aainstitutetext: Institute of Mathematical Sciences, Ewha Womans University,
Seoul 03760, Koreabbinstitutetext: Scranton Honors Program, Ewha Womans
University, Seoul 03760, Korea

# Islands in the Proliferating de Sitter Space

Jong-Hyun Baek a,b and Kang-Sin Choi<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study two-dimensional de Sitter universe which evolves and proliferates
according to Ginsparg–Perry–Bousso–Hawking mechanism, using Jackiw-Teitelboim
gravity coupled to a conformal matter. Black holes are generated by quantum
gravity effects from pure de Sitter space and then evaporate to yield multiple
disjoint de Sitter spaces. The backreaction from the matter CFT is taken into
account for the dilaton as a function of temperature of the CFT. We discuss
the evaporation of black holes and calculate the finite temperature entropy of
an inflating region using the island formula. We find that the island moves
toward the apparent horizon of the black hole as temperature increases. The
results are applied to the case of multiple evaporating black holes.

###### Keywords:

de Sitter, JT gravity, black hole entropy, islands, multiverse

##  1 Introduction

De Sitter space is a key ingredient to understand the history of the universe.
It can be a simple model of the inflationary phase in the very early universe.
Similar to a black hole, de Sitter space possesses a cosmological horizon and
has a characteristic temperature Gibbons:1977mu . An observer in the static
patch sees a thermal equilibrium at the dS temperature. Another aspect of
interest is that de Sitter space has a semiclassical instability, leading to
the nucleation of black holes Ginsparg:1982rs . A gravitational instanton
effect can generate a Schwarzschild black hole, which is extremal in size and
can evaporate subsequently.

This black hole generating process can be regarded as a spontaneous topology
change, where a spatial section of de Sitter space $S^{3}$ undergoes a
transition to $S^{1}\times S^{2}$ according to the Bousso–Hawking mechanism
Bousso:1997wi . The latter geometry is described by the Schwarzschild-de
Sitter (SdS) solution Kottler  in the Nariai limit Nariai . For evaporation of
black holes, we may see this geometry as an $S^{2}$ fibration over $S^{1}$,
that is, the radius of $S^{2}$ is not uniform along the $S^{1}$. As the black
holes evaporate, the $S^{2}$ factor shrinks to points, pinching off the
geometry to a number of $S^{3}$’s corresponding to pure de Sitter spaces
Bousso:1998bn ; Bousso:2002fq .

In this paper we study this process in two-dimensions by utilizing Jackiw-
Teitelboim (JT) gravity Teitelboim:1983ux ; Jackiw:1984je . In JT gravity, the
essential features of higher-dimensional theory is captured by the dilaton,
which can be roughly identified with the radius of $S^{2}$ in this case. Thus
fluctuations of the dilaton describe deviation from the Nariai limit, where
the black hole horizon has the same size as the cosmological one
Maldacena:2019cbz ; Cotler:2019nbi .

We couple a conformal matter to gravity and consider an equilibrium state with
the black hole at finite temperature. The back-reaction from the semiclassical
energy momentum tensor of the matter CFT changes the causal structure of the
geometry and correspondingly the Penrose diagram. We use the back-reacted
dilaton solution and impose the boundary condition such that the dilaton
approaches the one for pure de Sitter at future boundary and calculate the
entanglement entropy of a non gravitating region in the context of evaporating
black holes.

The island formula Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni
computes the von Neumann entropy of system $R$ through

$S(\rho_{R})=\text{min ext}_{I}\,S_{\text{gen}}(I\cup R),$ (1)

where the generalized entropy is given by

$S_{\text{gen}}(I\cup R)=\frac{A(\partial I)}{4G_{N}}+S_{\text{mat}}(I\cup
R).$ (2)

Here the island $I$ is a gravitating region and $A(\partial I)$ is the area of
the boundary of $I$ and $S_{\text{mat}}(I\cup R)$ is the von Neunmann entropy
of the system $I\cup R$. The generalized entropy is extremized over the
configurations of $I$ and then minimized over the choices. The island rule was
applied in the eternal black hole in Almheiri:2019yqk  and was derived using
the semiclassical gravitational path integral with the replica method
Penington:2019kki ; Almheiri:2019qdq ; Almheiri:2020cfm .

We study the purity of the newly generated de Sitter space, which can be seen
as follows. An observer in the “Milne” patch (an inflating region near the
spacelike future infinity $\cal{I}^{+}$, see Fig. 1) can collect the radiation
from the evaporating black hole. The matter entropy increases as the CFT
temperature increases until the island appears purifying the entangled pairs
of radiation. Above the Page temperature, the generalized entropy becomes
relevant and it decreases over temperature. This shows that the final dS space
after evaporation is a pure state. (It is not necessary to consider dS space
as the final state. Minkowski space(s) are considered in Refs. Chen:2020tes ;
Hartman:2020khs  in different contexts.)

In section 2, we review the JT de Sitter gravity and its origin from the four-
dimensional Einstein gravity. Higher dimensional Nariai geometry is reduced to
two-dimensions to obtain the global de Sitter space. In section 3, we explain
the backreacted dilaton solution and the boundary condition at future
infinity. The changes in causal structure and the Penrose diagrams due to
backreaction are described. We also discuss how evaporation of black hole in
Bousso-Hawking mechanism can be modeled with the solution. In section 4, the
entropy of an inflating region is calculated using the island formula. High
and low temperature limits are considered and numerical method is used in the
intermediate temperature regime. In section 5, the nucleation of multiple
black holes are considered and attaching the new de Sitter spaces at future
boundary is discussed. The entropy formula is generalized to incorporate the
multiple cover of dS and the change in the Casimir energy. The computation of
the entanglement entropy of multiple regions shows that each new de Sitter
space is pure. In section 6, we conclude with a summary and future directions.

##  2 JT gravity in de Sitter space

We briefly review the de Sitter JT gravity. The four-dimensional
Schwarzschild-de Sitter black hole is described by the metric,

$ds^{2}=-f(\tilde{r})d\tilde{t}^{2}+f(\tilde{r})^{-1}d\tilde{r}^{2}+\tilde{r}^{2}d\Omega^{2},\qquad
f(\tilde{r})=1-\frac{2\mu}{\tilde{r}}-\frac{{\Lambda}}{3}\tilde{r}^{2},$ (3)

where $\mu$ is a mass parameter and ${\Lambda}$ is a cosmological constant. If
$\mu=0$, the metric reduces to pure de Sitter space in the static patch. The
function $f(\tilde{r})$ has two positive roots for
$0<\mu<\frac{1}{3\sqrt{{\Lambda}}}$, one of them corresponding to the black
hole horizon and the other to the cosmological horizon. When the two roots
coincide, the geometry is described by the Nariai solution Nariai .

In the near-Nariai limit of the SdS black hole, the metric takes the
factorized form of dS${}_{2}\times S^{2}$,

$ds^{2}=-\left(1-\frac{r^{2}}{L^{2}}\right)dt^{2}+\left(1-\frac{r^{2}}{L^{2}}\right)^{-1}dr^{2}+L^{2}\phi\,d\Omega_{2}^{2},$
(4)

where the $L$ is the radius of $dS_{2}$ and $\phi$ is the dilaton, which can
be written as

$\phi=\phi_{0}+\Phi.$ (5)

The constant $\phi_{0}$ corresponds to the radius of the $S^{2}$ in the Nariai
limit, and $\Phi$ to fluctuation away from it.

Dimensional reduction of Einstein gravity with a positive cosmological
constant along the $S^{2}$ gives the action of de Sitter JT gravity
Maldacena:2019cbz ; Cotler:2019nbi ; Svesko:2022txo .

$I_{\text{JT}}=\frac{\phi_{0}}{16\pi G_{N}}\int
d^{2}x\sqrt{-g}R+\frac{1}{16\pi G_{N}}\int
d^{2}x\sqrt{-g}\Phi\left(R-\frac{2}{L^{2}}\right)+I_{\text{CFT}}[g],$ (6)

where the boundary terms are omitted, and $I_{\text{CFT}}$ is the action of
the matter CFT, which couples to the metric. Here $G_{N}$ is the two-
dimensional Newton’s constant related to the four-dimensional one,
$G_{N}=G_{N}^{(4)}/(4\pi L^{2})$.

The dilaton constrains the metric to dS2. In conformal gauge and lightcone
coordinates, $x^{\pm}=L({\tau}\pm\theta)$, the metric is written

$ds^{2}=\frac{L^{2}}{\cos{\tau}^{2}}(-d{\tau}^{2}+d\theta^{2})=-e^{2{\omega}}dx^{+}dx^{-},\qquad
e^{-2{\omega}}=\cos^{2}{\tau},$ (7)

where ${\tau}\in(-\pi/2,\pi/2)$ and $\theta\in(-\pi/2,3\pi/2)$. The metric
equations of motion are given by

$e^{2{\omega}}\partial_{\pm}\Big{[}e^{-2{\omega}}\partial_{\pm}\Phi\Big{]}=-8\pi
G_{N}\langle T_{\pm\pm}\rangle,$ (8)

and

$2\partial_{+}\partial_{-}\Phi-\frac{1}{L^{2}}e^{2{\omega}}\Phi=16\pi
G_{N}\langle T_{+-}\rangle,$ (9)

where $\langle T_{ab}\rangle$ is the expectation value of the stress tensor of
the matter CFT.

In the absence of the energy–momentum tensor, the sourceless dilaton $\Phi$ is
given by

$\Phi_{\text{sl}}({\tau},\theta)=\bar{\phi}L\frac{\cos\theta}{\cos\tau}=\bar{\phi}r,$
(10)

for a constant $\bar{\phi}$ and it has negative values for
$\pi/2<\theta<3\pi/2$. Near future infinity ${\tau}=\pi/2$, the dilaton goes
to $-\infty$ in the black hole or the crunching region and to $+\infty$ in the
inflating region with $|\theta|<\pi/2$. This solution describes two
dimensional dS space obtained from SdS4 and drawn in Fig. 1. It inherits the
singularity from the back hole in the higher dimensional solution.

BH$\tau=\frac{\pi}{2}$$\tau=-\frac{\pi}{2}$Milne0$\frac{\pi}{2}$-$\frac{\pi}{2}$$\pi$$\parallel$$\parallel$$\cal{I}^{+}$
Figure 1: Penrose diagram for the two-dimensional de Sitter space.
Singularities are at $\tau=\pm\pi/2$ and $\cos\theta<0.$ The black and blue
dots are the black hole horizon and cosmological horizon, respectively. Two
sides are indentified.

There is a different way to obtain dS2, that is a circle reduction from dS3.
In this case, the range of the $\theta$ coordinate is a segment, half in size
with no identification of both ends. This dS2 contains no singularity and the
dilaton is always positive since $-\pi/2\leq\theta\leq\pi/2$. It’s called the
half reduction, while the above dS2 from SdS4 is called the full reduction in
Svesko:2022txo .

From the two-dimensional point of view, the nucleation of black holes from
pure dS2 and subsequent evaporation back into a number of pure de Sitter
spaces with no black hole can be understood as a transition in the $\theta$
coordinate from a segment to a circle and then into a number of segments. As
black holes evaporate the dilaton $\Phi$ goes to minus infinity at the black
hole horizon, pinching off the circle into segments.

In Bousso:1998bn , this process was described using the perturbation of the
radius in the transverse $S^{2}$. The perturbation changes along the $\theta$
direction of dS2 and can be decomposed into Fourier modes with mode number
$n$. More than one black holes can nucleate if a higher fluctuation $n>1$ mode
of the perturbation dominates, with probability $e^{-n^{2}}$. It may be that
de Sitter spaces have proliferated by the mechanism before and during
inflationary phases, giving rise to eternal inflation.

##  3 Solutions with back-reaction from CFT

In this section, we review the solution of Balasubramanian, Kar and Ugajin
(BKU) Balasubramanian:2020xqf  with a slight generalization of different left,
right temperatures. It describes de Sitter space with a black hole in thermal
equilibrium with the matter CFT at fixed temperature $T=1/(\beta L)$. The
back-reaction from the matter deforms the causal structure of the spacetime
and the Penrose diagram. We explain how the black hole evaporation can be
understood as we increase the temperature. We mainly focus on the hyperbolic
patches of de Sitter space but for studies in the static patch, see
Sybesma:2020fxg ; Aalsma:2021bit ; Kames-King:2021etp ; Aalsma:2022swk .

###  3.1 Back-reacted dilaton

We consider a thermal state on a flat space such that the stress tensor is of
the form

${\tau}_{\pm\pm}=\frac{c}{48\pi
L^{2}}\left(\frac{2\pi}{{\beta}_{\pm}}\right)^{2}-\frac{c}{48\pi L^{2}},$ (11)

where $c$ is the central charge of the CFT and ${\beta}_{\pm}$ are the
left/right inverse temperature and the second term is the Casimir energy due
to the periodicity of $\theta\simeq\theta+2\pi$. They are to be further
discussed in Sections 4.1 and 5.3.

The expectation value of the stress tensor on a curved space can be obtained
by using the trace anomaly and the conservation equation of stress tensor,

$\begin{split}\langle
T_{\pm\pm}\rangle&=\frac{c}{12\pi}\Big{(}\partial_{\pm}^{2}{\omega}-(\partial_{\pm}{\omega})^{2}\Big{)}+{\tau}_{\pm\pm},\\\
\langle
T_{+-}\rangle&=-\frac{c}{12\pi}\partial_{+}\partial_{-}{\omega},\end{split}$
(12)

where ${\omega}$ is the exponent of the conformal factor in (7). For the state
(11), we have

$\langle T_{\pm\pm}\rangle=\frac{c}{48\pi
L^{2}}\left(\frac{2\pi}{{\beta}_{\pm}}\right)^{2},\qquad\langle
T_{+-}\rangle=-\frac{c}{48\pi L^{2}\cos^{2}{\tau}}.$ (13)

The Casimir energy is canceled by the contribution from the warp factor
$\omega$. These are going to describe the thermal radiation from the black
hole. Note that the temperature $1/(\beta_{\pm}L)$ is different from the
Gibbons–Hawking temperture of de Sitter space, $T_{GH}=1/(2\pi L)$.

With the stress tensor expectation value, a solution to (8) and (9) can be
found

$\Phi({\tau},\theta)={\alpha}\frac{\cos\theta}{\cos{\tau}}-\frac{1}{2}\left[\Big{(}(K_{+}+K_{-}){\tau}+(K_{+}-K_{-})\theta\Big{)}\tan{\tau}+K_{+}+K_{-}\right]+\frac{cG_{N}}{3},$
(14)

where

$K_{\pm}=\frac{cG_{N}}{3}\left(\frac{2\pi}{{\beta}_{\pm}}\right)^{2}.$ (15)

Since we are interested in the transition to pure dS2 after the black hole has
evaporated, we may impose the boundary condition at late time,
${\tau}\rightarrow\pi/2$, that the dilaton approaches the sourceless solution
(10), presumably in a new coordinate. The effects of the back-reaction can be
seen as the $SO(1,1)$ boost on the coordinates $r$ and $t$
Balasubramanian:2020xqf ,

$\frac{r^{\prime}}{L}=b_{+}\frac{\cos\theta}{\cos{\tau}}-b_{-}\tan{\tau},\qquad\tanh\frac{t^{\prime}}{L}=b_{+}\frac{\sin{\tau}}{\sin\theta}-b_{-}\frac{1}{\tan\theta}$
(16)

where the rotation parameter $b_{\pm}=(b\pm b^{-1})/2$ with $b\geq 1$. This
transformation is a subgroup of the isometry $SO(1,2)$ of dS2. Thus we expect
the dilaton of the form Balasubramanian:2020xqf ,

$\displaystyle\Phi_{0}({\tau},\theta)$ $\displaystyle=$
$\displaystyle\bar{\phi}r^{\prime}=\bar{\phi}L\left[b_{+}\frac{\cos\theta}{\cos{\tau}}-b_{-}\tan{\tau}\right].$
(17)

in the limit $b=1$, this solution is simplified to (10).

For $\tau\to\pi/2$, at leading order in $\tau$, the dilaton solution (14) is
expanded as

$\Phi({\tau},\theta)=\frac{\frac{\pi}{4}(K_{+}+K_{-})-\frac{1}{2}(K_{+}-K_{-})\theta-{\alpha}\cos\theta}{{\tau}-\frac{\pi}{2}}+\cdots$
(18)

In order to match the leading order coefficients of $\Phi_{0}$ and $\Phi$, we
need to take $K_{+}=K_{-}=K$, with $K$ defined as in (15) but with ${\beta}$.
Then the parameters are related

$K=\frac{cG_{N}}{3}\left(\frac{2\pi}{{\beta}}\right)^{2}=\frac{\bar{\phi}L}{\pi}\left(b-\frac{1}{b}\right),\qquad{\alpha}=\frac{\bar{\phi}L}{2}\left(b+\frac{1}{b}\right).$
(19)

Thus the limit $b=1$ corresponds to the zero temperature limit
$\beta\to\infty$. The dilaton satisfying the boundary condition is given by

$\begin{split}\phi&=\phi_{0}+\Phi({\tau},\theta)\\\
&\equiv\phi_{0}+\frac{\bar{\phi}L}{2}\left[\left(b+\frac{1}{b}\right)\frac{\cos\theta}{\cos{\tau}}-\frac{2}{\pi}\left(b-\frac{1}{b}\right)({\tau}\tan{\tau}+1)\right]+\frac{cG_{N}}{3}.\end{split}$
(20)

Note that the dilaton has dependence on the temperature through the parameter
$b$ by the relation (19). As $b$ gets larger, the effect of back-reaction
becomes stronger.

###  3.2 Back-reacted horizons

The dilaton (20) defines the black hole and the cosmological apparent horizons
by $\partial_{{\tau}}\phi=\partial_{\theta}\phi=0$. They are located at
$({\tau},\theta)=(0,\pi)$ for black hole and $({\tau}_{0},0)$ for cosmological
apparent horizon at large enough $b$, satisfying

$\sin{\tau}_{0}=\frac{b^{2}-1}{b^{2}+1}.$ (21)

It was shown in Balasubramanian:2020xqf  that the dialton at the black hole
apparent horizon, $\Phi(0,\pi)$ in (20), decreases as the temperature of the
CFT increases. This was interpreted as the evaporation of the black hole. The
dilaton at the cosmological event horizon, which coincides with the
cosmological apparent horizon at high temperature, reaches at the upper limit
in the high temperature limit:
$\phi({\tau}_{0},0)|_{b=\infty}=\phi_{0}+\bar{\phi}L$. The back-reaction also
changes the Penrose diagram as in Fig. 2, where the future singularity is
elongated and curves downward. This effect is due to the change in the range
of $\theta$ for which the dilaton $\Phi$ becomes negative as
${\tau}\rightarrow\pi/2$.

We can also see changes in the horizons by coordinate transformations. Before
turning on the CFT temperature ${\beta}$, the de Sitter metric and dilaton are
written in the static patch

$ds^{2}=\left(1-\frac{r^{2}}{L^{2}}\right)dt^{2}+\left(1-\frac{r^{2}}{L^{2}}\right)^{-1}dr^{2},\qquad\Phi=\bar{\phi}r.$
(22)

The cosmic horizon is at $r=L$ and the black hole horizon at $r=-L$. With the
same metric, the Milne patch corresponds to $r>L$, where it includes the
future infinity ${\cal I}^{+}$ at $r\to\infty$.

In the new coordinates (16), the horizons satisfy

$\frac{r^{\prime}_{H}}{L^{\prime}}=b_{+}\frac{\cos\theta}{\cos\tau}-b_{-}\tan\tau=\pm
1,\qquad\left(\frac{r}{L}=\frac{\cos\theta}{\cos{\tau}}\right).$ (23)

In terms of $r$, we have

$\frac{r}{L}=\frac{\pm 1+b_{-}\tan\tau}{b_{+}}.$ (24)

At zero temperature $\beta\to\infty$, $b_{-}\to 0,b_{+}\to 1$ from (19). Thus
it coincides with the original horizons $r/L=\pm 1$, regardless of the value
of $\tau$.

Consider the cosmic horizon choosing the positive sign. Since, for
sufficiently sizable $\tau>0$, the location of the new horizon is,

$\frac{r}{L}=\frac{b_{-}\tan\tau+1}{b_{+}}>1,\quad|\theta|<\frac{\pi}{2},$
(25)

which means that the new horizon is located at

$r=L^{\prime}=\frac{b_{-}\tan\tau+1}{b_{+}}L$ (26)

which is greater than $L$ for ${\tau}$ near $\pi/2$. That is, we have a
growing cosmic horizon for nonzero $\beta$.

To see that the horizon of the black hole shrinks, we note that near
${\tau}\rightarrow\pi/2$, the dilaton $\Phi$ diverges and changes sign at
$\theta=\theta_{0}$, where

$\cos\theta_{0}=\frac{b_{-}}{b_{+}}=\frac{b^{2}-1}{b^{2}+1}.$ (27)

So at ${\tau}=\pi/2$, the range of $\theta$ for singularity is
($\theta_{0},2\pi-\theta_{0})$, for which the dilaton $\Phi$ is negative. The
future event horizon of the black hole is the null lines extended from the end
points of the interval. The black hole horizon is measured at
$\theta^{\prime}=\pi+\theta$, thus

$\left|\frac{r}{L}\right|=\left|\frac{-b_{-}\tan\tau+1}{b_{+}}\right|,$ (28)

which is less than 1 for sizable $\tau$. Thus we have a shrinking black hole
horizon.

RI$\pi$$0$ Figure 2: Backreacted de Sitter at finite temperature. The dotted
lines are Penrose diagram at zero temperature. The black dot is the apparent
horizon of the black hole and red dots are the cosmological horizons.

We have mentioned that as the CFT temperature increases, the causal structure
changes and the black hole interior grows as in Fig. 2. However, the black
hole apparent horizon remain at $({\tau},\theta)=(0,\pi)$ and the dilaton
value at that point decreases over temperature, corresponding to the
evaporation of the black hole.

On the other hand, the cosmological apparent and event horizon is moved to
$({\tau}_{0},0)$ near ${\tau}=\pi/2$. It seems that the future light cone of
the cosmological horizon gets smaller in the Penrose diagram. But the physical
size of the horizon measured by the dilaton shows that it does not decrease.

###  3.3 Bousso-Hawking process

The black hole that we have seen thus far is an eternal black hole in
equilibrium at finite temperature $T=1/(\beta L)$. In the zero temperature
limit $\beta\to\infty$, there is no radiation from the black hole and only the
background radiation of dS2 at $T_{GH}$.

Ultimately, we are interested in the description of an evaporating black hole
in de Sitter space and the emergence of pure dS2 at the end of evaporation. We
assume adiabatic evolution. The BKU back-reacted solution can be thought of as
a local and instantaneous description of the black hole, which is in thermal
equilibrium at a given CFT temperature. As time goes on, the black hole
evaporates and the CFT temperature also increases so we may regard it as a
parameter playing the role of time evolution. More realistic description of
the decay of AdS black hole is done in Almheiri:2019psf .

Another remark is that after the evaporation, the resulting geometry should be
pure dS2. Although the Penrose diagram changes due to back-reaction, near the
future infinity $\cal{I}^{+}$ we can rotate the coordinates by (16) and
transform the metric and dilaton into the Milne form (22). This is consistent
with the boundary condition (17). So one can glue new dS2 space along the
future boundary of the existing one. However, for an observer to be able to
travel into the new de Sitter space within finite time, a subtlety arises. We
shall come back to this issue in Section. 5.1.

##  4 Entropy computation

In this section we calculate the entropy of a region in the future infinity
$\cal{I}^{+}$ during the evaporation process. We consider two limits of low
and high temperature. Islands appear in both cases. In the low temperature
limit, we find the Page $\theta_{R}$, which corresponds to the size of the
inflating region. In the high temperature approximation, the Page temperature
is computed.

###  4.1 The von Neumann entropy

To track the finite temperature effect of the black hole, we consider the
“Hartle–Hawking” (HH) vacuum in a “Schwarzschild”-like coordinate, in which
the energy-momentum tensor is (11). A stationary observer in the
“Schwarzschild”-like coordinate $x^{\pm}=L({\tau}\pm\theta)$ observes the
thermal radiation from the black hole. To calculate the entropy, we transform
the coordinates such that the expectation value of the energy-momentum tensor
(11) vanishes

$\langle HH|\tau_{\pm\pm}(y^{\pm})|HH\rangle=0.$ (29)

This is different from Bunch–Davies vacuum, whose temperature is related to
the curvature of de Sitter space. Henceforth, the braket
$\langle\,\cdot\,\rangle$ denotes the expectation value for this
Hartle–Hawking vacuum.

The following transformation

$y^{\pm}=\exp\left(-\frac{2\pi}{BL}x^{\pm}\right),$ (30)

with a notation of relative temperature $T_{R}=1/(BL)$,

$T_{R}\equiv\frac{1}{BL}\equiv\sqrt{\frac{1}{(\beta L)^{2}}-\frac{1}{(2\pi
L)^{2}}}=\sqrt{T^{2}-T^{2}_{\rm GH}},$ (31)

gives a vacuum state via

$\left(\frac{\partial y^{\pm}}{\partial
x^{\pm}}\right)^{2}\langle{\tau}_{\pm\pm}(y^{\pm})\rangle=\langle{\tau}_{\pm\pm}(x^{\pm})\rangle+\frac{c}{24\pi}\\{y^{\pm},x^{\pm}\\},$
(32)

and $\\{y^{\pm},x^{\pm}\\}=-2\pi^{2}/(BL)^{2}$. The metric (7) is written in
$y^{\pm}$ coordinates,

$ds^{2}=-\Omega(y)^{-2}dy^{+}dy^{-},\qquad\Omega(y)^{2}=\left(\frac{2\pi}{BL}\right)^{2}y^{+}y^{-}\cos^{2}\left(\frac{-B}{4\pi}\log
y^{+}y^{-}\right).$ (33)

At zero temperature $T=0$, this reduces to the same transformation as in
Hartman:2020khs .

The von Neumann entropy is given by the single interval entropy of 2d CFT in a
flat space together with a Weyl transformation by $\Omega(y)$ Calabrese:2004eu
; Calabrese:2009qy

$\displaystyle S_{\text{mat}}(x_{1},x_{2})$
$\displaystyle=\frac{c}{6}\log\left[\frac{-(y_{1}^{+}-y_{2}^{+})(y_{1}^{-}-y_{2}^{-})}{\epsilon^{2}_{\text{uv}}\Omega(y_{1})\Omega(y_{2})}\right]$
$\displaystyle=\frac{c}{6}\log\left[\frac{B^{2}L^{2}(\cosh\frac{2\pi}{B}(\theta_{1}-\theta_{2})-\cosh\frac{2\pi}{B}({\tau}_{1}-{\tau}_{2}))}{2\epsilon_{\text{uv}}^{2}\pi^{2}\cos{\tau}_{1}\cos{\tau}_{2}}\right]$
$\displaystyle=\frac{c}{6}\log\left[\frac{B^{2}L^{2}\left(\sinh\frac{\pi}{BL}{\Delta}x^{+}\sinh\left(-\frac{\pi}{BL}{\Delta}x^{-}\right)\right)}{\epsilon_{\text{uv}}^{2}\pi^{2}\cos\left(\frac{x_{1}^{+}+x_{1}^{-}}{2L}\right)\cos\left(\frac{x_{2}^{+}+x_{2}^{-}}{2L}\right)}\right],$
(34)

where we defined $\Delta x^{\pm}=x_{1}^{\pm}-x_{2}^{\pm}$. It is the most
convenient to work in the global coordinates, so we view the formula as a
function of $x^{\pm}$.

One may note that the entropy formula at zero temperature in
Balasubramanian:2020xqf  can be reproduced from (4.1) by taking
$T=1/({\beta}L)\to 0$, i.e. $B\to 2\pi i$. When the relative temperature
$T_{R}$ is positive, in the absence of the warp factor, the entropy formula is
periodic under the imaginary time

$\tau\simeq\tau+iB,$ (35)

which is the property of a thermal two-point correlation function
Gibbons:1977mu . Indeed, for high temperature $B\to\beta$ and this is the
desired property of the black hole of temperature $T=1/(\beta L)$. For $T\leq
T_{\rm GH}$, we have imaginary $T_{R}$ making the entropy non-thermal. This is
interpreted that, if the black hole temperature $T$ is lower than the
background temperature $T_{\rm GH}$ of de Sitter space, there is no thermal
radiation from the black hole.

###  4.2 Entropy with islands

We use (4.1) for the matter entropy for an inflating region
$R=(x^{\prime}_{R},x_{R})$ near the future infinity $\cal{I}^{+}$. The region
$R$ is in the asymptotically flat part of the geometry. For an island located
in the region $I=(x^{\prime}_{I},x_{I})$, the generalized entropy is given by

$S_{\text{gen}}(I\cup R)=\frac{A(\partial
I)}{4G_{N}}+S_{\text{mat}}(x_{I},x^{\prime}_{R})+S_{\text{mat}}(x_{R},x^{\prime}_{I}),$
(36)

up to UV counterterms, and the area of the island boundary $A(\partial I)$ is
given by the value of the dilaton at those points $\phi(x_{I})$ and
$\phi(x_{I^{\prime}})$. The arguments of $S_{\text{mat}}$ goes from the left
to right end point of the intervals.

For simplicity, we take the region $R$ in a symmetric manner such that
${\tau}_{1}={\tau}_{2}={\tau}_{R}$ and $\theta_{1}=-\theta_{2}=\theta_{R}$.
Also we require ${\tau}_{R}$ close to $\pi/2$ so that $R$ becomes a non-
gravitating region. Then, the matter entropy becomes

$S_{\text{mat}}(R)=\frac{c}{3}\log\left[\frac{BL}{\pi\epsilon}\sinh\frac{2\pi}{B}\theta_{R}\right]-\frac{c}{3}\log(\cos{\tau}_{R}).$
(37)

And the generalized entropy is

$\displaystyle S_{\text{gen}}(I\cup R)$
$\displaystyle=\frac{\phi_{0}}{2G_{N}}+\frac{\Phi({\tau}_{I},\theta_{I})}{2G_{N}}+\frac{c}{3}\log\left[\frac{BL}{\pi\epsilon}\sinh\frac{\pi}{B}({\tau}_{I}-{\tau}_{R}+\theta_{I}-\theta_{R})\right]$
(38)
$\displaystyle~{}~{}~{}+\frac{c}{3}\log\left[\frac{BL}{\pi\epsilon}\sinh\frac{\pi}{B}(-{\tau}_{I}+{\tau}_{R}+\theta_{I}-\theta_{R})\right]-\frac{c}{3}\log\big{[}\cos{\tau}_{I}\cos{\tau}_{R}\big{]},$

where the back-reacted dilaton $\Phi({\tau}_{I},\theta_{I})$ is given by (20).

The entanglement entropy is found by minimizing the generalized entropy with
respect to $\theta_{I}$ and extremizing it with respect to $\tau_{I}$. The
condition for the derivative with respect to $\theta_{I}$ to vanish is

$\partial_{\theta_{I}}S_{\text{gen}}=-\frac{\bar{\phi}L}{2G_{N}}b_{+}\frac{\sin\theta_{I}}{\cos{\tau}_{I}}+\frac{\pi
c}{3B}\left(\coth\frac{\pi}{B}({\tau}_{IR}+\theta_{IR})+\coth\frac{\pi}{B}(-{\tau}_{IR}+\theta_{IR})\right)=0,$
(39)

where ${\tau}_{IR}\equiv{\tau}_{I}-{\tau}_{R}$ and similarly for
$\theta_{IR}$. The derivative of $S_{\text{gen}}$ with respect to ${\tau}_{I}$
gives

$\displaystyle\partial_{{\tau}_{I}}S_{\text{gen}}$
$\displaystyle=\frac{\bar{\phi}L}{2G_{N}}\left(b_{+}\frac{\cos\theta_{I}\sin{\tau}_{I}}{\cos^{2}{\tau}_{I}}-\frac{2}{\pi}b_{-}(\tan{\tau}_{I}+{\tau}_{I}\sec^{2}{\tau}_{I})\right)$
$\displaystyle~{}~{}~{}+\frac{ac}{6}\left(\coth\frac{\pi}{B}({\tau}_{IR}+\theta_{IR})-\coth\frac{\pi}{B}(-{\tau}_{IR}+\theta_{IR})\right)+\frac{c}{3}\tan{\tau}_{I}=0.$
(40)

###  4.3 Low temperature limit

For large ${\beta}$, we have $2\pi/B\to i$ and the matter entropy (37) can be
written as

$S_{\text{mat,l}}(R)=\frac{c}{3}\log\frac{2L\theta_{R}}{\epsilon}-\frac{c}{3}\log(\cos{\tau}_{R}).$
(41)

In this limit, we also have $b\approx 1$ from (19), the derivative (39)
simplifies to

$\partial_{\theta_{I}}S_{\text{gen}}\simeq-\frac{\bar{\phi}L}{2G_{N}}\frac{\sin\theta_{I}}{\cos{\tau}_{I}}+\frac{c}{3}\frac{2\theta_{IR}}{\theta_{IR}^{2}-{\tau}_{IR}^{2}}=0.$
(42)

Note that the ${\beta}$ dependence drops out. In this case the back-reaction
is negligible, so we expand $\theta_{I}$ and ${\tau}_{I}$ around $\pi/2$, and
also set ${\tau}_{R}=\pi/2$ and $\theta_{R}=\pi/2-\epsilon_{\theta}$ for
region $R$. The condition becomes a quadratic equation for ${\tau}_{I}$ and
the solution is

$\displaystyle{\tau}_{I}$
$\displaystyle=\frac{\pi}{2}+\frac{-4cG_{N}+\sqrt{(4cG_{N})^{2}-6(\bar{\phi}L)^{2}\epsilon_{\theta}^{2}}}{\bar{\phi}L\epsilon_{\theta}}$
$\displaystyle\cong\frac{\pi}{2}-\frac{3\bar{\phi}L}{4cG_{N}}\epsilon_{\theta},$
(43)

where $\epsilon_{\theta}\ll cG_{N}/\bar{\phi}L$ has been assumed.

In the ${\tau}_{I}$ derivative of $S_{\text{gen}}$ (4.2), for large ${\beta}$,
the second term in the first line can be ignored since $b\approx 1$. So we
have a ${\beta}$ independent equation again,

$\partial_{{\tau}_{I}}S_{\text{gen}}\simeq\frac{\bar{\phi}L}{2G_{N}}\frac{\cos\theta_{I}\tan{\tau}_{I}}{\cos{\tau}_{I}}-\frac{c}{3}\frac{2{\tau}_{IR}}{\theta^{2}_{IR}-{\tau}_{IR}^{2}}+\frac{c}{3}\tan{\tau}_{I}=0.$
(44)

Expanding $\theta_{I}$ and ${\tau}_{I}$ around $\pi/2$ and using (4.3) for
${\tau}_{I}$, we find

$\displaystyle\theta_{I}$
$\displaystyle=\frac{\pi}{2}+\frac{2cG_{N}\epsilon_{\theta}}{\sqrt{(4cG_{N})^{2}-6(\bar{\phi}L)^{2}\epsilon_{\theta}^{2}}}$
$\displaystyle\cong\frac{\pi}{2}+\frac{1}{2}\epsilon_{\theta}.$ (45)

Plugging (4.3) and (4.3) into the generalized entropy (38) at large ${\beta}$,
we have

$\displaystyle S_{\text{gen,l}}$
$\displaystyle=\frac{\phi_{0}}{2G_{N}}+\frac{\bar{\phi}L}{2G_{N}}\frac{\cos\theta_{I}}{\cos{\tau}_{I}}+\frac{c}{3}\log\frac{L^{2}(-{\tau}_{IR}^{2}+\theta_{IR}^{2})}{\epsilon^{2}\cos{\tau}_{I}}-\frac{c}{3}\log(\cos{\tau}_{R})$
$\displaystyle\cong\frac{\phi_{0}}{2G_{N}}-\frac{c}{3}+\frac{c}{3}\log\left[\frac{3L^{2}}{\epsilon^{2}}\left(\frac{cG_{N}}{\bar{\phi}L}-\frac{\bar{\phi}L}{4cG_{N}}\right)\left(\frac{\pi}{2}-\theta_{R}\right)\right]-\frac{c}{3}\log(\cos{\tau}_{R}).$
(46)

The Page $\theta_{R,\text{Page}}$ can be found by equating $S_{\text{gen,l}}$
in (4.3) and $S_{\text{mat,l}}$ in (41). Assuming $\phi_{0}\gg
cG_{N}\gg\bar{\phi}L$, we find

$\theta_{R,\text{Page}}\cong\frac{\pi}{2}\left(1-\frac{2\epsilon\bar{\phi}}{3cG_{N}}e^{-\frac{3\phi_{0}}{2cG_{N}}}\right),$
(47)

which is very close to but smaller than $\pi/2$.

###  4.4 High temperature limit

We now consider the case of small ${\beta}$. In this limit, we have
$B\to\beta$. The matter and generalized entropy in (37) and (38) can be
approximated to

$\displaystyle S_{\text{mat,h}}$ $\displaystyle=\frac{2\pi
c}{3\beta}\theta_{R}+\frac{c}{3}\log\frac{{\beta}L}{2\pi\epsilon}-\frac{c}{3}\log(\cos{\tau}_{R}),$
$\displaystyle S_{\text{gen,h}}$
$\displaystyle=\frac{\phi_{0}}{2G_{N}}+\frac{\bar{\phi}L}{4G_{N}}b\left(\frac{\cos\theta_{I}}{\cos{\tau}_{I}}-\frac{2}{\pi}({\tau}_{I}\tan{\tau}_{I}+1)\right)+\frac{c}{6}$
$\displaystyle+\frac{2\pi
c}{3{\beta}}\theta_{IR}+\frac{2c}{3}\log\frac{{\beta}L}{2\pi\epsilon}-\frac{c}{3}\log(\cos{\tau}_{I}\cos{\tau}_{R}).$
(48)

From (19), the parameter $2b_{+}\cong 2b_{-}\cong
b\cong\frac{4\pi^{3}cG_{N}}{3\bar{\phi}L}\frac{1}{{\beta}^{2}}$ for small
${\beta}$ so the dilaton is the dominant contribution to $S_{\text{gen,h}}$
since it is proportional to $b\propto 1/{\beta}^{2}$. The condition (4.2) can
be expanded in ${\beta}$

$\frac{\bar{\phi}Lb}{4G_{N}}\left(\cos\theta_{I}\sin\tau_{I}-\frac{2}{\pi}(\sin\tau_{I}\cos\tau_{I}+\tau_{I})\right)+\frac{c}{3}\sin\tau_{I}\cos\tau_{I}=0.$
(49)

and it is easy to see that ${\tau}_{I}=0$ is a solution. The condition (39)
reduces to

$\frac{\bar{\phi}L}{4G_{N}}b\left(-\frac{\sin\theta_{I}}{\cos{\tau}_{I}}\right)+\frac{2\pi
c}{3{\beta}}=0,$ (50)

and assuming $\theta_{I}\approx\pi$, we find that

$\theta_{I}\cong\pi-\frac{2{\beta}}{\pi^{2}}.$ (51)

The location of $({\tau}_{I},\theta_{I})$ shows that the island shrinks to the
location of the apparent horizon of the blackhole, $(0,\pi)$, as temperature
increases. Using ${\tau}_{I}$ and $\theta_{I}$, we obtain

$S_{\text{gen,h}}=\frac{\phi_{0}}{2G_{N}}-\frac{\pi^{2}c}{3{\beta}^{2}}(\pi+2)+\frac{c}{6}+\frac{2\pi
c}{3{\beta}}(\pi-\theta_{R})+\frac{2c}{3}\log\frac{{\beta}L}{2\pi\epsilon}-\frac{c}{3}\log\cos{\tau}_{R}.$
(52)

The Page temperature Page:1993wv  is computed by setting the above entropy
equal to the matter entropy in (4.4).

$T_{\text{Page}}\cong\sqrt{\frac{3\phi_{0}}{2(\pi+2)\pi^{2}cG_{N}}}+\frac{\pi-2\theta_{R}}{(\pi+2)\pi}.$
(53)

###  4.5 Intermediate temperature and the Page curve

It is likely that the Page temperature is in the intermediate range. For this,
we resort to numerical computation, to calculate the von Neumann entropy.

A typical page curve is drawn in Fig. 3 with benchmark values
$\phi_{0}=100,c=10,\bar{\phi}=50,L=1,\theta_{R}=1,G_{N}=1/4,\epsilon=0.01$.

Figure 3: Page curve for the evaporating black hole in dS2 for a choice of
parameters. The temperature of the black hole plays the role of time.

The matter entropy (37) increases as temperature increases, reflecting that of
radiation from the decaying black hole. It is clear that for high enough
temperature, the entanglement entropy (38) is dominated by the dilaton term,
as explained below Eq. (4.4), decreasing in temperature. If we decrease
$\phi_{0}/G_{N}$, the Page temperature decreases, similar to (53). This
dominant behavior is insensitive to other parameters than the temperature,
thus the general curve should have a similar form.

Interestingly, numerical analysis revealed that the minimum of general entropy
is obtained for very small $\tau_{I}$ of order $0.1$ over large ranges of
temperatures ${\cal O}(1)<\beta<{\cal O}(100)$. Assuming this, the
extremization condition (4.2) is approximated as

$\tau_{I}\simeq\frac{\frac{ac}{6}\left(\coth\frac{\pi}{B}(-\frac{\pi}{2}+\theta_{I}-\theta_{R})-\coth\frac{\pi}{B}(\frac{\pi}{2}+\theta_{I}-\theta_{R})\right)}{\frac{\bar{\phi}L}{4G_{N}}\left(b_{+}\cos\theta_{I}-\frac{4}{\pi}b_{-}\right)-\frac{c}{3}}.$
(54)

##  5 The de Sitter multiverse

We consider multiple copies of global de Sitter spaces and calculate the
entropy. Also, we address a question that to what extent the island formula is
local. We expect it natural that a state in each individual de Sitter space
should be pure even if multiple de Sitter spaces are generated from one de
Sitter space; otherwise a wavefunction in one de Sitter space should be
entangled with one in another space. There is other possibility discussed in
Levine:2022wos .

###  5.1 Multiverse solutions

The BKU backreacted geometry, considered in Section 3, is characterized by the
dilaton (20) that is periodic under $\theta\simeq\theta+2\pi$. We may extend
it to the multiple covers by allowing multiple periodicity $2\pi n$ with the
same metric (7). This gives us $n$ copies of de Sitter spaces having $2n$
black hole horizons and $2n$ cosmic horizons in the interval
$\theta\in(0,2n\pi)$.

Black holes and de Sitter spaces are symmetric with respect to the axes at
$\theta=k\pi,$ with $k$ an integer. It is equivalent to the fact that each
black hole backreacts at the same temperature. Otherwise, the black holes at
different temperatures enforce the dilaton with different values of $b$ to be
matched along some domain wall, which is not possible in general. We just
leave a comment that different stress tensors for left and right black holes
yields asymmetric radiation and corresponds to the asymmetric dilaton profile
as in (14).

After black holes have evaporated, we are left with pure de Sitter spaces. To
describe this, we attach newly created de Sitter spaces to the existing
multiple de Sitter space. The Penrose diagram for a typical example is
depicted in Fig. 4. For this, we may attempt to excise and glue the geometry
as long as (i) the metric and (ii) the dilaton takes the same values at the
interface Hartman:2020khs ; Aguilar-Gutierrez:2021bns . Instead of attaching
dS spaces, one can consider other spaces such as Minkowski or anti de Sitter
space and glue them at the Milne “expanding” patch and/or even the similar
“crunching” patch of the black hole region.

We may attempt to glue two de Sitter spaces of different sizes by matching the
metric in the global coordinates,

$ds^{2}=\begin{cases}\displaystyle\frac{L_{e}^{2}}{\cos^{2}\tau}(-d\tau^{2}+d\theta^{2}),&~{}~{}\tau<\tau_{c},~{}~{}~{}~{}(k-\frac{1}{2})\pi<\theta<(k+\frac{1}{2})\pi,\\\
\displaystyle\frac{L_{l}^{2}}{\cos^{2}(\tau-\tau_{c}+\tau_{l})}(-d\tau^{2}+d\theta^{2}),&~{}~{}\tau>\tau_{c},~{}~{}~{}~{}(k-\frac{1}{2})\pi<\theta<(k+\frac{1}{2})\pi,\\\
\end{cases}$

for a fixed integer $k$, satisfying

$\frac{L_{l}^{2}}{\cos^{2}\tau_{l}}=\frac{L_{e}^{2}}{\cos^{2}\tau_{c}},$

where $\tau_{c}$ is the the position of a constant ${\tau}$ cut of the two
dS2’s. The gravity action for each dS space should have different cosmological
constants $2/L_{e}^{2}$ and $2/L_{l}^{2}$. However the warp factor of the
neighboring region do not match at the interface $\theta=(k\pm 1/2)\pi$, for
$\tau_{c}<\pi/2$.

${\cal I}_{1}^{-}$$\parallel$$\parallel$$R_{1}$$R_{2}$$R_{2}$${\cal
I}_{1}^{+}$${\cal I}_{2}^{+}$${\cal I}_{2}^{+}$ Figure 4:  Schematic diagram
of two black holes generated at ${\tau}=0$ from pure dS. A topology change of
the spatial section from an interval to a circle occurs with two copies of
global de Sitter spaces by GPBH mechanism. After the black holes evaporate
completely, two disjoint dS spaces emerge. We excise small regions at the
spacelike interfaces of de Sitter spaces and glue them together along the
green lines so that an observer can move to a new de Sitter space within
finite time.

Instead, we may cut the upper edge of the Milne patch of the existing de
Sitter at $r^{\prime}=1/\epsilon_{\rm s}$ with small enough $\epsilon_{\rm
s}$, where $r^{\prime}$ is the coordinate in the Milne patch after the boost
transformation (16) with the dilaton (17). This removes ${\cal I}^{+}$. Then
we prepare another dS2 whose opposite Milne patch is cut at the same distance,
removing ${\cal I}^{-}$. By identifying the rest of coordinates, we have two
connected de Sitter spaces. This is done in the exact same way as for the
boundary of AdS to introduce a cutoff for UV divergence (The Penrose diagram
for AdS is rotated by 90 degree in dS case). This also remedies the infinite
time evolution problem between two consecutive de Sitter spaces, because the
original boundary of the Milne patch ${\cal I}^{+}$ is not reachable at finite
time from a point in the same de Sitter space.

###  5.2 Multiple islands and locality

We calculate von Neumann entropy for the $n$-copied multiverse. Let us first
consider the simplest case of single interval $n=1$. The generalized entropy
(38) depends on the difference between the nearest endpoints
$\\{x^{\pm}_{I},x^{\prime\pm}_{R}\\}$ and
$\\{x^{\pm}_{R},x^{\prime\pm}_{I}\\}$ (we denote the left and right end of the
interval by primed and unprimed coordinates, respectively),

$S_{\text{gen}}(I\cup
R)=\frac{\phi(x_{I})}{4G_{N}}+S_{\text{mat}}(x_{I},x^{\prime}_{R})+S_{\text{mat}}(x_{R},x^{\prime}_{I})+\frac{\phi(x^{\prime}_{I})}{4G_{N}}.$
(55)

We assumed that $R$ is in the flat space so that there is no corresponding
area (dilaton) terms. Of course this comes from the fact that the whole
universe is pure and the corresponding entropy is zero, so that

$S_{\text{gen}}(I\cup R)=S_{\text{gen}}\left((I\cup R)^{c}\right),$

in a Cauchy slice including the intervals $I$ and $R$. As long as the formula
(55) is guaranteed to be correct, it is local in the sense that it does not
depend on the details of the shape of the intervals $I$ and $R$, but only on
the endpoints of them. Formally the formula is sufficiently local and we
heuristically use it to calculate the entropy in the multiple covers of the de
Sitter space.

$x^{\prime}_{R_{1}}$$x_{R_{1}}$$R_{1}$$x^{\prime}_{I_{1}}$$x_{I_{1}}$$I_{1}$$x^{\prime}_{R_{2}}$$x_{R_{2}}$$R_{2}$$x^{\prime}_{I_{2}}$$x_{I_{2}}$$I_{2}$
Figure 5: Islands for the $n=2$ covering space. We may choose different CFT
intervals $R_{1},R_{2}$ that they may collect radiation from the adjacent
black holes at a different rate. Corresponding islands $I_{1}$ and $I_{2}$ are
formed. However the formula is formally local so that we may think of this as
two independent copies of a single, periodic dS space.

For later convenience, we define an entropy of symmetric interval such that

$\tilde{S}_{\text{gen}}(x_{I},x_{R})\equiv\frac{\phi(x_{I})}{2G_{N}}+2S_{\text{mat}}(x_{I},x_{R}).$
(56)

That is, the entropy in the case
$x_{R}^{\prime}=-x_{R},x_{I}^{\prime}=-x_{I}$. In the argument, we emphasized
the coordinates of the endpoints of the intervals $I$ and $R$, instead of the
set $I\cup R$.

The extrimization conditions of $S_{\text{gen}}(I\cup R)$ for
$\theta_{I},\tau_{I},\theta_{I}^{\prime},\tau_{I}^{\prime}$ can be written in
terms of the symmetric entropy, $\tilde{S}_{\text{gen}}$, as

$\displaystyle\frac{\partial S_{\text{gen}}(I\cup R)}{\partial\tau_{I}}$
$\displaystyle=\frac{1}{4G_{N}}\frac{\phi(x_{I})}{\partial\tau_{I}}+\frac{\partial
S_{\text{mat}}(x_{I},x^{\prime}_{R})}{\partial\tau_{I}}=\frac{1}{2}\frac{\partial\tilde{S}_{\text{gen}}(x_{I},x_{R}^{\prime})}{\partial\tau_{I}}=0,$
(57) $\displaystyle\frac{\partial S_{\text{gen}}(I\cup
R)}{\partial\theta_{I}}$
$\displaystyle=\frac{1}{4G_{N}}\frac{\phi(x_{I})}{\partial\theta_{I}}+\frac{\partial
S_{\text{mat}}(x_{I},x^{\prime}_{R})}{\partial\theta_{I}}=\frac{1}{2}\frac{\partial\tilde{S}_{\text{gen}}(x_{I},x_{R}^{\prime})}{\partial\theta_{I}}=0,$
(58) $\displaystyle\frac{\partial S_{\text{gen}}(I\cup
R)}{\partial\tau_{I}^{\prime}}$
$\displaystyle=\frac{1}{4G_{N}}\frac{\phi(x^{\prime}_{I})}{\partial\tau_{I}^{\prime}}+\frac{\partial
S_{\text{mat}}(x^{\prime}_{I},x_{R})}{\partial\tau^{\prime}_{I}}=\frac{1}{2}\frac{\partial\tilde{S}_{\text{gen}}(x^{\prime}_{I},x_{R})}{\partial\tau_{I}^{\prime}}=0,$
(59) $\displaystyle\frac{\partial S_{\text{gen}}(I\cup
R)}{\partial\theta^{\prime}_{I}}$
$\displaystyle=\frac{1}{4G_{N}}\frac{\phi(x^{\prime}_{I})}{\partial\theta_{I}^{\prime}}+\frac{\partial
S_{\text{mat}}(x^{\prime}_{I},x_{R})}{\partial\theta^{\prime}_{I}}=\frac{1}{2}\frac{\partial\tilde{S}_{\text{gen}}(x^{\prime}_{I},x_{R})}{\partial\theta_{I}^{\prime}}=0.$
(60)

The first two equations do not affect the last two and vice versa. In other
words, the extrimization of the island boundary separately takes place for
each cross-interval.

In the end, the extremized entropy becomes the average of the symmetric ones

$\text{ext}_{I}S_{\text{gen}}(I\cup
R)=\text{ext}_{I}\frac{1}{2}\left(\tilde{S}_{\text{gen}}(x_{I},x_{R}^{\prime})+\tilde{S}_{\text{gen}}(x_{I}^{\prime},x_{R})\right).$
(61)

Therefore, for asymmetric $R$ where $x^{\prime}_{R}\neq-x_{R}$, we may obtain
the entropy by averaging the symmetric ones. In the symmetric case of
$x^{\prime}_{R}=-x_{R}$, we naturally have symmetric island
$x^{\prime}_{I}=-x_{I}$ so that two of the above are nontrivial.

Now consider the case of two covers $n=2$. The situation is depicted in Fig.
5. We take two intervals $R_{1}=(x^{\prime}_{R_{1}},x_{R_{1}})$ and
$R_{2}=(x^{\prime}_{R_{2}},x_{R_{2}})$ in the Milne patches. In this case each
observers collects the radiations from both adjacent, the left- and the
right-, black holes. The islands $I_{1}$ and $I_{2}$ can form inside the black
holes, as in Fig. 5. The generalized entropy is

$\begin{split}S_{\text{gen}}(I_{1}\cup R_{1}\cup I_{2}\cup
R_{2})&=\frac{\phi(x_{I_{1}})}{4G_{N}}+S_{\text{mat}}(x_{I_{1}},x^{\prime}_{R_{1}})+S_{\text{mat}}(x_{R_{1}},x^{\prime}_{I_{2}})+\frac{\phi(x^{\prime}_{I_{2}})}{4G_{N}}\\\
&+\frac{\phi(x_{I_{2}})}{4G_{N}}+S_{\text{mat}}(x_{I_{2}},x^{\prime}_{R_{2}})+S_{\text{mat}}(x_{R_{2}},x^{\prime}_{I_{3}})+\frac{\phi(x^{\prime}_{I_{3}})}{4G_{N}},\end{split}$
(62)

where $x^{\prime}_{I_{3}}$ is identified with $x^{\prime}_{I_{1}}$.

The radiation collected in the region $R_{1}$ comes from both the black holes
containing $I_{1}$ and $I_{2}$, respectively. However, the entropy is only
dependent on the cross-interval
$x_{I_{1}}-x^{\prime}_{R_{1}},x_{R_{1}}-x^{\prime}_{I_{2}}$. Consequently, the
radiation falling in each $R_{1}$ and $R_{2}$ becomes pure separately. Each
symmetric entropy (56) provides a building block of the interval, containing
half $R_{i}$ and its neighboring half $I_{i}$.

This possibility of locality in purification can be understood in the
following way. At a given interval $R_{i}$, only the nearest black holes
affect the entropy. If black holes are sufficiently separated in the spacelike
direction, they should be causally disconnected. Of course, we assume that the
intervals $R_{i}$ and $I_{i},I_{i+1}$ are sufficiently large so that the
entropy is dominated by two-point correlation functions, not the higher.
Another point to mention is that in the Penrose diagram, the bifurcate
horizons disappear due to the growth of black hole inside. This means that
left and right one sided black holes are causally disconnected.

We may consider $R_{2}\to\emptyset$ limit. Shrinking $R_{2}$ makes $I_{1}$ and
$I_{2}$ smaller at the same time, moving $x_{I_{2}}$ to the left and
$x^{\prime}_{I_{1}}$ to the right, while leaving other coordinates
$x^{\prime}_{I_{2}}$ and $x_{I_{1}}$ intact. At high temperature, these points
approach to the apparent horizons of the corresponding black holes. We should
be careful, however, that the intervals $(I\cup R)^{c}$ should be kept small
enough that the two-point correlation function should be dominant in the
entropy formula.

The generalization is straightforward. The entropy is

$\begin{split}S_{\text{gen}}\left(\bigcup_{k=1}^{n}(I_{k}\cup
R_{k})\right)&=\sum_{k=1}^{n-1}\left[\frac{\phi(x_{I_{k}})}{4G_{N}}+S_{\text{mat}}(x_{I_{k}},x^{\prime}_{R_{k}})+S_{\text{mat}}(x_{R_{k}},x^{\prime}_{I_{(k+1)}})+\frac{\phi(x^{\prime}_{I(k+1)})}{4G_{N}}\right]\\\
&=\sum_{k=1}^{n-1}\frac{1}{2}\left[\tilde{S}_{\text{gen}}(x_{I_{k}},x^{\prime}_{R_{k}})+\tilde{S}_{\text{gen}}(x_{R_{k}},x^{\prime}_{I_{(k+1)}})\right],\end{split}$
(63)

with the identification $x^{\prime}_{I_{n}}\equiv x^{\prime}_{I_{1}}$. Again
the $\tilde{S}_{\text{gen}}$ is the symmetric entropy (56). The extremization
can be done in each $R_{k}$ as above.

###  5.3 Finite temperature entropy

The Casimir energy changes in the energy momentum tensor Aguilar-
Gutierrez:2021bns ; Levine:2022wos  due to the change in $\theta$ periodicity.
The stress tensor of a thermal state with $2\pi n$ periodicity of $\theta$
coordinate is given by

${\tau}_{\pm\pm}=\frac{c}{48\pi
L^{2}}\left(\frac{2\pi}{{\beta}}\right)^{2}-\frac{c}{48\pi n^{2}L^{2}}.$ (64)

The energy momentum tensor in the dS2 background is

$\langle T_{\pm\pm}\rangle=\frac{c}{48\pi
L^{2}}\left(\frac{2\pi}{{\beta}}\right)^{2}+\frac{c}{48\pi
L^{2}}\left(1-\frac{1}{n^{2}}\right).$ (65)

The dilaton takes the form

$\Phi({\tau},\theta)={\alpha}\frac{\cos\theta}{\cos{\tau}}-K_{n}\left({\tau}\tan{\tau}+1\right)+\frac{cG_{N}}{3},$
(66)

with

$K_{n}=\frac{cG_{N}}{3}\left(\frac{2\pi}{{\beta}}\right)^{2}+\frac{cG_{N}}{3}\left(1-\frac{1}{n^{2}}\right).$
(67)

When $n=1$, $K_{n}$ reduces to $K$ defined in the previous section. Imposing
the boundary condition at ${\tau}=\pi/2$ that the dilaton describes
asymptotically pure de Sitter space implies

$K_{n}=\frac{\bar{\phi}L}{\pi}\left(b-\frac{1}{b}\right),\qquad{\alpha}=\frac{\bar{\phi}L}{2}\left(b+\frac{1}{b}\right),$
(68)

where $b$ is the parameter introduced in (17).

To compute the entropy, we perform a similar transformation as (30)

$y^{\pm}=\exp\left(-\frac{2\pi}{B_{n}L}x^{\pm}\right),\qquad\frac{2\pi}{B_{n}}=\sqrt{\left(\frac{2\pi}{{\beta}}\right)^{2}-\frac{1}{n^{2}}},$
(69)

so that the energy momentum tensor vanishes in $y_{n}^{\pm}$ coordinates. The
matter entropy for an interval $R_{i}=(x^{\pm}_{1,R_{i}},x^{\pm}_{2,R_{i}})$
is same as before but with $B_{n}$,

$\displaystyle S_{\text{mat},n}(R_{i})$
$\displaystyle=\frac{c}{6}\log\left[\frac{2L^{2}}{\epsilon_{\text{uv}}^{2}}\left(\sinh\frac{\pi}{B_{n}L}{\Delta}x^{+}_{R_{i}}\sinh\left(-\frac{\pi}{B_{n}L}{\Delta}x^{-}_{R_{i}}\right)\right)\right]$
$\displaystyle-\frac{c}{6}\log\left[\left(\frac{2\pi}{B_{n}}\right)^{2}\cos\left(\frac{x_{1,R_{i}}^{+}+x_{1,R_{i}}^{-}}{2L}\right)\cos\left(\frac{x_{2,R_{i}}^{+}+x_{2,R_{i}}^{-}}{2L}\right)\right]$
(70)

Both small ${\beta}$ limit and large $n$ limit lead to the same entropy
formula as in the previous section, exception that the relative temperature
now satisfies $T_{R}^{2}=T^{2}-(T_{\rm GH}/n)^{2}$. Also, in the large
${\beta}$ limit it reduces to well-known multiverse entropy formula in Ref.
Aguilar-Gutierrez:2021bns . We are interested in the entropy of $n$ regions in
inflating patches as in Fig. 4. Thus
$S_{\text{total}}=\sum_{i=1}^{n}S_{\text{mat},n}(R_{i})$.

The generalized entropy in $n$ black hole case is given in (63). We have taken
all $R_{i}$’s as $n$ copies of a single region. Since $B_{n}\rightarrow\beta$
at high temperature, we just have $n$ identical islands of previous section.

${\tau}_{I_{i}}\cong
0,\qquad\theta_{I_{i}}=(2i-1)\pi-\frac{2{\beta}}{\pi^{2}}\qquad(i=1,...,n)$
(71)

The Page temperature is

$T_{\text{Page}}\cong\sqrt{\frac{3\phi_{0}}{2(\pi+2)\pi^{2}cG_{N}}}+\frac{\pi-2\theta_{R_{i}}}{(\pi+2)\pi},$
(72)

for each black hole.

##  6 Discussion

We have considered the Ginsparg–Perry–Bousso–Hawking mechanism, which provides
a good scenario for proliferating de Sitter multiverses. Gravitational
instanton effect can attach a handle to the spatial section of de Sitter
space, connecting the two poles of dS to form the Nariai limit of the
Schwarzschild de Sitter space. In two-dimensions, by coupling the JT gravity
to a conformal matter, we let the black hole evaporate at finite CFT
temperature above the Gibbons-Hawking temperature, using the
Balasubramanian–Kar–Ugajin solution.

The BKU solution includes the backreaction from the semiclassical expectation
value of the stress tensor for a thermal state in the global conformal
coordinates of de Sitter space. We have computed the entanglement entropy of
an interval in the expanding patch of dS2 by a coordinate transformation to a
vacuum state and demonstrated that it becomes zero at the end due to the
purification of the states by islands.

Then, the generation of multiple de Sitter spaces was considered and the
entropy of them was investigated, including the change in the Casimir energy.
We showed that after the black holes have evaporated completely, the entropy
of each new universe vanishes. It implies that each universe is locally pure
and does not know the existence of other multiverses, though other
possibilities exist.

We close the discussion by comparing two versions of de Sitter multiverse
evolution scenarios. In the Bousso–Hawking (BH) model Bousso:1998bn ;
Bousso:2002fq  the multiverse is described by fluctuation of Foruier modes
along the $S^{1}$ direction, which is the spatial section of dS2 space. That
is, the dilaton is given as

$\phi=\frac{1}{\Lambda_{2}}\left(1-2\epsilon_{\rm
BH}\sum_{n}\sigma_{n}(\tau)\cos n\theta\right),$ (73)

with a small perturbation parameter $\epsilon_{\rm BH}$ and modified
cosmological constant $1/\Lambda_{2}=(1-N/(3\Lambda^{2}))/\Lambda^{2}$. $N$ is
the number of scalar fields. They contribute to the one-loop effects in the
effective action and their energy momentum tensor gives rise to the
fluctuation $\sigma_{n}(\tau)$. The profile $\sigma_{n}(\tau)$ is determined
by the solution to the two-dimensional Einstein equation.

The Bousso-Hawking solution is very similar to the BKU solution in the zero
temperature limit and for $n=1$. Indeed the classical solution matches
$\sigma_{1}(\tau)=\cos^{-1}\tau$ and both solutions consider the fluctuation
around it. However the classical solution of dilaton (10) can only describe
the sourceless part. In the BKU solution, the constant energy momentum tensor
from matter source modifies the dilaton but the sourced solution is
independent on $\theta$ and only depends on ${\tau}$. Thus the higher Fourier
mode in $\theta$ is not possible in BKU solution.

For future directions, we mention that though we have not studied in this
article, one may consider the entanglement between the disjoint de Sitter
spaces. The entropy formula has been proposed in Balasubramanian:2020coy ;
Balasubramanian:2021wgd . It would also be interesting to understand the
application of the entanglement islands in the recent proposal of the de
Sitter holography Susskind:2021dfc ; Susskind:2021esx ; Susskind:2021omt ;
Shaghoulian:2021cef ; Shaghoulian:2022fop .

###### Acknowledgements.

This work is supported by the grant NRF-2018R1A2B2007163 of National Research
Foundation of Korea.

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