{R}$
$\displaystyle=\varphi_{0}-\delta\varphi_{R}\,,\qquad\varphi_{I}=\varphi_{0}+\delta\varphi_{I},$

where we have defined

$\varphi_{0}=\arccos{\frac{\pi
cG_{N}}{6\phi_{r}}\quantity(1-\frac{1}{n^{2}})}.$ (72)

Making these substitutions in (32), we get

$\displaystyle S_{\text{gen}}((R\cup I)^{c})\approx$
$\displaystyle\frac{c}{3}\log\left[\frac{2n^{2}(\cos{}(\frac{\delta\sigma_{I}-\delta\sigma_{R}}{n})-\cos{}(\frac{\delta\varphi_{I}+\delta\varphi_{R}}{n}))}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\delta\sigma_{I}\delta\sigma_{R}}\right]$
(73)
$\displaystyle\qquad+2\phi_{r}\frac{\cos(\varphi_{0}+\delta\varphi_{I})}{\delta\sigma_{I}}-\frac{\pi
c}{12}\quantity(1-\frac{1}{n^{2}})\frac{1}{\delta\sigma_{I}}+2\phi_{0}\,,$

where we have set $4G_{N}=1$. Next, if we expand
$\cos(\varphi_{0}+\delta\varphi_{I})$ about $\varphi_{0}$ and assume that the
sum $\delta\varphi_{I}+\delta\varphi_{R}$ and the difference
$\delta\sigma_{I}-\delta\sigma_{R}$ are small, we arrive at

$\displaystyle S_{\text{gen}}((R\cup I)^{c})\approx$
$\displaystyle\frac{c}{3}\log\left[\frac{(\delta\varphi_{I}+\delta\varphi_{R})^{2}-(\delta\sigma_{I}-\delta\sigma_{R})^{2}}{\epsilon_{\mathrm{rg}}\epsilon_{\mathrm{uv}}\delta\sigma_{I}\delta\sigma_{R}}\right]-2\phi^{\prime}_{r}\frac{\delta\varphi_{I}}{\delta\sigma_{I}}+2\phi_{0},$
(74)

where we have defined

$\phi^{\prime}_{r}=\phi_{r}\sqrt{1-\quantity(\frac{\pi
c(1-n^{-2})}{24\phi_{r}})^{2}}.$ (75)

This is identical in form to Eq. (36), and so the rest of the analysis
proceeds as before, but with $\phi_{r}\rightarrow\phi_{r}^{\prime}$. While
$\phi_{r}^{\prime}$ now depends on $n$ explicitly, any $n$-dependence only
enters at $O((c/\phi_{r})^{2})$.

Next, we consider multiverse models in the presence of flat or AdS2 bubbles,
like the ones in Sec. 3.2. The task is to show that the backreacted dilaton
solutions continuously join up along bubble interfaces. Let us first examine
the flat case. With a judicious choice of integration constants, the solution
of Eq. (65) with $V(\phi)=2$ and the source (67) for $\phi$ in a flat bubble
centred about $\varphi=0$ reads (cf. Eq. (14))

$\phi=\phi_{r}-K_{n}+2\frac{\cos\varphi-\cos\sigma}{\cos\varphi+\cos\sigma}-K_{n}\frac{\sigma\sin\sigma+\varphi\sin\varphi}{\cos\varphi+\cos\sigma},$
(76)

where we have defined

$K_{n}=\frac{cG_{N}}{3}\left(1-\frac{1}{n^{2}}\right).$ (77)

In particular, the flat solution (76) and the dS solution (68) coincide along
$\sigma=|\varphi|$, and so they can be continuously joined together.

A consequence of the backreaction is that $\mathcal{I}^{+}$ in a flat bubble
develops segments where $\phi\rightarrow-\infty$. By examining the behaviour
of (76) as one approaches the lines $\sigma=\pi\pm\varphi$, one concludes that
$\varphi<|\varphi_{f}|$ is the portion of $\mathcal{I}^{+}$ on which
$\phi\rightarrow\infty$, where

$\tan\varphi_{f}=\frac{4}{\pi K_{n}}.$ (78)

This is illustrated in Fig. 22. Therefore, should islands still develop, we
expect that the endpoints of a region $R$ inside of a flat bubble should be
placed just to the interior of
$(\sigma,\varphi)=(\pi-\varphi_{f},\pm\varphi_{f})$. We were unable to locate
extrema of $S_{\mathrm{gen}}((R\cup I)^{c})$ by placing
$(\sigma_{I},\varphi_{I})$ perturbatively to the past of
$(\pi-\varphi_{f},\varphi_{f})$ and mirroring our earlier analysis, and so
additional work would be needed to conclusively determine whether or not
islands develop.

Figure 22: Penrose diagram for dS${}_{2}^{n}$ with $n=2$ and a flat bubble
centred at $\varphi=0$. The domain of dependence of the part of
$\mathcal{I}^{+}$ on which $\phi\rightarrow\infty$ is shaded in blue, but the
metric is still flat in the blue hatched region.

For an AdS2 bubble, the solution of Eq. (65) with $V(\phi)=-2\phi$ and the
source (67) for $\phi$ reads (cf. Eq. (18))

$\phi=-\phi_{r}\frac{\cos\sigma}{\cos\tilde{\varphi}}-K_{n}(\tilde{\varphi}\tan\tilde{\varphi}+1),$
(79)

where $\tilde{\varphi}=0$ corresponds to the centre of the bubble. Its value
along the line $\sigma=\tilde{\varphi}$ for $0<\tilde{\varphi}<\pi/2$
coincides with the value of the dS dilaton (68) along the line
$\sigma=\pi+\varphi$ for a constant shift $\tilde{\varphi}=\varphi+\pi$.
Similarly, the AdS dilaton’s value along the line $\sigma=-\tilde{\varphi}$
for $-\pi/2<\tilde{\varphi}<0$ coincides with that of (68) along the line
$\sigma=\pi-\varphi$ for a constant shift $\tilde{\varphi}=\varphi-\pi$.
Therefore, with the appropriate coordinate translations, the backreacted dS
and AdS dilaton profiles continuously join up along AdS2 bubble walls.

In our previous analysis, we were only able to locate islands with endpoints
in AdS2 bubbles numerically in the regime $\phi_{r}/G_{N}\ll c$. Therefore, we
cannot conclusively say whether or not islands develop when we take
backreaction into account, which requires $\phi_{r}/G_{N}>c$ for $n>1$ per Eq.
(70).

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