CPHT-RR066.102023, October 2023

Closed FRW holography: A time-dependent
ER=EPR realization

Victor<EMAIL_ADDRESS>Hervé
<EMAIL_ADDRESS>François
<EMAIL_ADDRESS>and Nicolaos<EMAIL_ADDRESS>
1 CPHT, CNRS, Ecole polytechnique, IP Paris,
F-91128 Palaiseau, France

2 Department of Physics, University of Cyprus,
Nicosia 1678, Cyprus

Abstract

We extend a recent de Sitter holographic proposal and entanglement entropy
prescription to generic closed FRW cosmologies in arbitrary dimensions, and
propose that for large classes of bouncing and Big Bang/Big Crunch
cosmologies, the full spacetime can be encoded holographically on two
holographic screens, associated to two antipodal observers. In the expanding
phase, the two screens lie at the apparent horizons. In the contracting phase,
there is an infinite number of possible trajectories of the holographic
screens, which can be grouped in equivalence classes. In each class the
effective holographic theory can be derived from a pair of “parent” screens on
the apparent horizons. A number of cases including moduli dominated
cosmologies escape our discussion, and it is expected that two antipodal
observers and their associated screens do not suffice to reconstruct these
cosmologies. The leading contributions to the entanglement entropy between the
screens arise from a minimal extremal trapped or anti-trapped surface lying in
the region between them. This picture entails a time-dependent realization of
the ER=EPR conjecture, where an effective geometrical bridge connecting the
screens via the minimal extremal surface emerges from entanglement. For the
Big Crunch contracting cases, the screens disentangle and the geometrical
bridge closes off when the minimal extremal trapped sphere hits the Big Crunch
singularity at a finite time before the collapse of the Universe.
Semiclassical, thermal corrections are incorporated in the cases of radiation
dominated cosmologies.

> ###### Contents

  1. 1 Introduction
  2. 2 Basics of closed FRW cosmology
    1. 2.1 Cosmological evolutions
    2. 2.2 Penrose diagrams
    3. 2.3 Horizons
  3. 3 Holographic proposal for cosmologies with $|\gamma|\geq 1$
    1. 3.1 Bousso bound
    2. 3.2 Equivalent configurations
    3. 3.3 Bilayer proposal
  4. 4 Time-dependent ER=EPR
    1. 4.1 The two-screen system in cosmologies with $|\gamma|\geq 1$
    2. 4.2 The single-screen system in bouncing cosmologies with $\gamma<-1$
    3. 4.3 The single-screen system in Big Bang/Big crunch cosmologies with $\gamma\geq 1$
    4. 4.4 Maximin spheres
    5. 4.5 Equivalence of holographic constructions
    6. 4.6 Discussion
  5. 5 Big Bang/Big Crunch cosmologies with  $1/(n-1)\leq\gamma<1$
  6. 6 Entanglement entropy at the semiclassical level
  7. 7 Conclusion
  8. A Scale factor in closed FRW cosmology
  9. B Area extremization in a causal diamond
    1. B.1 Extrema of the area defined in the Penrose diagram
    2. B.2 Minima, maxima and saddle points of the area defined in a causal diamond
    3. B.3 Minima, maxima and saddle points as solutions of an extremization problem
  10. C Thermodynamics and FRW cosmology

##  1 Introduction

The holographic principle [1, 2] has given us powerful tools to study quantum
gravitational systems. There are by now robust non-perturbative formulations
of string theory and M-theory on certain backgrounds such as Matrix theory [3]
and the Anti-de Sitter space/conformal field theory (AdS/CFT) correspondence
[4, 5, 6]. Extending such holographic formulations to more generic cases, one
would hope to gain insight into difficult problems of quantum gravity
associated with black hole physics, the information loss paradox and the
initial singularity in cosmology.

Indeed thus far, most concrete holographic examples involve asymptotically AdS
or flat backgrounds characterized by conditions of “asymptotic coldness” [7,
8], a property that requires the fluctuations of the metric and other bulk
fields to tend to zero as the boundary of space is approached. Thanks to this
property, asymptotic, weakly coupled gravity regions can be established, where
string theoretic tools and symmetries can be applied in order to formulate
precise observables and construct the holographic dictionary. Still very
little is known about the exact nature of the holographic dual theories
describing cosmological backgrounds, where fluctuations persist across the
entire region of space and a spatial boundary may be absent. Such cosmological
backgrounds include accelerating Universes dominated by dark energy or a
positive cosmological constant, closed FRW cosmologies with Big Bang and Big
Crunch singularities, as well as our own Universe with early inflationary and
high temperature phases. It is thus important to understand how to formulate
holography for cosmological backgrounds and develop the holographic dictionary
describing the bulk spacetime.

A significant step towards this direction has been the recent understanding of
a deep connection between geometry and quantum entanglement [9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], culminating with the formulation
of a well defined prescription for the calculation of the fine grained entropy
of gravitational systems [10, 11, 14, 15, 17, 18, 22, 23]. The calculation
involves extremizing a generalized entropy formula with both geometrical area
and semiclassical entropy contributions. Starting with [10, 11], the
prescription has been applied with great success to compute the entanglement
entropy of strongly coupled CFT systems, using the AdS/CFT correspondence, and
later on the entropy of Hawking radiation produced during the evaporation of a
black hole [23, 24, 25, 26, 27, 28]. There have been also interesting attempts
to apply it in the context of cosmology, see e.g. [29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40]. Recently, it has also been applied in the context of
static patch holography for de Sitter space [41, 42, 43, 44] in order to
compute the fine grained entropy of subsystems of the dual holographic theory
[45, 46, 47, 48].111See also [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]
for related interesting work.

Recall that the static patch holographic proposal for de Sitter space entails
anchoring holographic screens on the two cosmological horizons associated with
comoving observers at antipodal points of the spatial sphere. The cosmological
horizons bound the two static causal patches of these observers, respectively.
Until recently, only very general properties of the holographic dual theory
could be established, based on symmetries [41, 42, 43, 44, 61, 62, 63, 64] and
the holographic covariant entropy bound [65, 66]. In parallel, the importance
of observers in quantum cosmology has been a subject of great interest,
leading to important results on the gravity side that may be applicable in the
context of holography [67, 68, 69, 70].

Two prescriptions for holographic entropy computations have been put forward,
the monolayer and the bilayer proposals [45, 46, 47], uncovering new
interesting properties of the holographic dual system and its subsystems. In
the monolayer proposal, the leading geometrical contributions and the quantum
corrections to the generalized entropy arise solely from the region between
the cosmological horizons, lying in the exterior of the two static patches. In
the bilayer proposal, the generalized entropy receives geometrical and quantum
contributions from the exterior as well as the two interior regions. In [48],
we argued that the bilayer prescription is the consistent one and leads to a
stronger holographic conjecture, where the entire de Sitter spacetime, the two
interior regions and the exterior region, can be reconstructed in terms of the
holographic dual on the horizons. In a direct manifestation of the ER=EPR
paradigm [16], quantum entanglement connects the two static patches via
effective bridges that span the exterior causal diamond region associated with
the two screens. In particular, the entanglement wedge of the screen with the
larger quantum area extends and covers the exterior causal diamond region
[48]. On the other hand, the entanglement wedge of the screen with the smaller
quantum area confines to its interior region. The entanglement wedge of the
full two-screen system consists of complete bulk Cauchy slices, which account
for the full set of bulk degrees of freedom.

In this paper, we extend the causal patch holographic proposal and the bilayer
holographic entropy prescription to generic closed FRW cosmologies in
arbitrary dimensions, which are characterized by a more involved horizon
structure.222See [71, 72, 73, 74, 75, 76, 77, 78, 79] for other related
attempts. We consider both bouncing, non-singular cosmologies, for which the
perfect fluid index satisfies $-1<w<2/n-1$, where $n$ is the number of spatial
dimensions, as well as Big Bang/Big Crunch cosmologies with $2/n-1<w\leq
4/n-1$. Given a pair of comoving observers at two antipodal points of the
spatial sphere, one can identify particle and event horizons delimiting the
corresponding causal patches, and two apparent horizons bounding the regions
of trapped and anti-trapped surfaces in between them. Unlike the de Sitter
case ($w=-1$) and the Big Bang/Big Crunch case $w=4/n-1$, the apparent
horizons are timelike and lie inside the causal patches of the observers. They
divide complete bulk Cauchy slices into three parts (except for those passing
through the bifurcation point, which are divided into two parts), two interior
parts with the topology of spherical caps and a third part in between the
apparent horizons with the topology of a barrel. In the expanding phase of the
cosmologies, the covariant entropy conjecture [65, 66] motivates us to place a
holographic screen on each of the two apparent horizons, respectively – see
also, [73, 76, 78]. In the contracting phase of the cosmologies, on the other
hand, the screens can be pushed farther into the region between the apparent
horizons, which is a region of trapped spheres, as long as they remain in the
causal patch of the corresponding observer, respectively. So in contrast to
the de Sitter case, there is an infinite number of possible screen
trajectories in the contracting phase of the cosmology, leading to different
holographic constructions in terms of two-screen quantum mechanical systems.
As we will argue, however, these holographic constructions can be grouped in
equivalent classes. In each class, the holographic effective theory on the two
screens can be derived from the theory on a pair of “parent” screens on the
apparent horizons. All equivalent screen configurations lead to the same
predictions for the fine grained entropy of certain gravitational subsystems,
which emerge holographically and are associated to single screen subsystems.

The two screen trajectories bound the left and right interior regions in the
causal patches of the observers, respectively. A third exterior region between
the trajectories provides an effective bridge between the single screen
subsystems, signaling quantum entanglement, in accordance with the ER=EPR
conjecture [16]. Unlike the de Sitter case, the two screens are not always out
of causal contact in these more generic cases. Indeed, the screens can
exchange energy and information for a certain period of time when they lie in
the overlap region of the two causal patches. Therefore, the holographic
degrees of freedom on them can interact via time dependent interaction terms.
Another crucial difference with the de Sitter case is that the area of the
screens does not remain constant as these evolve along their trajectories.
This suggests that the evolution of the holographic theory is not unitary, but
instead amounts to a sequence of mappings between Hilbert spaces of varying
dimensionality – see [80, 81] for similar discussions.

We show how the bilayer prescription for holographic entanglement entropy
calculations can be beautifully extended to these general closed FRW
cosmologies and adapted to incorporate quantum corrections at the
semiclassical level. We then proceed to compute the fine grained entropy of
the two-screen system and the single screen subsystems to leading geometrical
order, and determine the entanglement wedges. For the two-screen system, we
find that the entropy vanishes to leading order, and the entanglement wedge
comprises of complete bulk Cauchy slices passing through the screens. Based on
this result, we argue that the entropy vanishes to all orders, in the cases
where the bulk state on full Cauchy slices is pure. The entropy of the single
screen subsystems receives geometrical contributions from the exterior region
only. In the contracting (expanding) phase of the cosmology, we find that the
minimal extremal homologous surface is the minimal area trapped (anti-trapped)
sphere, which lies on the boundary of the causal diamond in the exterior
region. This minimal area sphere solves a constrained extremization problem
[48], where Lagrange multipliers and auxiliary fields are introduced enforcing
all extremal surfaces to lie in the exterior causal diamond. Our
investigations lead us to conjecture that the entire cosmological spacetime
can be holographically encoded on the two screens and reconstructed from the
holographic data. Moreover, the computations of the entropies provide a nice
time-dependent realization of the ER=EPR conjecture. Quantum entanglement
creates an effective geometrical bridge via the minimal trapped or anti-
trapped sphere in the exterior causal diamond. Indeed, the entanglement wedges
of the single screen subsystems extend in the exterior region via lightlike
segments joining the screen with the minimal extremal sphere. Therefore, the
latter is nothing but the intersection of the two entanglement wedges. In the
contracting phase of the Big Bang/Big Crunch cosmologies, the minimal extremal
sphere hits the Big Crunch singularity at a certain time before the screens
reach the Big Crunch singularity. Just at this moment the bridge closes off
(since the minimal area sphere collapses to zero size), leading to
disentanglement. In the expanding phase of the bouncing cosmologies, the
entropy of a single screen saturates an upper bound despite the fact that the
areas of the screens grow to infinity in the far future.

On the other hand for Big Bang/Big Crunch cosmologies with $4/n-1<w\leq 1$,
the observers at the antipodal points are separated by particle and/or event
horizons and remain out of causal contact for the entire cosmological
evolution. We argue that the screens in these cases must be placed on the
boundaries of the causal patches, and they never coincide. The two-screen
system is always in a mixed state, and the entropies of subsystems do not
receive exterior geometrical contributions. Only the interior causal patch
regions can be encoded holographically in these cases, on each screen,
respectively. Two observers do not suffice to reconstruct the full bulk
spacetime.

We then proceed to study semiclassical and thermal corrections at the 1-loop
level, focusing on radiation dominated closed Big Bang/Big Crunch cosmologies.
We show how to incorporate the $(G\hbar)^{0}$ corrections due to the coarse
grained, thermal entropy carried by the gas of particles in the bulk. The
first corrected entropy of a single screen subsystem behaves similarly to the
leading geometrical entropy. In the contracting phase of the cosmology, the
first order corrected entropy decreases and eventually gets dominated by the
quantum corrections when the classical extremal surface reaches the Big Crunch
singularity.

Let us stress that in this work we extend and apply the bilayer proposal for
holographic entropy computations and study the consequences for the closed FRW
cosmological spacetimes under study, without providing a detailed proof or
derivation. It would be interesting to derive direct evidence for this
proposal by studying gravitational replica path integrals in the cosmological
context.

The plan of the paper is as follows. In Section 2, we review some basic
properties of closed FRW cosmologies and their horizon structures. We then
proceed with a proposal to extend the static patch holographic proposal of de
Sitter space to closed FRW cosmologies in arbitrary dimensions in Section 3.
We discuss both bouncing and Big Bang/Big Crunch cosmologies for which the
Penrose diagrams are taller than wide. In the expanding phase of the
cosmologies, we argue that the holographic screens must be placed on the two
apparent horizons, in the causal patches of the comoving observers at two
antipodal points of the spatial sphere. In the contracting phase of the
cosmology, the screens can be pushed farther into the region of trapped
spheres between the apparent horizons, leading to different holographic
constructions. The latter can be grouped in equivalent classes, with the
effective holographic theory on them derived from the theory on a “parent”
two-screen system at the apparent horizons. We then proceed to generalize the
bilayer entropy prescription to these more general cases and specify the
extremization problems enforcing all extremal surfaces to lie in the two
interior and exterior causal diamond regions. In Section 4, we compute the
leading geometrical fine grained entropy of the two-screen system and the
single screen subsystems. These lead to an interesting interpretation of the
behavior of the fine grained entropy of a single screen in terms of a time-
dependent version of the “ER=EPR” relation. For Big Bang/Big Crunch
cosmologies with Penrose diagrams wider than tall, we argue that the screens
must be placed on the cosmological horizons in Section 5, and argue that the
two-screen system cannot encode the full spacetime in these cases. In Section
6, we incorporate semiclassical, thermal corrections in radiation dominated
cosmologies, applying our prescription in Section 3. Our conclusions are
presented in Section 7. In Appendix A, we derive the expression of the scale
factor of closed FRW cosmologies in arbitrary dimensions, as a function of the
conformal time and in Appendix C, we review how thermodynamical relations in
radiation dominated FRW cosmologies can be derived from an effective action
corrected at the 1-loop level. In Appendix B, we specify the area
extremization problem in the causal diamond in the exterior region and
determine the minimal extremal sphere homologous to a single screen.

##  2 Basics of closed FRW cosmology

In order to motivate in the next section our proposals for holographic systems
dual to closed FRW cosmological universes, we begin our discussion with a
brief survey of basic features of closed FRW cosmologies, the associated
Penrose diagrams and horizon structures.

###  2.1 Cosmological evolutions

We consider an $(n+1)$-dimensional closed FRW cosmology, where $n\geq 2$. In
conformal gauge, the spacetime metric can be written as

${\rm d}s^{2}=a^{2}(\eta)\left(-{\rm d}\eta^{2}+{\rm
d}\Omega^{2}_{n}\right)\\!,$ (2.1)

where $\eta$ is the conformal time, $a(\eta)$ is the scale factor and ${\rm
d}\Omega^{2}_{n}$ is the metric of the unit $n$-dimensional sphere Sn. In this
work, we restrict to an FRW cosmology induced by a single perfect fluid of
energy density $\rho$ and pressure $p$, satisfying the state equation

$p=w\rho,$ (2.2)

where the constant $w\in[-1,1]$ is the perfect fluid index. As reviewed in
Appendix A, the energy density satisfies

$\rho={C\over a^{n(1+w)}},$ (2.3)

where $C>0$ is a constant. The evolution of the scale factor depends
drastically on $w$, which admits a critical value

$w_{\rm c}=-1+{2\over n}\in(-1,0].$ (2.4)

Throughout this work, we will focus on the generic case where $w\neq w_{\rm
c}$, which yields333The scale factor evolution for $w=w_{\rm c}$ can be found
in Eq. (A.19).

$a(\eta)=a_{0}\\!\left(\sin{\eta\over|\gamma|}\right)^{\gamma},\leavevmode\nobreak\
\leavevmode\nobreak\ \quad\eta\in\big{[}0,|\gamma|\pi\big{]},$ (2.5)

where we have defined

$a_{0}=\left({n(n-1)\over 16\pi C}\right)^{1\over n(w_{\rm
c}-w)},\qquad\gamma={2\over n(w-w_{\rm c})}={2\over n(1+w)-2}.$ (2.6)

Qualitatively, the cosmological evolution is as follows:

$\bullet$ When $w$ increases from $-1$ to $w_{\rm c}$, the parameter $\gamma$
decreases from $-1$ to $-\infty$,

$-1\leq w<w_{\rm c}\qquad\Longrightarrow\qquad-1\geq\gamma>-\infty.$ (2.7)

Since $\gamma<0$, the cosmological evolution bounces. The scale factor
decreases from an infinite value at $\eta=0$ and reaches its minimum $a_{0}$
at $\eta=|\gamma|\pi/2$. It then expands and becomes infinite at finite
conformal time $\eta=|\gamma|\pi$. The evolution is nowhere singular. The
particular case $w=-1$ ($\gamma=-1$) corresponds to a cosmological evolution
induced by a positive cosmological constant, i.e. a de Sitter spacetime.

$\bullet$ When $w$ increases from $w_{\rm c}$ to 1, the parameter $\gamma$
decreases from $+\infty$ to $1/(n-1)$,

$w_{\rm c}<w\leq 1\qquad\Longrightarrow\qquad+\infty>\gamma\geq{1\over n-1}.$
(2.8)

Since $\gamma>0$, the scale factor increases from a Big Bang singularity at
$\eta=0$ where it vanishes, and reaches its maximum $a_{0}$ at
$\eta=|\gamma|\pi/2$. It then decreases up to a Big Crunch singularity at
$\eta=|\gamma|\pi$, where it vanishes. Semiclassical bulk computations can be
trusted for a great part of the cosmological evolution as long as
$a_{0}>>l_{p}$. However, near the singularities, the geometrical description
breaks down, and so obtaining a holographic dual picture seems necessary to
understand the physics. The case $w=1$ ($\gamma=1/(n-1)$) corresponds to a
cosmological evolution induced by moduli fields. $w=1/n$ ($\gamma=2/(n-1)$)
corresponds to an evolution induced by radiation. Finally, $w=0$
($\gamma=2/(n-2)$ for $n\geq 3$) corresponds to an evolution induced by
massive non-relativistic matter.

###  2.2 Penrose diagrams

The spacetime metric (2.1) can be written in the explicitly SO($n$)-symmetric
form

${\rm d}s^{2}=a^{2}(\eta)\left(-{\rm d}\eta^{2}+{\rm
d}\theta^{2}+\sin^{2}(\theta){\rm d}\Omega_{n-1}^{2}\right)\\!,$ (2.9)

where $\theta\in[0,\pi]$ is a polar angle and ${\rm d}\Omega^{2}_{n-1}$ is the
metric of the unit $(n-1)$-dimensional sphere Sn-1. The causal structure of
spacetime can be clarified by drawing a Penrose diagram, which is the
rectangle parametrized by the coordinates $(\theta,\eta)$. Slices of constant
conformal time $\eta$ are spheres Sn. Moreover, every generic point
$(\theta,\eta)$ of the diagram corresponds to a sphere Sn-1. In the particular
cases where $\theta=0$ or $\theta=\pi$, Sn-1 reduces to a point, which is the
north pole or the south pole of Sn, respectively. When $|\gamma|\geq 1$, the
diagrams are taller than wide. In the bouncing case, they are shown in Figures
1(a) to 1(d)

 $\phantom{2|\gamma|}\pi$$0$$0$$\pi$

(a) $\gamma=-1$ i.e. $w=-1$.

 $\phantom{2}|\gamma|\pi$$0$$0$$\pi$

(b) $-1>\gamma>-2$ i.e. $-1<w<-1+{1\over n}$.

 $\phantom{|\gamma|}2\pi$$0$$0$$\pi$

(c) $\gamma=-2$ i.e. $w=-1+{1\over n}$.

 $x^{-}$$x^{+}$$\phantom{2}|\gamma|\pi$$0$$0$$\pi$

(d) $-2>\gamma>-\infty$ i.e. $-1+{1\over n}<w<w_{\rm c}$.

 $\phantom{2|\gamma|}\pi$$0$$0$$\pi$

(e) $\gamma=1$ i.e. $w={4\over n}-1$.

 $\phantom{2}\phantom{||}\gamma\pi$$0$$0$$\pi$

(f) $1<\gamma<2$ i.e. ${4\over n}-1>w>{3\over n}-1$.

 $\phantom{|\gamma|}2\pi$$0$$0$$\pi$

(g) $\gamma=2$ i.e. $w={3\over n}-1$.

 $x^{-}$$x^{+}$$\phantom{2}\phantom{||}\gamma\pi$$0$$0$$\pi$

(h) $\phantom{-}2<\gamma<+\infty$ i.e. ${3\over n}-1>w>w_{\rm c}$.

Figure 1: Penrose diagrams of closed FRW spacetimes, when $|\gamma|\geq 1$,
with the corresponding values of the fluid index $w\in[-1,w_{\rm
c})\cup(w_{\rm c},4/n-1]$. Cases (a) to (d) correspond to bouncing
cosmologies, while cases (e) to (h) correspond to Big Bang/Big Crunch
cosmologies. The causal patches of the pode and antipode are delimited by
their cosmological horizons, shown in dashed lines. The causal patch of the
pode appears in blue. The apparent horizons of the pode and antipode are the
diagonal line segments. They divide the Penrose diagrams in four triangular
domains in which the Bousso wedges are indicated. The axes of the light-cone
coordinates $x^{+}$, $x^{-}$ are shown in cases (d) and (h).

for various values of $\gamma$ in the range $-1\geq\gamma>-\infty$, i.e.
$-1\leq w<w_{\rm c}$. The diagrams are identical in the Big Bang/Big Crunch
case when $1\leq\gamma<+\infty$, i.e. $4/n-1\geq w>w_{\rm c}$, up to wavy
lines indicating the singularities in Figures 1(e) to 1(h). When
$1/(n-1)\leq\gamma<1$, i.e. $1\geq w>-1+4/n$, the Penrose diagram is wider
than tall, as shown in Figure 2.

$\gamma\pi$$0$$0$$\pi$ Figure 2: Penrose diagram of a closed FRW spacetime,
when $1/(n-1)\leq\gamma<1$. The cosmological evolution is of Big Bang/Big
Crunch type and the fluid index satisfies $1\geq w>{4\over n}-1$. The causal
patches and apparent horizons of the pode and antipode, as well as the Bousso
wedges, are shown as explained in Figure 1.

###  2.3 Horizons

In the following, we will refer to the north and south poles of Sn, which are
located at $\theta=0$ and $\theta=\pi$, as the pode and the antipode [44]. Let
us consider a pair of comoving observers sitting respectively at the pode and
the antipode. Their worldlines on the Penrose diagrams are the left and right
vertical lines. Each observer has a “particle horizon,” which delimits the
part of spacetime that he/she will ever be able to send a signal to. He/she
also has an “event horizon,” which delimits the part of spacetime that he/she
will ever be able to receive a signal from. In the Penrose diagrams, these
horizons for the observers at the pode and the antipode correspond to segments
on the lines:

| pode | antipode
---|---|---
 particle horizon | $\eta=\theta$ | $\eta=\pi-\theta$
 event horizon | $\eta=|\gamma|\pi-\theta$ | $\eta=|\gamma|\pi-\pi+\theta$
(2.10)

The particle and event horizons of the observers delimit their respective
causal patches. Their boundaries, or cosmological horizons, are shown in
dashed lines in Figs 1 and 2, while the causal patch of the pode is depicted
in blue. When $|\gamma|\geq 1$, the causal patches of the pode and antipode
overlap, while for $1/(n-1)\leq\gamma<1$ they are disconnected.

As will be seen in the next section, another notion, which is that of
“apparent horizons” of the observers, enters naturally in the discussion of
holographic dual descriptions of bulk cosmological evolutions. Strictly
speaking, apparent horizons are the boundaries of the union of all trapped
surfaces. However, in the present work, we extend the definition of apparent
horizons to also include the boundaries of the union of all anti-trapped
surfaces. Trapped (anti-trapped) surfaces are spacelike codimension 2 surfaces
whose areas decrease (increase) locally along any future timelike direction.
In the case of an FRW cosmology, thanks to the SO($n$) symmetry of the metric
(2.9), the apparent horizons can be determined by looking for the domains of
the Penrose diagram where all points $(\theta,\eta)$ correspond to trapped or
anti-trapped spheres Sn-1.

From Eq. (2.9), the area of Sn-1 located at $(\theta,\eta)$ is given by

${\cal
A}(\theta,\eta)=\omega_{n-1}\,a_{0}^{n-1}\left(\sin{\eta\over|\gamma|}\right)^{\gamma(n-1)}(\sin\theta)^{n-1},$
(2.11)

where $\omega_{n-1}$ is the area of the unit Sn-1. To discuss how it varies
locally along any future timelike direction starting from $(\theta,\eta)$, it
is convenient to define light-cone coordinates

$x^{+}=\eta+\theta,\qquad x^{-}=\eta-\theta.$ (2.12)

A sphere Sn-1 is trapped if its area decreases when both $x^{+}$ and $x^{-}$
increase. Since

${\partial{\cal A}\over\partial
x^{\pm}}=\operatorname{sign}(\gamma)\,{n-1\over
2}\,\omega_{n-1}\,a_{0}^{n-1}\left(\sin{\eta\over|\gamma|}\right)^{\gamma(n-1)-1}(\sin\theta)^{n-2}\,\sin\\!\Big{(}\theta\pm{\eta\over\gamma}\Big{)},$
(2.13)

this is the case when both inequalities

$\operatorname{sign}(\gamma)\sin\\!\Big{(}\theta\pm{\eta\over\gamma}\Big{)}\leq
0$ (2.14)

are satisfied. Similarly, the sphere Sn-1 is anti-trapped if its area
increases when both $x^{+}$ and $x^{-}$ increase, which is true when the
reversed inequalities are met. The apparent horizons are the sets of points
that saturate either of these bounds. As a result, they are the two diagonal
lines of the Penrose diagrams,

${\eta\over|\gamma|}=\theta,\qquad{\eta\over|\gamma|}=\pi-\theta.$ (2.15)

The rectangular Penrose diagrams are thus divided in four triangular regions,
as shown in Figures 1 and 2. In each triangle, the constant signs of
$\partial{\cal A}/\partial x^{+}$ and $\partial{\cal A}/\partial x^{-}$ are
symbolized by so-called Bousso wedges [65]. The latter are 90∘ wedges, whose
sides indicate in which directions along the $x^{+}$ and $x^{-}$ axes the area
of the Sn-1 located at the tip decreases. For the bouncing (Big Bang/Big
Crunch) cosmologies, the lower triangle is the region of trapped (anti-
trapped) surfaces, while the upper triangle is the region of anti-trapped
(trapped) surfaces. We define the “apparent horizon of the pode” as the
boundary of the left triangle,

$\left\\{\\!\begin{array}[]{ll}\displaystyle{\eta\over|\gamma|}=\pi-\theta,&\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{if}\quad\displaystyle{\pi\over
2}\leq{\eta\over|\gamma|}\leq\pi,\\\
\displaystyle{\eta\over|\gamma|}=\theta,&\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{if}\quad\displaystyle
0\leq{\eta\over|\gamma|}\leq{\pi\over
2},\phantom{\\!\\!\overset{\displaystyle|}{|}}\end{array}\right.$ (2.16)

and the apparent horizon of the antipode as the boundary of the right
triangle,

$\left\\{\\!\begin{array}[]{ll}\displaystyle{\eta\over|\gamma|}=\theta,&\quad\mbox{if}\quad\displaystyle{\pi\over
2}\leq{\eta\over|\gamma|}\leq\pi,\\\
\displaystyle{\eta\over|\gamma|}=\pi-\theta,&\quad\mbox{if}\quad\displaystyle
0\leq{\eta\over|\gamma|}\leq{\pi\over
2}.\phantom{\\!\\!\overset{\displaystyle|}{|}}\end{array}\right.$ (2.17)

They are timelike when $|\gamma|>1$, spacelike when $1/(n-1)\leq\gamma<1$ and
lightlike when $|\gamma|=1$. In the latter case, they coincide with the
cosmological horizons of the pode and antipode. In all cases, we define the
bifurcate horizon as the intersection of the apparent horizon of the pode and
antipode. It is the Sn-1 at $(\theta,\eta)=(\pi/2,|\gamma|\pi/2)$.

##  3 Holographic proposal for cosmologies with $|\gamma|\geq 1$

In this section, we motivate our proposals for generalizing the de Sitter
static-patch holography conjecture [44] to closed FRW cosmologies, when
$|\gamma|\geq 1$. We argue that one possibility is to place a holographic
screen on the apparent horizon of the observer at the pode and another screen
on the apparent horizon of the antipode. An infinite number of other choices
can be considered in the contracting phases of the cosmological evolutions,
which may be equivalent. In Section 5, we will explain why, in the case where
$1/(n-1)\leq\gamma<1$, we do not expect that two observers at the pode and
antipode and associated screens suffice to describe holographically the bulk
cosmological evolution.

We will reach our conclusions by considering an arbitrary foliation ${\cal F}$
of spacetime, with SO$(n)$-symmetric Cauchy slices.444 We focus on
SO$(n)$-symmetric slices, exploiting the symmetries of the FRW metric, but
more general slices may also be considered. In these cases, the holographic
screens on the apparent horizons will not be spherical, but the Bousso bound
can still be applied. $\Sigma$ will denote a generic slice of ${\cal F}$. In
this section, we will assume that the state on $\Sigma$ is pure, and thus with
vanishing entropy. As will be seen in Section 6, our considerations also apply
to the case of a universe filled with radiation, for which the state on
$\Sigma$ is taken to be a mixed thermal state, and thus with entropy vanishing
at leading order $(G\hbar)^{-1}$ only. Our proposal, which relies on the
Bousso covariant-entropy conjecture [65] as well as the bilayer prescription
of Refs [45, 46, 47], will be tested in the forthcoming sections.

###  3.1 Bousso bound

Define ${\cal S}_{1}$ the Sn-1 sphere at the intersection of $\Sigma$ and the
apparent horizon of the pode, which is given in Eq. (2.16). We denote
$\Sigma_{1}$ the part of $\Sigma$ located to the left of ${\cal S}_{1}$ and
shown in bold red in the Penrose diagram in Figure 3.

$x^{-}$$x^{+}$${\cal
S}_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma$
(a) Case of a bouncing cosmology, $\gamma\leq-1$.

$x^{-}$$x^{+}$${\cal
S}_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma$
(b) Case of a Big Bang/Big Crunch cosmology, $\gamma\geq 1$.

Figure 3: An SO$(n)$-symmetric Cauchy slice $\Sigma$. The intersection of
$\Sigma$ with the apparent horizon of the pode is the Sn-1 sphere ${\cal
S}_{1}$. $\Sigma_{1}$ is the part of $\Sigma$ to the left of ${\cal S}_{1}$.
The coarse-grained entropy on $\Sigma_{1}$ is less than or equal to the
coarse-grained entropy on the future-directed light-sheet of ${\cal S}_{1}$,
which is shown in blue. When the part of $\Sigma$ located between the apparent
horizons of the pode and antipode lies in the region of anti-trapped surfaces,
${\cal S}_{1}$ and the screen on it must be retained on the apparent horizon
of the pode.

We first argue that the coarse-grained entropy on $\Sigma_{1}$ is expected to
be bounded from above by the area of ${\cal S}_{1}$ divided by $4G\hbar$. To
this end, consider a codimension 2 spacelike surface in spacetime and define a
light-sheet as a codimension 1 surface generated by light-like rays that begin
at the surface, extend orthogonally from it and are of non-positive expansion
[65]. Applying this definition to ${\cal S}_{1}$, we see from the orientation
of the Bousso wedge in the left triangular region of the Penrose diagram that
the future-directed light-sheet of ${\cal S}_{1}$ corresponds to the blue line
segment. The latter is parallel to the $x^{-}$ axis, starts from ${\cal
S}_{1}$ and ends on the cosmic line of the pode. Thanks to the second law of
thermodynamics, the coarse-grained entropy on $\Sigma_{1}$ is bounded from
above by the coarse-grained entropy passing through this future-directed
light-sheet. Then, applying the Bousso covariant-entropy conjecture [65], this
light-sheet being of non-positive expansion, the entropy passing through it is
expected to be bounded from above by the area of ${\cal S}_{1}$ divided by
$4G\hbar$. Combining the two statements above, we conclude that the coarse-
grained entropy on $\Sigma_{1}$ is bounded from above by the area of ${\cal
S}_{1}$ divided by $4G\hbar$. Moreover, $\Sigma_{1}$ and ${\cal S}_{1}$ in
particular are located in the causal patch of the pode. Hence, in order to
describe the state on $\Sigma_{1}$ in terms of a holographic dual theory, it
is natural to place a screen on the apparent horizon of the pode and identify
it with the boundary sphere ${\cal S}_{1}$. Notice, though, that the screen
may encode more information than that contained on $\Sigma_{1}$.

Unlike the de Sitter case $\gamma=-1$, when $|\gamma|>1$, the causal patch of
the pode extends in all four triangular regions of the Penrose diagram, as
seen in Figures 1(b) to 1(d) and 1(f) to 1(h). Since a screen associated with
the observer at the pode must be retained in the causal patch of the pode, one
may wonder whether Bousso’s rule allows ${\cal S}_{1}$ and the screen on it to
be pushed farther along $\Sigma$, in order to describe holographically a
greater part of it. Since $\Sigma$ is a spacelike slice, its part located
between the apparent horizons of the pode and antipode lies entirely either in
the top or bottom triangular region of the Penrose diagram. It is either in
the region of expansion or contraction of the cosmology. If it lies in the
expanding phase (the region of anti-trapped surfaces) the orientation of the
Bousso wedges shown in Figure 3 does not allow ${\cal S}_{1}$ to be pushed
farther to the right along $\Sigma$. Hence, we keep ${\cal S}_{1}$ and the
screen on it on the apparent horizon of the pode.

The situation when the part of $\Sigma$ between the apparent horizons lies in
the contracting phase (the region of trapped surfaces) is more involved. It
turns out to be relevant to define $\eta_{2}$ as the conformal time at which
$\Sigma$ intersects the apparent horizon of the antipode. Let us discuss the 2
types of cosmologies separately:

$\bullet$ Bouncing case $\gamma<-1$: Define $\eta_{\rm c}$ the conformal time
at which the apparent horizon of the antipode intersects the particle horizon
of the pode. We have

$\eta_{\rm c}={|\gamma|\over 1+|\gamma|}\,\pi\,,$ (3.18)

as indicated in Figure 4(a).

$\eta_{\rm c}$$\eta_{2}$$\eta_{2}$$|\gamma|\pi$$0$${\cal S}_{1}$${\cal
S}_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma$
(a) Case of a bouncing cosmology, $\gamma\leq-1$.

$\eta^{\prime}_{\rm c}$$\eta_{2}$$\eta_{2}$$\gamma\pi\phantom{||}$$0$${\cal
S}_{1}$${\cal
S}_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma$
(b) Case of a Big Bang/Big Crunch cosmology, $\gamma\geq 1$.

Figure 4: When the part of $\Sigma$ between the apparent horizons of the pode
and antipode lies in the region of trapped surfaces, ${\cal S}_{1}$ and the
screen on it can follow any timelike (possibly locally lightlike) trajectory
in the purple region, with its end points fixed.

When $\eta_{2}\in[\eta_{\rm c},|\gamma|\pi/2]$, the orientation of the Bousso
wedge in the bottom triangular region shows that ${\cal S}_{1}$, and the
screen it supports, may be pushed up to any point (actually sphere Sn-1) of
$\Sigma$ between the two apparent horizons. However, they cannot be pushed
farther along $\Sigma$, in the right triangular region, due to the orientation
of the Bousso wedge in this domain. Indeed, the coarse-grained entropy on
$\Sigma_{1}$ is still expected to be bounded from above by the area of ${\cal
S}_{1}$ divided by $4G\hbar$, while $\Sigma_{1}$ and ${\cal S}_{1}$ still
remain in the causal patch of the pode. Hence, the state on the entire
$\Sigma_{1}$ may be reconstructed from the state of the screen on ${\cal
S}_{1}$. We arrive at the same conclusions when $\eta_{2}\in[0,\eta_{\rm c}]$,
except that ${\cal S}_{1}$ can be pushed at most up to the particle horizon of
the pode, in order for the screen and the whole $\Sigma_{1}$ to stay in the
causal patch of the pode. In short, ${\cal S}_{1}$ can be placed along
$\Sigma$ in the purple region in Figure 4(a). Therefore, letting the Cauchy
slice $\Sigma$ evolve throughout the foliation ${\cal F}$, we have infinitely
many choices of trajectories for the screen associated with the observer at
the pode. ${\cal S}_{1}$ first follows any timelike (possibly locally
lightlike) curve that starts from the lower left corner of the Penrose diagram
in Figure 4(a), ends at the bifurcate horizon and lies in the purple region.
It then follows the apparent horizon of the pode.555Notice that given any
other foliation of the whole spacetime with SO$(n)$-symmetric Cauchy slices,
each slice intersects once and only once the trajectory of the screen. This
follows from the fact that the trajectory is nowhere spacelike.

$\bullet$ Big Bang/Big Crunch case $\gamma>1$: The analysis is similar in
every respect to the previous case. Let $\eta^{\prime}_{\rm c}$ be the
conformal time at which the apparent horizon of the antipode intersects the
event horizon of the pode. This definition leads to

$\eta^{\prime}_{\rm c}={\gamma^{2}\over 1+|\gamma|}\,\pi=|\gamma|\pi-\eta_{\rm
c},$ (3.19)

as indicated in the Penrose diagram in Figure 4(b). When
$\eta_{2}\in[\gamma\pi/2,\eta^{\prime}_{\rm c}]$, the Bousso wedge in the
upper triangular region allows ${\cal S}_{1}$ and the screen to be pushed up
to any point (actually sphere Sn-1) along $\Sigma$ between the two apparent
horizons. They cannot be pushed farther, in the right triangular region. For
the same reasons as before, the state on $\Sigma_{1}$ may be described
holographically from the state of the screen on ${\cal S}_{1}$. When
$\eta_{2}\in[\eta^{\prime}_{\rm c},\gamma\pi]$, the conclusion remains the
same except that ${\cal S}_{1}$ can be pushed at most up to the event horizon
of the pode. This has to be the case for $\Sigma_{1}$ and the screen to stay
in the causal patch of the pode. Another reason is that the light-sheet of
${\cal S}_{1}$ in the direction of the positive $x^{-}$ axis would otherwise
terminate at the Big Crunch singularity (instead of the cosmic line of the
pode), and so part of the coarse-grained entropy on $\Sigma_{1}$ would end up
at the singularity, without passing through the light-sheet. In summary,
${\cal S}_{1}$ can be placed on $\Sigma$ in the purple region in Figure 4(b).
When $\Sigma$ varies throughout ${\cal F}$, the screen first follows the
apparent horizon of the pode up to the bifurcate horizon, and then any
timelike (possibly locally lightlike) curve in the purple region, up to the
upper left corner of the Penrose diagram.${}^{\ref{foot}}$

Everything said so far for the observables accessible from the pode, as well
as the holographic screen on ${\cal S}_{1}$, can be readily adapted to the
case of the observer at the antipode and a second screen. Let ${\cal S}_{2}$
be the Sn-1 sphere at the intersection of $\Sigma$ and the apparent horizon of
the antipode, which is defined in Eq. (2.17). Denote $\Sigma_{2}$ the part of
$\Sigma$ located to the right of ${\cal S}_{2}$ in the Penrose diagram. The
state on $\Sigma_{2}$ may be reconstructed from the state of a second screen
located on ${\cal S}_{2}$. Moreover, ${\cal S}_{2}$ and its screen can be
pushed along $\Sigma$ in the region of trapped surfaces, provided they remain
in the causal patch of the observer at the antipode. In this way, they follow
a timelike (possibly locally lightlike) curve connecting the bifurcate horizon
and the origin (in the bouncing case) or end (in the Big Bang/Big Crunch case)
of the cosmic line of the observer at the antipode.

Notice that the trajectories of ${\cal S}_{1}$ and ${\cal S}_{2}$ intersect at
the bifurcate horizon and possibly elsewhere in the region of trapped
surfaces. When this occurs on a Cauchy slice $\Sigma$, the union of the
coincident screens should describe holographically the entire slice $\Sigma$,
since the latter is nothing but the union $\Sigma_{1}\cup\Sigma_{2}$. More
generally, when ${\cal S}_{1}$ and thus ${\cal S}_{2}$ on $\Sigma$ are in the
region of trapped surfaces, two cases can occur:

$\bullet$ If ${\cal S}_{1}$ is to the right of ${\cal S}_{2}$ in the Penrose
diagram, $\Sigma$ is still the union of $\Sigma_{1}$ and $\Sigma_{2}$ (with
some overlap) and thus the two-screen system should describe the state on the
entire slice $\Sigma$.

$\bullet$ If ${\cal S}_{1}$ is to the left of ${\cal S}_{2}$, the two-screen
system may still describe more than the states on $\Sigma_{1}$ and
$\Sigma_{2}$. To see why, let us apply Bousso’s rule to the complement of
$\Sigma_{1}$ in $\Sigma$. From the Bousso wedges in the region of trapped
surfaces and the right triangular region in Figure 4, we see that the coarse-
grained entropy on the complement of $\Sigma_{1}$ is bounded by the area of
${\cal S}_{1}$ divided by $4G\hbar$. The same remark applies to the complement
of $\Sigma_{2}$ in $\Sigma$ and ${\cal S}_{2}$. Since $\Sigma$ is now equal to
the union of these complementary slices (with some overlap), additivity of
coarse-grained entropy guaranties that the two-screen system has enough number
of degrees of freedom to describe the state on the entire $\Sigma$.

As can be seen from the Bousso wedges in Figure 3, the above argument
involving the complement of $\Sigma_{1}$ in $\Sigma$ and the screen on ${\cal
S}_{1}$ does not apply when ${\cal S}_{1}$ is in the left triangular region.
Hence, when this is the case, in order to describe the state on the greatest
part of the Cauchy slice $\Sigma$, the screen should be placed on the apparent
horizon of the pode so that $\Sigma_{1}$ is the largest. The argument
involving the complement of $\Sigma_{1}$ in $\Sigma$ does not apply either if
one tries to push ${\cal S}_{1}$ and the screen in the region of anti-trapped
surfaces.

###  3.2 Equivalent configurations

In order to investigate the possibility for various configurations of the
screens to be equivalent (in a sense that needs to be specified), let us begin
with a few remarks. The observer at the pode and the screen on ${\cal S}_{1}$
evolve along their own trajectories, which start and end at the same points in
the Penrose diagram in Figure 4. As a result, they share the same causal
patches shown in Figure 1. The same applies to the observer at the antipode
and the screen on ${\cal S}_{2}$. When one of the screens is in the causal
patch of the other, they can exchange energy and information, possibly via
time dependent interaction terms. The exchanges can be reciprocal when both
screens are in each other’s causal patch.666This is in contrast to the example
of the eternal AdS black hole, for which there are no interactions between the
two copies of the dual CFT.

Moreover, on each slice $\Sigma$ of ${\cal F}$, let us denote ${\cal S}_{\rm
L}$ the leftmost sphere between ${\cal S}_{1}$, ${\cal S}_{2}$. We also define
$\Sigma_{\rm L}$ the part of $\Sigma$ on the left of ${\cal S}_{\rm L}$.
Likewise, ${\cal S}_{\rm R}$ is the rightmost sphere between ${\cal S}_{1}$,
${\cal S}_{2}$, while $\Sigma_{\rm R}$ is the part of $\Sigma$ on the right of
${\cal S}_{\rm R}$. We claim that the states on $\Sigma_{\rm L}$ and
$\Sigma_{\rm R}$ may be reconstructed from the states of the screens on ${\cal
S}_{\rm L}$ and ${\cal S}_{\rm R}$, respectively. This is a trivial statement
when ${\cal S}_{1}$ is to the left of ${\cal S}_{2}$. Moreover, when ${\cal
S}_{1}$ is to the right of ${\cal S}_{2}$, the sphere ${\cal S}_{2}$ is
located on $\Sigma_{1}$, implying the second screen to be in the causal patch
of the pode, and thus in the causal patch of the first screen. In fact, both
screens are in the causal patch of the other. Since they are in causal
contact, they can exchange information, enabling ${\cal S}_{2}$ to serve as an
alternative location, $\tilde{\cal S}_{1}$, for a screen associated with the
observer at the pode. In this case, the screen on $\tilde{\cal S}_{1}$ is
expected to contain the information necessary to reconstruct the state on
$\tilde{\Sigma}_{1}$, the complement of $\Sigma_{2}$ in $\Sigma$. We thus have
${\cal S}_{\rm L}={\cal S}_{2}$, while $\Sigma_{\rm L}$ is the complement of
$\Sigma_{2}$ in $\Sigma$. Similarly, ${\cal S}_{\rm R}={\cal S}_{1}$ and
$\Sigma_{\rm R}$ is the complement of $\Sigma_{1}$ in $\Sigma$. Another way of
arguing that the roles of the screens can be swapped is as follows. If ${\cal
S}_{1}$ is to the right of ${\cal S}_{2}$, we must have ${\cal S}_{1}={\cal
S}_{2}$ on an earlier Cauchy slice of the foliation. On this earlier slice,
the degrees of freedom of the coincident screens can be exchanged with each
other. Hence, it is permitted to impose that, subsequently, the screen
associated with the pode is the leftmost, while that associated with the
antipode is the rightmost. In the following, we will adopt this convention.

At this stage, we have infinitely many possible choices of foliations ${\cal
F}$ and trajectories for ${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$.
Anticipating with results presented in Section 4, we now argue that these
configurations can be grouped in equivalence classes. In each class, the
effective holographic theory on the screens may be derived from a “parent”
setup, where both screens evolve along the apparent horizons, with the slicing
of spacetime corresponding to a new foliation. To arrive to these conclusions,
consider a Cauchy slice $\Sigma$ of ${\cal F}$, where ${\cal S}_{\rm L}$ is in
the purple region in Figure 5.

$x^{-}$$x^{+}$${\cal S}_{\rm L}$${\cal S}_{\rm
L}^{\prime}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Delta_{\rm
L}$${\cal S}_{\rm R}$${\cal S}_{\rm
R}^{\prime}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Delta_{\rm
R}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
L}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
R}$ (a) Case of a bouncing cosmology, $\gamma\leq-1$.

$x^{-}$$x^{+}$${\cal S}_{\rm L}$${\cal S}_{\rm
L}^{\prime}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Delta_{\rm
L}$${\cal S}_{\rm R}$${\cal S}_{\rm
R}^{\prime}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Delta_{\rm
R}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
L}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
R}$ (b) Case of a Big Bang/Big Crunch cosmology, $\gamma\geq 1$.

Figure 5: The screens on ${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$ can be
moved to any Sn-1 spheres along $\Delta_{\rm L}$ and $\Delta_{\rm R}$, which
are parallel to the $x^{+}$ and $x^{-}$ axes in the region of trapped
surfaces. The spheres of greatest areas along $\Delta_{\rm L}$, $\Delta_{\rm
R}$ are ${\cal S}^{\prime}_{\rm L}$, ${\cal S}^{\prime}_{\rm R}$, on the
apparent horizons.

Define $\Delta_{\rm L}$ the blue line segment that runs parallel to the
$x^{+}$ axis from ${\cal S}_{\rm L}$ to the apparent horizon of the pode.
Likewise, $\Delta_{\rm R}$ is parallel to the $x^{-}$ axis from ${\cal S}_{\rm
R}$ to the apparent horizon of the antipode. All “points” of $\Delta_{\rm L}$
and $\Delta_{\rm R}$ are spheres Sn-1, with different areas. Let us displace
the screens initially on ${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$ to any
spheres of $\Delta_{\rm L}$ and $\Delta_{\rm R}$, respectively. Notice that
the region between these two spheres and lying on any Cauchy slice of
spacetime has the topology of a barrel. In the spirit of the ER=EPR relation
[16], we expect that the existence of this region is a manifestation of
entanglement between the two boundary screens in the holographic picture.
Moreover, we will see in Section 4 that applying the bilayer prescription
presented below, one always obtains the same result for the fine-grained
entanglement entropy between the two screens, whatever their places on
$\Delta_{\rm L}$ and $\Delta_{\rm R}$. In addition, we will see that each such
choice leads to the same predictions concerning the fine-grained entropy of
certain gravitational bulk slices, which may be reconstructed from the
holographic theory on the screens. However, from the Bousso wedge in the
region of trapped surfaces, we see that the spheres on $\Delta_{\rm L}$ and
$\Delta_{\rm R}$ having the greatest areas, which we will refer to as ${\cal
S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm R}^{\prime}$ respectively, are at the
boundaries of $\Delta_{\rm L}$ and $\Delta_{\rm R}$ on the apparent horizons.
As a result, the screens on ${\cal S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm
R}^{\prime}$ must have more degrees of freedom than their counterparts on
${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$. This suggests that the effective
holographic theory on ${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$ can be obtained
by integrating out some degrees of freedom of the “parent” system on ${\cal
S}_{\rm L}^{\prime}\cup{\cal S}_{\rm R}^{\prime}$. This is reminiscent of an
RG flow in local quantum field theories.777However, as in the de Sitter case
$\gamma=-1$, we do not expect the holographic dual theory to these closed FRW
cosmologies to be local. Since the construction of ${\cal S}_{\rm L}^{\prime}$
and ${\cal S}_{\rm R}^{\prime}$ guaranties that they are spacelike separated,
we can choose a Cauchy slice $\Sigma^{\prime}$ of spacetime that passes
through them. Repeating all steps for every slice $\Sigma$ belonging to ${\cal
F}$, we generate a new foliation ${\cal F}^{\prime}$ of slices
$\Sigma^{\prime}$ intersecting the apparent horizons of the pode and antipode
at ${\cal S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm R}^{\prime}$, respectively.
It is always possible to choose ${\cal F}^{\prime}$ so that
$\Sigma^{\prime}=\Sigma$ for every slice $\Sigma$ where ${\cal S}_{\rm L}$ and
${\cal S}_{\rm R}$ are on the apparent horizons, i.e. ${\cal S}_{\rm L}={\cal
S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm R}={\cal S}_{\rm R}^{\prime}$. This is
in particular the case in the expanding phase of the cosmological evolution.
However, in general, the other slices $\Sigma^{\prime}$ of the foliation
${\cal F}^{\prime}$ do not belong to ${\cal F}$.

The state of the two-screen system on ${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$
can be found by evolving the screens along their arbitrary trajectories, slice
after slice in ${\cal F}$. Alternatively, we may evolve the screens along the
apparent horizons up to ${\cal S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm
R}^{\prime}$, slice after slice in ${\cal F}^{\prime}$, and then along
$\Delta_{\rm L}$ and $\Delta_{\rm R}$ up to ${\cal S}_{\rm L}$ and ${\cal
S}_{\rm R}$, while maintaining them spacelike separated.888In this second
stage, the screens do not lie in general on Cauchy slices of ${\cal F}$ or
${\cal F}^{\prime}$. Note that this is possible if we allow non-
differentiable, but continuous, trajectories. The final states of the two-
screen system in these two cases should be unitarily equivalent, with the same
entanglement, von Neumann entropies between the two screens.

Let us focus on a parent configuration. For each slice $\Sigma^{\prime}$ of
the foliation ${\cal F}^{\prime}$, let us denote $\eta_{\rm L}^{\prime}$ and
$\eta_{\rm R}^{\prime}$ the conformal times of the screens on the apparent
horizons defined in Eqs (2.16) and (2.17). The areas of the spheres ${\cal
S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm R}^{\prime}$ are given by

${\cal A}({\cal S}_{\rm
L}^{\prime})=\omega_{n-1}\,a_{0}^{n-1}\left(\sin{\eta_{\rm
L}^{\prime}\over|\gamma|}\right)^{(\gamma+1)(n-1)},\qquad{\cal A}({\cal
S}_{\rm R}^{\prime})=\omega_{n-1}\,a_{0}^{n-1}\left(\sin{\eta_{\rm
R}^{\prime}\over|\gamma|}\right)^{(\gamma+1)(n-1)}.$ (3.20)

They are constant in the de Sitter case $\gamma=-1$, which means that there is
a perfect compensation between the fact that the areas decrease as we approach
the worldlines of the observers, and the fact that they increase when the
conformal time approaches 0 or $|\gamma|\pi$. In the generic bouncing case
$\gamma<-1$, the increasing effect dominates: the area of ${\cal S}_{\rm
L}^{\prime}$ and ${\cal S}_{\rm R}^{\prime}$ start infinite at $\eta_{\rm
L}^{\prime}=\eta_{\rm R}^{\prime}=0$, reach their minimum values at $\eta_{\rm
L}^{\prime}=\eta_{\rm R}^{\prime}=|\gamma|\pi/2$ and then grow to infinity at
$\eta_{\rm L}^{\prime}=\eta_{\rm R}^{\prime}=|\gamma|\pi$. In the Big Bang/Big
Crunch case $\gamma\geq 1$, the areas of ${\cal S}_{\rm L}^{\prime}$ and
${\cal S}_{\rm R}^{\prime}$ grow from zero at $\eta_{\rm L}^{\prime}=\eta_{\rm
R}^{\prime}=0$, reach their maximum values at $\eta_{\rm L}^{\prime}=\eta_{\rm
R}^{\prime}=\gamma\pi/2$ and then decrease until they vanish at $\eta_{\rm
L}^{\prime}=\eta_{\rm R}^{\prime}=\gamma\pi$. In all cases, the total area of
the screens in Planck units should be a measure of the number of the degrees
of freedom on them, and the exponential of this a measure of the
dimensionality of the Hilbert space of the dual holographic theory. See Refs
[61, 62] for such proposals in the context of de Sitter holography. As a
result, the evolution of the holographic theory is not unitary, in the sense
that it corresponds to a sequence of maps between Hilbert spaces of different
dimensionalities. For the Big Bang/Big Crunch case, the dimensionality of the
Hilbert space is maximum when both screens are at the bifurcation point. Near
the singularities the dimensionality of the Hilbert space is very small, with
the holographic dual having very few degrees of freedom. In the bouncing
cases, the maximum dimensionality occurs when the screens are at the future or
past null infinity.

###  3.3 Bilayer proposal

We are ready to present our proposal for computing entanglement entropies
between complementary subsystems of the pair of screens, as well as the bulk
regions that may be reconstructed from such holographic subsystems.

Let us first repeat our notations and complete them. For any cosmological
evolution satisfying $|\gamma|\geq 1$, consider a foliation ${\cal F}$ of
spacetime, with SO$(n)$-symmetric Cauchy slices. In the Penrose diagram in
Figure 4, choose a nowhere spacelike curve that starts at the bottom left
corner, ends at the top left corner, is located in the purple region of the
contracting era, and follows the apparent horizon of the pode in the expanding
era. Similarly, choose a second curve, swapping the roles of left and right.
As shown in Figure 6,

${\cal S}_{\rm L}$${\cal S}_{\rm
R}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
E}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
L}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
R}$ (a) Case of a bouncing cosmology, $\gamma\leq-1$.

${\cal S}_{\rm L}$${\cal S}_{\rm
R}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
E}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
L}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Sigma_{\rm
R}$ (b) Case of a Big Bang/Big Crunch cosmology, $\gamma\geq 1$.

Figure 6: Trajectories of leftmost screen on ${\cal S}_{\rm L}$ (purple) and
rightmost screen on ${\cal S}_{\rm R}$ (orange). The interior region of the
pode is purple shaded, while the interior region of the antipode is orange
shaded. The exterior region is white.

each Cauchy slice $\Sigma$ of ${\cal F}$ intersects these trajectories at two
Sn-1 spheres, ${\cal S}_{\rm L}$ being the leftmost and ${\cal S}_{\rm R}$ the
rightmost. We denote $\Sigma_{\rm L}$ the part of $\Sigma$ to the left of
${\cal S}_{\rm L}$, $\Sigma_{\rm R}$ the part to the right of ${\cal S}_{\rm
R}$ and $\Sigma_{\rm E}$ the part between $\Sigma_{\rm L}$ and $\Sigma_{\rm
R}$. $\Sigma_{\rm L}$ and $\Sigma_{\rm R}$ have the topologies of spherical
caps and their boundaries are respectively ${\cal S}_{\rm L}$ and ${\cal
S}_{\rm R}$. The topology of $\Sigma_{\rm E}$ is that of a barrel and its
boundary is ${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$. We refer to the “interior
region of the pode,” which is shown in purple in Figure 6, the domain spanned
by all Cauchy slices $\Sigma_{\rm L}$. Likewise the “interior region of the
antipode,” which is shown in orange, is covered by all slices $\Sigma_{\rm
R}$. The remaining domain of spacetime, which we call the “exterior region,”
is spanned by all slices $\Sigma_{\rm E}$ and appears in white in Figure 6. We
will label these three regions as ${\rm L}$, ${\rm E}$ and ${\rm R}$,
respectively. Note that given $\Sigma\in{\cal F}$, all Cauchy slices
$\hat{\Sigma}$ of spacetime that pass through ${\cal S}_{\rm L}$ and ${\cal
S}_{\rm R}$ have equivalent causal structures: denoting $\hat{\Sigma}_{i}$,
$i\in\\{{\rm L},{\rm E},{\rm R}\\}$, the part of $\hat{\Sigma}$ located in
region $i$, the causal diamonds of $\Sigma_{i}$ and $\hat{\Sigma}_{i}$ are the
same.

Classical entropy: For any subregion $A$ of ${\cal S}_{\rm L}\cup{\cal S}_{\rm
R}$ on the Cauchy slice $\Sigma$, denote

$A_{\rm L}=A\cap{\cal S}_{\rm L},\qquad A_{\rm E}=A,\qquad A_{\rm
R}=A\cap{\cal S}_{\rm R}.$ (3.21)

For $i\in\\{{\rm L},{\rm E},{\rm R}\\}$, let $\chi_{i}$ be a codimension-2
surface of minimal extremal area that is homologous to $A_{i}$ and lies on a
Cauchy slice $\hat{\Sigma}_{i}$.999Notice that the homology constraint implies
that $\chi_{i}$ is anchored on the boundaries of $A_{i}$, namely
$\partial\chi_{i}=\partial A_{i}$. At leading order in $G\hbar$, the von
Neumann entropy $S$ of the subsystem on $A$, i.e. the entanglement entropy
between $A$ and its complement in ${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$,
satisfies

$S(A)={{\cal A}(\chi_{\rm L})+{\cal A}(\chi_{\rm E})+{\cal A}(\chi_{\rm
R})\over 4G\hbar}+{\cal O}((G\hbar)^{0}),$ (3.22)

where ${\cal A}(\chi_{i})$ is the area of $\chi_{i}$.

Let us stress that since $\chi_{i}$ must lie on some Cauchy slice
$\hat{\Sigma}_{i}$, one must look for it within the causal diamond of
$\Sigma_{i}$. Since this region has a boundary, extremizing the area
functional of surfaces homologous to $A_{i}$ requires the introduction of
Lagrange multipliers and auxiliary fields enforcing all extremal surfaces to
lie in the diamond, including its boundary [48]. If there exists a single
extremal-area surface in region $i$, it is $\chi_{i}$. If several exist,
$\chi_{i}$ is one with the smallest area.

Entanglement wedge: Let $\hat{\cal C}_{i}$ be the codimension-1
surface101010All codimensions are with respect to the whole
$(n+1)$-dimensional spacetime. on $\hat{\Sigma}_{i}$ bounded by $\chi_{i}$ and
$A_{i}$, i.e. satisfying $\partial\hat{\cal C}_{i}=\chi_{i}\cup A_{i}$.
Assuming entanglement wedge reconstruction [20], we have the following:

$\bullet$ The state on $\hat{\cal C}_{\rm L}\cup\hat{\cal C}_{\rm
E}\cup\hat{\cal C}_{\rm R}$ is dual to the state of the holographic subsystem
on $A$. Generically, they are mixed states.

$\bullet$ In particular, the von Neumann entropy of the state on $\hat{\cal
C}_{\rm L}\cup\hat{\cal C}_{\rm E}\cup\hat{\cal C}_{\rm R}$, i.e. the
entanglement entropy between $\hat{\cal C}_{\rm L}\cup\hat{\cal C}_{\rm
E}\cup\hat{\cal C}_{\rm R}$ and its complement in $\hat{\Sigma}$, equals the
von Neumann entropy of the holographic subsystem on $A$, i.e. the entanglement
entropy between $A$ and its complement in ${\cal S}_{\rm L}\cup{\cal S}_{\rm
R}$.

$\bullet$ The “entanglement wedge” of $A$, which is a spacetime region
reconstructible from the dual holographic subsystem on $A$, is the union of
the three causal diamonds of $\hat{\cal C}_{\rm L}$, $\hat{\cal C}_{\rm E}$,
$\hat{\cal C}_{\rm R}$.111111Notice that the union of causal diamonds is not
the causal diamond of a union. The former is a subset of the latter.

Notice that the choice of Cauchy slice $\hat{\Sigma}_{i}$ containing
$\chi_{i}$ is not unique, and so for the $\hat{\cal C}_{i}$’s. However, this
does not introduce any ambiguity in the definition of the entanglement wedge,
as the causal diamonds of all these surfaces $\hat{\cal C}_{i}$ are the same.
On the contrary, when several surfaces of minimal extremal areas exist in some
region $i$, meaning that they have equal areas, the causal diamonds of the
corresponding surfaces $\hat{\cal C}_{i}$ are in general distinct. In this
case, one is led to include quantum corrections to the classical, geometrical
entropy formula, in order to lift the degeneracy.

Semiclassical entropy: To take into account the first-order quantum
corrections, let us define the generalized entropy

$S_{\rm gen}(\chi_{\rm L},\chi_{\rm E},\chi_{\rm R})={{\cal A}(\chi_{\rm
L})+{\cal A}(\chi_{\rm E})+{\cal A}(\chi_{\rm R})\over 4G\hbar}+S_{\rm
semicl}\big{(}\hat{\cal C}_{\rm L}\cup\hat{\cal C}_{\rm E}\cup\hat{\cal
C}_{\rm R}\big{)},$ (3.23)

where $\chi_{i}$, $i\in\\{{\rm L},{\rm E},{\rm R}\\}$, is an arbitrary
codimension-2 surface homologous to $A_{i}$ and lying on a Cauchy slice
$\hat{\Sigma}_{i}$, while $\hat{\cal C}_{i}\subset\hat{\Sigma}_{i}$ is the
codimension-1 surface satisfying $\partial\hat{\cal C}_{i}=\chi_{i}\cup
A_{i}$. In the above expression, $S_{\rm semicl}$, which is of order
$(G\hbar)^{0}$, is the 1-loop contribution to the entropy of the bulk matter,
radiation and gravitational fields on $\hat{\cal C}_{\rm L}\cup\hat{\cal
C}_{\rm E}\cup\hat{\cal C}_{\rm R}$. The quantity $4G\hbar S_{\rm gen}$ can be
thought of as the “quantum area” of $\chi_{\rm L}\cup\chi_{\rm E}\cup\chi_{\rm
R}$, at the semiclassical level. As before, for any $i\in\\{{\rm L},{\rm
E},{\rm R}\\}$, the choice of Cauchy slice $\hat{\Sigma}_{i}$ is not unique,
and so for $\hat{\cal C}_{i}$. However, since the density matrices on
different $\hat{\cal C}_{i}$’s are unitarily related, $S_{\rm semicl}$ is
independent of the choice of $\hat{\cal C}_{i}$ and thus the definition of
$S_{\rm gen}$ is unambiguous.

To obtain the entanglement entropy at the semiclassical level, one has to
extremize the generalized entropy with respect to $\chi_{\rm L}$, $\chi_{\rm
E}$ and $\chi_{\rm R}$, and then select among the extrema the configuration
for which $S_{\rm gen}$ is minimal. This can be summarized by the expression,

$S(A)=\min{\rm ext}\;S_{\rm gen}(\chi_{\rm L},\chi_{\rm E},\chi_{\rm R})+{\cal
O}(G\hbar).$ (3.24)

Notice that in general the semiclassical entropy associated with bulk quantum
fields is not additive. More specifically, the semiclassical entropy on the
union of two surfaces is less than or equal to the sum of the entropies on
each surface [82]. Therefore, it is important to consider the 1-loop
correction on the union of surfaces in Eq. (3.23).

##  4 Time-dependent ER=EPR

In this section we compute the entanglement entropies associated with the two-
screen system on ${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$ and a single-screen
system on ${\cal S}_{\rm L}$ or ${\cal S}_{\rm R}$, for cosmologies with
$|\gamma|\geq 1$. The screens lie on SO$(n)$-symmetric Cauchy slices of the
foliation ${\cal F}$ and follow their respective trajectories, as described in
detail in Section 3.3 and shown in Figure 6. We denote the conformal
coordinates of ${\cal S}_{\rm L}$ by $(\theta_{\rm L},\eta_{\rm L})$ and those
of ${\cal S}_{\rm R}$ by $(\theta_{\rm R},\eta_{\rm R})$. The corresponding
lightcone coordinates are $(x^{+}_{\rm L},x^{-}_{\rm L})$ and $(x^{+}_{\rm
R},x^{-}_{\rm R})$. We mainly focus in this section on the classical,
geometrical contributions to the entropies, which are of order
$(G\hbar)^{-1}$, and comment on the effects of quantum corrections. In Section
6, we proceed to incorporate thermal corrections due to the coarse-grained
entropy carried by the thermal gas of particles permeating the bulk space in
radiation-dominated cosmologies, which are of order $(G\hbar)^{0}$.

###  4.1 The two-screen system in cosmologies with $|\gamma|\geq 1$

On a given Cauchy slice $\Sigma\in{\cal F}$, let us consider the two-screen
system on $A={\cal S}_{\rm L}\cup{\cal S}_{\rm R}$. We will calculate the
geometrical, classical contributions to the fine-grained entropy by applying
the rules of Section 3.3. Following the notations introduced there, we have
for this case $A_{\rm L}={\cal S}_{\rm L}$, $A_{\rm E}={\cal S}_{\rm
L}\cup{\cal S}_{\rm R}$ and $A_{\rm R}={\cal S}_{\rm R}$. Now, consider any
SO$(n)$-symmetric Cauchy slice $\hat{\Sigma}$ containing the spheres ${\cal
S}_{\rm L}$ and ${\cal S}_{\rm R}$. The parts of $\hat{\Sigma}$ and $\Sigma$
that lie in the pode interior region, $\hat{\Sigma}_{\rm L}$ and $\Sigma_{\rm
L}$, have a common causal diamond corresponding to the left blue triangle in
Figure 7,

${\cal S}_{\rm L}$${\cal S}_{\rm R}$ (a) For $\eta_{\rm L},\eta_{\rm
R}\in[0,|\gamma|\pi/2)$.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$ (b) For $\eta_{\rm L}=\eta_{\rm
R}=|\gamma|\pi/2$.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$ (c) For $\eta_{\rm L},\eta_{\rm
R}\in(|\gamma|\pi/2,|\gamma|\pi]$.

Figure 7: Entanglement wedge (blue shaded region) of the two-screen system on
${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$, when $\gamma\leq-1$. The entire
spacetime is covered by the entanglement wedge as the latter evolves during
the whole cosmological evolution.

for the bouncing cases. The situation is completely analogous for the Big
Bang/Big Crunch cosmologies. Since any such Cauchy slice $\hat{\Sigma}_{\rm
L}$ has the topology of a spherical cap bounded by $A_{\rm L}={\cal S}_{\rm
L}$, the surface homologous to $A_{\rm L}$ that is of minimal extremal area is
the empty set, $\chi_{\rm L}=\varnothing$. Indeed, the latter is a subset of
$\hat{\Sigma}_{\rm L}$ and it is homologous to $A_{\rm L}$ since the homology
condition $\partial\hat{\mathcal{C}}_{\rm L}=\varnothing\cup A_{\rm L}$ is
trivially satisfied for $\hat{\mathcal{C}}_{\rm L}=\hat{\Sigma}_{\rm L}$.
Moreover, the empty set is a surface of minimal extremal area since it cannot
be deformed and has a vanishing area. Hence, the classical contribution from
region L to the von Neumann entropy of the two-screen system is null. Since
$\hat{\mathcal{C}}_{\rm L}=\hat{\Sigma}_{\rm L}$, the part of the entanglement
wedge in L is the full causal diamond of $\hat{\Sigma}_{\rm L}$ and
$\Sigma_{\rm L}$, i.e. the left blue triangle in Figure 7.

The above arguments can be easily adapted to determine the classical
contributions to the von Neumann entropy from the interior region of the
antipode R. The empty set (which is trivially a subset of (any)
$\hat{\Sigma}_{\rm R}$) is the minimal extremal surface homologous to $A_{\rm
R}={\cal S}_{\rm R}$, $\chi_{\rm R}=\varnothing$. In addition, since
$\hat{\mathcal{C}}_{\rm R}=\hat{\Sigma}_{\rm R}$, the part of the entanglement
wedge in region R is the causal diamond of $\hat{\Sigma}_{\rm R}$ (and
$\Sigma_{\rm R}$), which corresponds to the right blue triangle in Figure 7.
Therefore, the classical contribution to the fine-grained entropy of the two-
screen system from region R is also vanishing.

Finally, consider the exterior region, for which $A_{\rm E}={\cal S}_{\rm
L}\cup{\cal S}_{\rm R}$. The slices $\hat{\Sigma}_{\rm E}$ in region E, which
are bounded by $S_{\rm L}\cup{\cal S}_{\rm R}$, span a common causal diamond
corresponding to the blue rectangle in Figure 7. The empty set is a subset of
minimal extremal area of $\hat{\Sigma}_{\rm E}$ and the homology constraint
$\partial\hat{\mathcal{C}}_{\rm E}=\varnothing\cup A_{\rm E}$ is trivially
satisfied for $\hat{\mathcal{C}}_{\rm E}=\hat{\Sigma}_{\rm E}$. Hence, we
obtain $\chi_{\rm E}=\varnothing$, leading to vanishing classical
contributions to the von Neumann entropy from the exterior region as well. The
entanglement wedge in region E is the full causal diamond of
$\hat{\Sigma}_{\rm E}$, i.e. the blue rectangle in Figure 7.

We conclude that the leading classical contributions to the von Neumann
entropy of the two-screen system are vanishing:

$S({\cal S}_{\rm L}\cup{\cal S}_{\rm R})=0+{\mathcal{O}}(G\hbar)^{0}.$ (4.25)

The full entanglement wedge is the union of the causal diamonds of
$\hat{\Sigma}_{\rm L}$, $\hat{\Sigma}_{\rm E}$ and $\hat{\Sigma}_{\rm R}$,
i.e. the union of the blue triangles and the rectangle in Figure 7. In
particular, the entanglement wedge comprises slices that are complete Cauchy
slices, with respect to the bulk cosmology. Assuming entanglement wedge
reconstruction [20], the state on the whole of $\hat{\Sigma}$ or $\Sigma$ can
be reconstructed from the holographic dual system on ${\cal S}_{\rm
L}\cup{\cal S}_{\rm R}$. At the end of Section 3.1, we already saw that the
two screen-system has an adequate number of degrees of freedom to describe the
state on the entire slice $\Sigma$, provided $\Sigma_{\rm E}$ lies in the
region of trapped surfaces. Here, we see that the bilayer proposal leads to
the conclusion that the state of the two-screen system is dual to the state on
$\Sigma$, irrespective of the location of the screens. Even if the latter are
on the apparent horizons that bound the region of anti-trapped surfaces, the
entanglement wedge extends and covers any $\hat{\Sigma}_{\rm E}$, leading to
full reconstruction of the states on any bulk Cauchy slice $\hat{\Sigma}$
containing the screens. This result was argued to hold for the de Sitter case
$\gamma=-1$ in Ref. [48]. As in the de Sitter case, each point of the
cosmological spacetime will have been inside the entanglement wedge at least
once, as the screens evolve with time. Therefore, we expect to be possible to
encode holographically on the two-screen system the state of the cosmology on
any slice of any complete foliation ${\cal F}$.

To go beyond the vanishing geometrical, classical contributions to the entropy
of the two-screen system, we can consider the generalized entropy $S_{\rm
gen}$ defined in Eq. (3.23). For the classical minimal extremal surfaces
$\chi_{\rm L}=\chi_{\rm E}=\chi_{\rm R}=\varnothing$, the non-negative 1-loop
contribution $S_{\rm semicl}$ is evaluated for $\hat{\cal C}_{\rm
L}\cup\hat{\cal C}_{\rm E}\cup\hat{\cal C}_{\rm R}$, which amounts to the
complete Cauchy slice $\hat{\Sigma}$. So, if the bulk cosmology is in a pure
state, the contribution $S_{\rm semicl}(\hat{\Sigma})$ is zero, implying that
the empty surfaces in the regions L, R, E lead to a minimal extremal
generalized entropy at the semiclassical level, since the latter vanishes in
this case. In fact, the von Neumann entropy of the dual system on ${\cal
S}_{\rm L}\cup{\cal S}_{\rm R}$ is expected to be exactly zero, with the two-
screen system in a pure state at all times $\eta_{\rm L}$, $\eta_{\rm R}$. On
the other hand, when the bulk cosmology is in a mixed state, as in the
radiation dominated cases, we have $S_{\rm semicl}(\hat{\Sigma})>0$. Hence, it
is necessary to extremize the generalized entropy $S_{\rm gen}$ in order to
find the correct surfaces $\chi_{i}$, $i\in\\{{\rm L},{\rm E},{\rm R}\\}$,
that yield the minimal extrema results. In this case, the von Neumann entropy
of the two-screen system is non-vanishing and of order $(G\hbar)^{0}$.

Finally, notice that the state on the slices of ${\cal F}$ remains pure if it
starts pure at past infinity. As we remarked at the end of Section 3.2, the
evolution of the two-screen system, along with the bulk cosmology, is not
unitary, in the sense that the number of the underlying degrees of freedom and
the dimensionality of the Hilbert space changes. It could be that this
evolution is isometric [80, 83] and linear, mapping pure states to pure
states. As we have seen, this possibility is indeed consistent with the
bilayer proposal.

###  4.2 The single-screen system in bouncing cosmologies with $\gamma<-1$

We proceed now to consider a single-screen system on ${\cal S}_{\rm L}$ or
${\cal S}_{\rm R}$, in bouncing cosmologies with $\gamma<-1$. The de Sitter
case $\gamma=-1$ has been analyzed in Refs [46, 47, 48] in great detail. In
the expanding phase, the screens must be placed on the apparent horizons. In
the initial contracting phase, they can be pushed farther into the region of
trapped surfaces between the apparent horizons, as explained in Section 3 and
shown in Figure 6(a). We will analyze the left screen subsystem located on
${\cal S}_{\rm L}$, but all results and conclusions can be adapted to apply
equally to the right screen on ${\cal S}_{\rm R}$.

In the pode interior region L, the minimal extremal surface homologous to
$A_{\rm L}={\cal S}_{\rm L}$ is the empty set, $\chi_{\rm L}=\varnothing$,
which can be taken to lie on any $\hat{\Sigma}_{\rm L}$, for the same reasons
explained in Section 4.1 for the two-screen system. Therefore, the classical
contribution from region L to the von Neumman entropy of the screen located on
${\cal S}_{\rm L}$ is zero. Since $\hat{{\cal C}}_{\rm L}=\hat{\Sigma}_{\rm
L}$, the part of the entanglement wedge in region L is the common causal
diamond of the $\hat{\Sigma}_{\rm L}$’s, corresponding to the blue triangular
domain in Figure 8.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (a) At early times, the causal diamond
in E is a rectangle with $M$ being the lower vertex. The area of $M$ is non-
zero and the classical contributions to the entropy of ${\cal S}_{\rm L}$ are
non-vanishing. As the screens approach future null infinity, the area of $M$
and the von Neumann entropy of ${\cal S}_{\rm L}$ increase.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (b) At large times $\eta_{\rm
L}=\eta_{\rm R}=|\gamma|\pi$, $M$ asymptotes to a fixed point and so the
entropy of ${\cal S}_{\rm L}$ saturates at a finite value.

Figure 8: Penrose diagram for a bouncing cosmology satisfying $\gamma<-1$,
when the screens are in the expanding phase. The dark blue region corresponds
to the entanglement wedge of the single-screen system located on ${\cal
S}_{\rm L}$. The causal diamond in the exterior region E is red shaded. The
minimal extremal sphere, denoted by the red dot $M$, corresponds to the lower
vertex.

The classical contribution to the entropy from the antipode interior region R
is also null. Indeed, since $A_{\rm R}=\varnothing$, we obtain $\chi_{\rm
R}=\varnothing$ as the minimal extremal homologous surface, which “lies” on
any Cauchy slice $\hat{\Sigma}_{\rm R}$. Indeed, we have
$\hat{\mathcal{C}}_{\rm R}=\varnothing$, as it satisfies trivially the
condition $\partial\hat{\mathcal{C}}_{\rm R}=\chi_{\rm R}\cup A_{\rm
R}=\varnothing$. Since the causal diamond of the empty set
$\hat{\mathcal{C}}_{\rm R}$ is empty, the entanglement wedge of the screen on
${\cal S}_{\rm L}$ does not extend in the antipode interior region R. These
conclusions hold regardless whether the screens lie in the expanding or the
contracting phase of the cosmology.

The classical contribution to the von Neumann entropy of the single-screen
system arises solely from the exterior region E. Let us first discuss the case
where the screens are located in the expanding phase of the cosmology and thus
lying on the two apparent horizons. The Cauchy slices $\hat{\Sigma}_{\rm E}$,
which are bounded by ${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$, span a common
causal diamond corresponding to the red region in Figure 8. Since $A_{\rm
E}={\cal S}_{\rm L}$ has no boundary, any homologous surface on a Cauchy slice
$\hat{\Sigma}_{\rm E}$ must be closed (see Footnote 9, where we have
$\partial\chi_{\rm E}=\partial A_{\rm E}=\varnothing$). We will look for
minimal extremal homologous surfaces in the causal diamond in region E that
are ${\rm SO}(n)$-symmetric. These are Sn-1 spheres represented by points on
the Penrose diagram, with areas depending on their positions. As remarked in
Section 3.3, since the causal diamond in region E has a boundary, the
extremization of the area functional involves Lagrange multipliers and
auxiliary fields, in order to impose all homologous extremal surfaces to lie
in this diamond, including its boundary [48].121212It would be interesting to
obtain direct evidence for the bilayer proposal and in particular the
constrained extremization problem from a replica bulk path-integral approach.
This would involve Lorentzian Schwinger-Keldysh path integrals. See e.g. Refs
[27, 28] for the black hole cases.

In Appendices B.3 and B.2, it is shown that all ${\rm SO}(n)$-symmetric
solutions to the constrained-extremization problem are the spheres whose areas
are minimal, maximal or saddle points. Moreover, these spheres turn out to lie
on the boundary of the causal diamond in region E. No sphere of extremal
surface exists in the interior of this diamond. The upper vertex of the causal
diamond is the unique sphere of maximal area, when the diamond has a
rectangular shape, as in Figure 15(c). On the other hand, when $x_{\rm
R}^{+}+x_{\rm L}^{-}>|\gamma|\pi$, the causal diamond acquires a fifth edge
along future null infinity, as in Figure 15(d). Since all spheres along this
edge have infinite areas, the latter are maximal. For $\gamma<-1$, there is a
unique sphere of minimal area corresponding to the lower vertex of the causal
diamond. It is depicted by the red dot $M$ in Figure 8, and also in Figures
15(c) and 15(d). $M$ is indeed the anti-trapped sphere with the smallest area,
as can be easily inferred from the Bousso wedges in the upper triangular
region (see Appendix B.2). Finally, the areas of the spheres ${\cal S}_{\rm
L}$ and ${\cal S}_{\rm R}$ are saddle points. Notice that the sphere $M$ of
minimal area coincides with the bifurcate horizon only when $\eta_{\rm
L}=\eta_{\rm R}=|\gamma|\pi/2$. As shown in Appendix B.1, the bifurcate
horizon is also a solution of the unconstrained-extremization problem for
arbitrary $\eta_{\rm L}$, $\eta_{\rm R}$. However, the latter lies outside the
causal diamond in region E, except in the particular case where $\eta_{\rm
L}=\eta_{\rm R}=|\gamma|\pi/2$.

We conclude that in the expanding phase of the bouncing cosmologies with
$\gamma<-1$, the minimal extremal surface $\chi_{\rm ext}$ is the lower vertex
of the causal diamond $M$. As a result, $\hat{\mathcal{C}}_{\rm E}$ is the
lightlike slice joining ${\cal S}_{\rm L}$ to $M$, which we denote
$\hat{\Sigma}_{{\cal S}_{\rm L}\mbox{\scriptsize-}M}$.131313The condition
$\partial\hat{\mathcal{C}}_{\rm E}=\chi_{\rm E}\cup{\cal S}_{\rm L}$ reduces
to $\partial\hat{\Sigma}_{{\cal S}_{\rm L}\mbox{\scriptsize-}M}=M\cup{\cal
S}_{\rm L}$. The entanglement wedge of ${\cal S}_{\rm L}$ then extends in the
exterior causal diamond region, acquiring a component consisting of
$\hat{\Sigma}_{{\cal S}_{\rm L}\mbox{\scriptsize-}M}$ which is its own causal
diamond. Indeed, since ${\cal S}_{\rm L}$ and $M$ are lightlike separated,
there is a unique $\hat{{\cal C}}_{\rm E}$. Since this Cauchy slice is
lightlike, its causal diamond consists only of itself. The conformal
coordinates of $M$ are given by

$(\theta_{M},\eta_{M})=\left(\frac{|\gamma|-1}{|\gamma|}\frac{\eta_{\rm
L}-\eta_{\rm R}}{2}+\frac{\pi}{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\frac{|\gamma|-1}{|\gamma|}\frac{\eta_{\rm L}+\eta_{\rm
R}}{2}+\frac{\pi}{2}\right),\,\,\,\,\eta_{\rm
L},\eta_{\rm,R}\geq|\gamma|\pi/2,$ (4.26)

and the minimal extremal area for $\eta_{\rm L},\eta_{\rm,R}\geq|\gamma|\pi/2$
is equal to

${\cal
A}_{\rm{min}}(M)=\omega_{n-1}a_{0}^{n-1}\sin^{\gamma(n-1)}\left(\frac{|\gamma|-1}{|\gamma|^{2}}\frac{\eta_{\rm
L}+\eta_{\rm
R}}{2}+\frac{\pi}{2|\gamma|}\right)\cos^{n-1}\left(\frac{|\gamma|-1}{|\gamma|}\frac{\eta_{\rm
L}-\eta_{\rm R}}{2}\right).$ (4.27)

Since the interior contributions vanish at the classical level, we deduce that
in the expanding phase, the leading classical entropy of the single-screen
subsystem ${\cal S}_{{\rm L}}$ is given as a function of $\eta_{{\rm L}}$ and
$\eta_{\rm R}$ by

$S_{{\cal S}_{{\rm L}}}(\eta_{\rm L},\eta_{\rm R})=\frac{{\cal
A}_{\rm{min}}(M)}{4G\hbar}+\mathcal{O}((G\hbar)^{0}).$ (4.28)

The entropy increases as $\eta_{\rm L}$ and $\eta_{\rm R}$ increase, and
eventually saturates a finite upper bound, as $\eta_{\rm L},\eta_{\rm
R}\to|\gamma|\pi$:

$S_{{\cal S}_{\rm L}}(\eta_{\rm L}=\eta_{\rm
R}=|\gamma|\pi)=\frac{1}{4G\hbar}\omega_{n-1}a_{0}^{n-1}\sin^{\gamma(n-1)}\left(\frac{\pi}{2|\gamma|}\right)+\mathcal{O}((G\hbar)^{0}).$
(4.29)

Notice that this result holds despite the growing to infinity of the area of
the screen ${\cal S}_{\rm L}$. At latter times, the entanglement entropy is a
small fraction of the maximal possible value. The extra degrees of freedom
added to the screens as these evolve along the apparent horizons remain
disentangled. This hints towards the fact that the effective bridge
manifesting the entanglement between the two screen system is realized by the
Cauchy slice with the smallest possible bottleneck, via the sphere $M$.

Finally, let us discuss the single screen system ${\cal S}_{\rm L}$, when the
screens are in the contracting phase of the cosmology, $\eta_{\rm L},\eta_{\rm
R}<|\gamma|\pi/2$. In this case, the screens can be pushed farther into the
lower triangular region, which is a region of trapped spheres. When
$(x^{+}_{\rm L}+x^{-}_{\rm R})/2\geq 0$, the exterior causal diamond region is
a rectangle, as shown in Figure 16(c). On the other hand, when $(x^{+}_{\rm
L}+x^{-}_{\rm R})/2<0$, it has a fifth edge along past lightlike infinity, as
in Figure 16(d). In both cases, there is a unique minimal extremal surface,
homologous to ${\cal S}_{\rm L}$, corresponding to the upper vertex of the
causal diamond, which is depicted by the red dot $M$ in Figures 16(c) and
16(d). This is shown in Appendices B.2 and B.3. The entanglement wedge
acquires a component in the exterior region E, comprising the lightlike
segment joining ${\cal S}_{\rm L}$ and $M$, $\hat{\Sigma}_{{\cal S}_{\rm
L}\mbox{\scriptsize-}M}$. The lightcone coordinates of $M$ can be expressed in
terms of the lightcone coordinates of the screens as follows:

$(x_{M}^{+},x_{M}^{-})=(x_{\rm R}^{+},x_{\rm L}^{-}).$ (4.30)

The minimal area is now given by

${\cal
A}_{\rm{min}}(M)=\omega_{n-1}a_{0}^{n-1}\sin^{\gamma(n-1)}\left(\frac{x_{\rm
R}^{+}+x_{\rm L}^{-}}{2|\gamma|}\right)\sin^{n-1}\left(\frac{x_{\rm
R}^{+}-x_{\rm L}^{-}}{2}\right),$ (4.31)

and the leading geometrical entropy satisfies Eq. 4.28, where now $\eta_{\rm
L},\eta_{\rm R}\leq\pi|\gamma|/2$. The entropy increases as the screens
approach the lower corners of the Penrose diagram as $\eta_{\rm L},\eta_{\rm
R}\to 0$, saturating an upper bound:

$S_{{\cal S}_{\rm L}}(\eta_{\rm L}=\eta_{\rm
R}=0)=\frac{1}{4G\hbar}\omega_{n-1}a_{0}^{n-1}\sin^{\gamma(n-1)}\left(\frac{\pi}{2|\gamma|}\right)+\mathcal{O}((G\hbar)^{0}).$
(4.32)

This maximal value of the fine grained entropy in the contracting phase is
equal to the maximal value achieved in the expanding phase, as the screens
approach future null infinity.

It can be easily seen that the conclusions above apply to the second screen
subsystem ${\cal S}_{{\rm R}}$. The leading geometrical entropy of ${\cal
S}_{{\rm R}}$ receives contributions from the exterior region only, with the
corresponding minimal extremal homologous surface being the sphere $M$.
Therefore, the leading geometrical entropy is equal to that of ${\cal S}_{{\rm
L}}$:

$S_{{\cal S}_{{\rm R}}}(\eta_{\rm L},\eta_{\rm R})=\frac{{\cal
A}_{\rm{min}}(M)}{4G\hbar}+\mathcal{O}((G\hbar)^{0}).$ (4.33)

As we have argued in subsection 4.1, the two screen system is in a pure state,
when the bulk state on $\Sigma$ is taken to be pure. Therefore, in this case
the entropies of the two screens have to be the same at all orders in
$G\hbar$.

We conclude this section by comparing with the de Sitter case $\gamma=-1$,
analyzed in [48]. In the de Sitter case there is a degeneracy of minimal area
spheres, which also solve the constrained-extremization problem at the
classical level. Indeed, for $\gamma=-1$, the apparent horizons become
lightlike and coincide with the cosmological horizons delimiting the pode and
antipode causal patches. The screens necessarily lie on the cosmological
horizons. The segments joining the screens with the bifurcate horizon belong
to the boundary of the exterior causal diamond region. All points along these
segments correspond to degenerate minimal extremal spheres. As argued in [48],
the classical degeneracy can be lifted by quantum corrections. The sphere of
smallest quantum area, which minimizes the generalized entropy to first order
in the quantum corrections, is the screen for which $|\eta-\pi/2|$ is bigger.
Here we see that this degeneracy can be also lifted by classical perturbations
on the geometry. If we perturb the geometry so that $\gamma<-1$, we find a
unique minimal extremal surface controlling the leading geometrical
contributions to the entanglement entropy between the screens.

###  4.3 The single-screen system in Big Bang/Big crunch cosmologies with
$\gamma\geq 1$

Next, we consider the single-screen subsystem $A={\cal S}_{\rm L}$ in Big
Bang/Big Crunch cosmologies with $\gamma\geq 1$. The analysis for the interior
regions is completely analogous to the bouncing cases presented in subsection
4.2. The minimal extremal surfaces homologous to $A_{\rm L}={\cal S}_{\rm L}$
and $A_{\rm R}=\varnothing$ are the empty sets, $\chi_{\rm L}=\varnothing$ and
$\chi_{\rm R}=\varnothing$, respectively. Therefore, the classical
contributions to the entanglement entropy from the interior regions L and R
are vanishing. The entanglement wedge has no component in region R, while it
contains the full causal diamond region of the $\hat{\Sigma}_{{\rm L}}$’s in
the pode interior region L, depicted by the blue triangle in Figure 9.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (a) At early times, the causal diamond
in E is a rectangle with $M$ being the top vertex. The area of $M$ is non-zero
and the classical contributions to the entropy of ${\cal S}_{\rm L}$ are non-
vanishing.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (b) When $(x^{+}_{\rm R}+x^{-}_{\rm
L})/2=\gamma\pi$, $M$ hits the Big Crunch singularity. The area of $M$ is zero
and the classical contributions to the entropy of ${\cal S}_{\rm L}$ vanish.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (c) The entropy of ${\cal S}_{\rm L}$
remains small of order $(G\hbar)^{0}$ when $(x^{+}_{\rm R}+x^{-}_{\rm
L})/2>\gamma\pi$, since $M$ is at the Big Crunch singularity.

Figure 9: Closed FRW cosmologies for $\gamma>1$. The two screens are denoted
by the black dots ${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$. We consider the
subsystem associated with the single screen ${\cal S}_{\rm L}$. The light red
shaded region is the causal diamond in the exterior region E, and the dark
blue region corresponds to the entanglement wedge of ${\cal S}_{\rm L}$. The
minimal extremal surface is denoted by the red dot $M$.

The classical geometrical contribution to the entropy of ${\cal S}_{\rm L}$
arises from the exterior region between the screens. It is given in terms of
the area of a minimal extremal surface, which lies on the boundary of the
causal diamond in the exterior region E, spanned by the $\hat{\Sigma}_{\rm
E}$’s, as explained in Appendices B.2 and B.3.

We will discuss in great detail the contracting phase of the cosmologies and,
so, the screen conformal times satisfy $\eta_{\rm L},\eta_{\rm
R}\geq\gamma\pi/2$. In the contracting phase, the screens can be pushed
farther into the upper triangular region, following timelike (and possibly
locally lightlike) trajectories, as explained in Section 3 and illustrated in
Figure 6(b). The upper triangular region is a region of trapped spheres. For
$(x^{+}_{\rm R}+x^{-}_{\rm L})/2<\gamma\pi$, the causal diamond in region E is
a rectangle with its upper vertex lying below the Big Crunch singularity, as
shown in Figure 9(a). The upper vertex of the rectangle hits the Big Crunch
singularity precisely when $(x^{+}_{\rm R}+x^{-}_{\rm L})/2=\gamma\pi$ – see
Figure 9(b). As we show in Appendices B.2 and B.3, the minimal extremal
surface $\chi_{\rm E}$, homologous to ${\cal S}_{\rm L}$, is precisely this
top vertex of the causal diamond. We depict this by the red dot $M$ in Figure
9. $M$ is indeed the trapped sphere with the smallest area, as can be easily
inferred from the structure of the Bousso wedges in the upper triangular
region. In addition, it solves the constrained-extremization problem, as
demonstrated explicitly in Appendix B.3. In terms of the screen lightcone
coordinates, the lightcone coordinates of $M$ are given by

$\left(x_{M}^{+},x_{M}^{-}\right)=\left(x_{\rm R}^{+},x_{\rm L}^{-}\right).$
(4.34)

and the minimal area by

${\cal A}_{min}(M)=\omega_{n-1}a_{0}^{n-1}\sin^{\gamma(n-1)}\left(\frac{x_{\rm
R}^{+}+x_{\rm L}^{-}}{2\gamma}\right)\sin^{n-1}\left(\frac{x_{\rm
R}^{+}-x_{\rm L}^{-}}{2}\right).$ (4.35)

The entanglement wedge acquires a component in the exterior region, comprising
of the lightlike segment $\hat{\Sigma}_{{\cal S}_{\rm L}\mbox{\scriptsize-}M}$
joining ${\cal S}_{\rm L}$ and $M$ – see Figure 9(a). Notice that when
$\eta_{\rm L}=\eta_{\rm R}=\eta_{c}=\pi|\gamma|/2$, the screens coincide at
the bifurcation point of the apparent horizons. The exterior causal diamond
becomes trivial, comprising the bifurcation point only. The bifurcate horizon
is an extremal of the unconstrained area functional, as shown in Appendix B.1.
Its area is equal to $\omega_{n-1}a_{0}^{n-1}$ yielding non-trivial classical
contributions to the entropy of ${\cal S}_{\rm L}$.

As the two screens follow their trajectories towards the Big Crunch
singularity, the top vertex $M$ of the causal diamond in E approaches the Big
Crunch singularity and its area decreases. Consequently the entropy of ${\cal
S}_{\rm L}$ decreases. $M$ hits the future singularity at $(x^{+}_{\rm
R}+x^{-}_{\rm L})/2=\gamma\pi$ – see Figure 9(b). Precisely at this time, the
area of $M$ vanishes, so that the classical contributions to the entropy from
the exterior region vanish.

When $(x^{+}_{\rm R}+x^{-}_{\rm L})/2>\gamma\pi$, the exterior causal diamond
acquires a fifth edge along the Big Crunch singularity slice. Along this edge,
the coordinates satisfy $\gamma\pi+\theta_{\rm L}-\eta_{\rm
L}\leq\theta\leq-\gamma\pi+\theta_{\rm R}+\eta_{\rm R}$ (and
$\eta=\gamma\pi$). All spheres on this segment are extremal and of vanishing
area. So they yield zero classical contribution to the entropy. To resolve
this degeneracy, we pick the left endpoint of this segment so as to minimize
the extent of the entanglement wedge in the exterior region [46], as in Figure
9(c). As we discussed in the case of the entanglement wedge of a single screen
system in the de Sitter case [48], it is not true that minimizing the extent
of the entanglement wedge is always the solution. The correct choice depends
on the structure of the quantum corrections. However, we can consider this
choice to give a lower bound on the extent of the entanglement wedge since all
other choices of extremal surfaces lead to an entanglement wedge that includes
this minimal choice.

Since the classical contributions from the interior region are always zero,
the bilayer proposal yields for the entropy of the single-screen system ${\cal
S}_{\rm L}$:

$S_{{\cal S}_{\rm L}}=\left\\{\begin{array}[]{cc}\frac{{\cal
A}_{min}(M)}{4G\hbar}+\mathcal{O}((G\hbar)^{0})&,(x^{+}_{\rm R}+x^{-}_{\rm
L})/2<\gamma\pi\\\ \mathcal{O}((G\hbar)^{0})&,(x^{+}_{\rm R}+x^{-}_{\rm
L})/2\geq\gamma\pi\end{array}\right..$ (4.36)

Notice that the classical entropy is always smaller or equal to the area of
the screen divided by $4G\hbar$ – the two are equal only at $\eta_{\rm
L}=\eta_{\rm R}=\gamma\pi/2$, when the screens coincide at the bifurcation
point. This is a consistency check for the bilayer proposal since the area of
the screen divided by $4G\hbar$ sets an upper bound for the von Neumann
entropy of the density matrix describing the state of the single screen
system.

Therefore, when $(x^{+}_{\rm R}+x^{-}_{\rm L})/2<\gamma\pi$, the entanglement
entropy is significantly large, of order $(G\hbar)^{-1}$, and decreasing. At
$(x^{+}_{\rm R}+x^{-}_{\rm L})/2=\gamma\pi$ the minimal extremal surface $M$
hits the Big Crunch singularity. As a result, the part of the entanglement
wedge of ${\cal S}_{\rm L}$ in E is pinched to the singularity. The effective
bridge connecting the screens closes off, leading to disentanglement (or a
significant decrease in the entanglement entropy), well before the screens
reach the Big Crunch singularity. The entanglement entropy becomes small, of
order $(G\hbar)^{0}$, even if the Universe has not yet collapsed.

The disentanglement phenomenon cannot occur when the two screens are in causal
contact, namely when they lie in the overlap region of the pode and antipode
causal patches. It is easy to see that in this case $M$ has to lie below the
intersection of the event horizons delimiting the pode and antipode causal
patches.

The leading classical contributions to the entropy remain vanishing for the
rest of the evolution of the screens along their trajectories. So the entropy
at latter times is dominated by the semiclassical contributions. Notice that
the slices connecting the screen with the minimal extremal surfaces reach up
to the Big Crunch singularity. So quantum corrections to the bulk theory
effective action will be important for the determination of the full
generalized entropy, including quantum corrections, but we expect the entropy
at these latter times to remain much smaller as compared to the value of order
$(G\hbar)^{-1}$ at earlier times. Indeed the number of degrees of freedom on
the screens decreases as these evolve towards the Big Crunch singularity and
the fine grained entropy must be bounded by this. Unlike the bouncing cases,
the entanglement entropy becomes very small (of order $(G\hbar)^{0}$) well
before the screens reach the Big Crunch singularity.

The situation for the right screen ${\cal S}_{\rm R}$ is completely analogous.
In particular, the classical entropy of ${\cal S}_{\rm R}$ is equal to that of
${\cal S}_{\rm L}$ since this is controlled by the same minimal extremal
sphere $M$. Notice also, that assuming the quantum extremal surface is common
to the two screens, their entropies, including quantum corrections, will
continue being equal assuming the bulk state on $\hat{\Sigma}$ is pure. This
is consistent with the fact that the two screen system is in a pure state in
this case, as argued in subsection 4.1.

Let us now discuss the initial expanding phase of the cosmology, during which
$\eta_{\rm L},\eta_{R}\leq\gamma\pi/2$. In this phase, the screens must lie on
the apparent horizons. The region between the apparent horizons is a region of
anti-trapped spheres. The exterior causal diamond region is a rectangle for
$(x_{{\rm L}}^{+}+x_{{\rm R}}^{-})/2\geq 0$, and now its lower vertex $M$, the
one closer to the Big Bang singularity, is the minimal extremal sphere that
leads to non-vanishing contributions to the entropy of ${\cal S}_{{\rm L}}$.
When $(x_{{\rm L}}^{+}+x_{{\rm R}}^{-})/2=0$, $M$ touches the initial
singularity and has vanishing area. On the other hand, when $(x_{{\rm
L}}^{+}+x_{{\rm R}}^{-})/2<0$, the exterior causal diamond acquires a fifth
edge along the Big Bang singularity, with all spheres along this segment being
extremal with vanishing area. Therefore, the entropy is initially small of
order $(G\hbar)^{0}$, with the degrees of freedom between the two screens
being effectively disentangled. The entanglement entropy is governed by
quantum, semiclassical corrections, which are expected to lift the classical
degeneracy and determine the precise extent of the entanglement wedge in
region E. At some point, the entanglement entropy becomes large, of order
$(G\hbar)^{-1}$, when the degrees of freedom on the two screens interact. Then
the entropy grows and reaches its maximal value when the screens are at the
bifurcate horizon.

Finally, let us briefly discuss the particular case $\gamma=1$, for which the
Penrose diagram is a square. This is the marginal case for the two distinct
classes of Big Bang/Big Crunch cosmologies. As in the de Sitter case
$\gamma=-1$, the apparent horizons are lightlike and coincide with the
cosmological horizons. The screens must be located on the cosmological
horizons, in order to remain in the causal patches of the pode and antipode,
respectively. The two screens at the horizons remain out of causal contact
during the contracting phase, up to the Big Crunch singularity. A sharp
difference with the de Sitter case $\gamma=-1$ is that now the areas of the
screens on the horizons are no longer constant but decrease with time, and
vanish at the Big Crunch singularity. The time evolution of the single screen
subsystem ${\cal S}_{{\rm L}}$ and the entanglement wedge structure is
illustrated in Figure 10.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (a) For $\pi<\eta_{\rm L}+\eta_{\rm
R}<3\pi/2$, the causal diamond region is a rectangle with $M$ being the top
vertex.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (b) For $\eta_{\rm L}+\eta_{\rm
R}=3\pi/2$, $M$ hits the Big Crunch singularity and has vanishing area.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$$M$ (c) For $\eta_{\rm L}+\eta_{\rm R}\geq
3\pi/2$, $M$ remains at the Big Crunch singularity.

Figure 10: Closed FRW cosmology for $\gamma=1$. The two screens at the
cosmological horizons are denoted by the black dots ${\cal S}_{\rm L}$ and
${\cal S}_{\rm R}$. The light red shaded region is the causal diamond in the
exterior region E. The sphere with the minimal extremal area is denoted by the
red dot $M$. The dark blue region is the entanglement wedge associated with
${\cal S}_{\rm L}$.

The behavior in the expanding phase can be obtained via time reversal of the
behavior in the contracting phase.

###  4.4 Maximin spheres

Let us consider the contracting phase of the cosmologies with $\gamma<-1$ and
$\gamma\geq 1$. When the causal diamond in the exterior region E is
rectangular, its four vertices are the only extremal spheres. Let us denote
${\cal S}_{\rm small}$ the screen with the smaller area and ${\cal S}_{\rm
big}$ the screen with the bigger area among the two. Then ${\cal S}_{\rm
small}$ is a maximin sphere. Indeed, consider the limiting Cauchy slice
consisting of the union of the two lower lightlike edges of the causal
rectangle, which join the screens with the bottom vertex, respectively. From
the structure of the Bousso wedges, it follows that, as we move along a
lightlike edge from the bottom vertex towards the screen at its end, the area
of the spheres decreases. Therefore, ${\cal S}_{\rm small}$ must be the
minimal area sphere on this limiting Cauchy slice. Since any other Cauchy
slice in the causal diamond in E ends on the two screens, the minimal area
sphere on it necessarily has smaller or equal area to the area of ${\cal
S}_{\rm small}$. Therefore, ${\cal S}_{\rm small}$ must be a maximin sphere:
among the set of minimal area spheres from all Cauchy slices, ${\cal S}_{\rm
small}$ is one with maximal area. On the other hand, ${\cal S}_{\rm big}$ is a
minimax sphere. Indeed, consider now the limiting Cauchy slice consisting of
the two upper edges of the causal rectangle, which join the screens with the
top vertex of the rectangle, respectively. ${\cal S}_{\rm big}$ is the maximal
area sphere on this limiting slice. Since any other Cauchy slice in the
exterior causal diamond ends on the screens, the maximal area sphere on it
necessarily has bigger or equal area to the area of ${\cal S}_{\rm big}$.
Therefore, this screen is a minimax sphere. Likewise, it is easy to see that
the top vertex, which is the minimal extremal sphere, is a minimin sphere and
the bottom vertex is a maximax sphere.

In the expanding phase of these cosmologies, similar arguments can be applied
to conclude that now the top vertex is a maximax sphere and the bottom vertex
is minimin sphere. The screen with the bigger conformal time, ${\cal S}_{\rm
big}$, is a minimax sphere, while the screen at the smaller conformal time,
${\cal S}_{\rm small}$, is a maximin sphere.

###  4.5 Equivalence of holographic constructions

As we already remarked in subsection 3.2, there are infinitely many possible
choices of trajectories for the two screens ${\cal S}_{\rm L}$ and ${\cal
S}_{\rm R}$ in the contracting phase of the cosmologies with $\gamma<-1$,
$\gamma>1$, leading to different holographic constructions. However, by
examining the full two-screen and single screen subsystems, evidence can be
obtained supporting the fact that these screen configurations can be grouped
in equivalence classes. In each class, the effective holographic theory on the
screens can be derived from a pair of “parent” screens on the apparent
horizons of the pode and antipode, yielding identical predictions for the
entropy of certain gravitational bulk systems, which can be reconstructed from
the single screen subsystems. In the following, we will assume entanglement
wedge reconstruction and unitary evolution among the Cauchy slices in the
entanglement wedge of a single screen subsystem (or any subsystem in general).
Recall also that it is consistent to assume unitary evolution among the Cauchy
slices $\hat{\Sigma}$ associated with a given two-screen configuration, and
also unitary evolution in the three causal diamonds associated with
${\hat{\Sigma}}_{\rm L}$, ${\hat{\Sigma}}_{\rm E}$ and ${\hat{\Sigma}}_{\rm
R}$. Indeed, the states and density matrices on these slices are associated
with a holographic quantum system of certain dimensionality (determined by the
total area of the screens), as well as its subsystems.

To determine the “parent” screens, we draw the lightlike segments $\Delta_{\rm
L}$ and $\Delta_{\rm R}$, which run parallel to the $x^{+}$ and $x^{-}$ axes,
respectively, and join the screens ${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$
with the apparent horizons of the pode and antipode, as in Figure 5. The left
“parent” screen ${\cal S}_{{\rm L}}^{\prime}$ lies at the intersection of
$\Delta_{\rm L}$ with the apparent horizon of the pode. Likewise, the right
“parent” screen ${\cal S}_{{\rm R}}^{\prime}$ lies at the intersection of
$\Delta_{\rm R}$ with the apparent horizon of the antipode. As mentioned in
subsection 3.2, all pairs of screens on $\Delta_{\rm L}$ and $\Delta_{\rm R}$
belong to the same equivalence class, and the effective holographic theory on
them can be derived from the theory on ${\cal S}_{{\rm L}}^{\prime}\cup{\cal
S}_{{\rm R}}^{\prime}$ by integrating out degrees of freedom. Indeed, ${\cal
S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm R}^{\prime}$ are the maximal area
spheres on $\Delta_{\rm L}$ and $\Delta_{\rm R}$, respectively, as follows
from the structure of the Bousso wedges in the region between the apparent
horizons in the contracting phase. So, as we propagate from ${\cal S}_{\rm
L}^{\prime}$ towards ${\cal S}_{\rm L}$ along $\Delta_{\rm L}$, the number of
degrees of freedom decreases. However, as we will see below, the von Neumann
entropy of the screens does not change, suggesting that the extra degrees of
freedom that are integrated out are redundant, as they do not participate in
the entanglement between the two screens and are not likely to be necessary
for the reconstruction of the dual bulk subsystems. The number of degrees of
freedom associated with a minimal extremal trapped sphere suffice to
reconstruct the state of these dual bulk subsystems.

We proceed now to argue that the screen configurations associated with
$\Delta_{\rm L}$ and $\Delta_{\rm R}$ lead to equivalent predictions, at least
for the bulk subsystems that are dual to the single screen subsystems and
emerge holographically. These bulk subsystems are associated with field
configurations on Cauchy slices in the entanglement wedge of a single screen.
First, we observe that the causal diamonds in E associated with pairs of
screens on $\Delta_{\rm L}$ and $\Delta_{\rm R}$ form a nested
series.141414The importance of causal diamonds in holography was recently
emphasised in [58]. Starting with ${\cal S}_{\rm L}^{\prime}$ and ${\cal
S}_{\rm R}^{\prime}$ and propagating along $\Delta_{\rm L}$ and $\Delta_{\rm
R}$, the exterior causal diamond of the new screen configuration is contained
in the exterior causal diamond of the previous one. Therefore, the exterior
causal diamond associated with ${\cal S}_{{\rm L}}$ and ${\cal S}_{{\rm R}}$
is contained in the exterior causal diamond of ${\cal S}_{\rm L}^{\prime}$ and
${\cal S}_{\rm R}^{\prime}$. The same holds for the exterior causal diamond
associated with any pair of screens on $\Delta_{\rm L}$ and $\Delta_{\rm R}$.

If the initial exterior causal diamond of ${\cal S}_{\rm L}^{\prime}$ and
${\cal S}_{\rm R}^{\prime}$ is rectangular, all subsequent exterior causal
diamonds associated with equivalent screens are also rectangular and share the
same top vertex $M$. But, $M$ corresponds to the minimal extremal sphere
$\chi_{\rm E}$, homologous to the left (and also the right) screen. This holds
irrespective of the choice of the pair of screens on $\Delta_{\rm L}$ and
$\Delta_{\rm R}$. The area of $M$ divided by $4G\hbar$ is equal to the leading
classical entropy of the left (and right) screen subsystem(s). In addition,
$M$ determines the extent of the left screen’s entanglement wedge in the
exterior region. We see that the fine grained entropies of the left and right
screen subsystems at the classical level remain constant as the screens move
along $\Delta_{L}$ and $\Delta_{R}$ and towards ${\cal S}_{{\rm L}}$ and
${\cal S}_{{\rm R}}$, respectively. In particular, the fine grained entropies
of ${\cal S}_{{\rm L}}$ and the parent screen ${\cal S}_{{\rm L}}^{\prime}$
are equal at the classical level. For any choice of pair of equivalent
screens, the entanglement wedge of the left screen subsystem consists of the
union of the left triangular causal diamond of the associated
$\hat{\Sigma}_{\rm L}$ slices and the lightlike segment connecting the left
screen and the top vertex $M$ in the exterior region.

One possibility to explain the constancy of the fine grained entropy as we
integrate out degrees of freedom is the following. The Hilbert space of the
“daughter” screen, whose dimensionality is smaller, should be realizable as a
vector subspace of the Hilbert space of the “parent” screen. Let us now choose
a suitable basis in which the density matrix of the “parent” screen is
diagonal. The non-trivial block of non-zero eigenvalues should correspond to
ket-bra operators associated with the subspace of the “daughter” screen and
should have a smaller size. This non-trivial block, which yields the non-
vanishing contributions to the von Neumann entropy, is nothing but the density
matrix of the “daughter” screen.

Let us focus on ${\cal S}_{{\rm L}}$ and the parent screen ${\cal S}_{{\rm
L}}^{\prime}$. In the exterior region, the entanglement wedge of ${\cal
S}_{{\rm L}}$ consists of the lightlike segment between ${\cal S}_{{\rm L}}$
and $M$, denoted by $\Delta_{{\cal S}_{{\rm L}}M}$. See Figure 11(b). The part
of the entanglement wedge of ${\cal S}_{{\rm L}}^{\prime}$ in E is the union
of $\Delta_{\rm L}$ and $\Delta_{{\cal S}_{{\rm L}}M}$ – see Figure 11(a).

${\cal S}_{\rm L}$${\cal S}_{\rm L}^{\prime}$${\cal S}_{\rm R}$${\cal S}_{\rm
R}^{\prime}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}M$
(a) “Parent” screen configuration ${\cal S}_{\rm L}^{\prime}\cup{\cal S}_{\rm
R}^{\prime}$, located on the apparent horizons of the pode and the antipode.

${\cal S}_{\rm L}$${\cal S}_{\rm L}^{\prime}$${\cal S}_{\rm R}$${\cal S}_{\rm
R}^{\prime}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}M$
(b) A “daughter” screen configuration ${\cal S}_{\rm L}\cup{\cal S}_{\rm R}$,
located anywhere along the lightlike segments joining ${\cal S}_{\rm
L}^{\prime}$ to $M$ and ${\cal S}_{\rm R}^{\prime}$ to $M$.

${\cal S}_{\rm L}$${\cal S}_{\rm L}^{\prime}$${\cal S}_{\rm R}$${\cal S}_{\rm
R}^{\prime}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}M$
(c) Minimal screen configuration, when both screens are located at the top
vertex $M$ of the exterior causal diamond.

Figure 11: Several screen configurations in a given equivalence class, defined
by a pair of screens ${\cal S}_{\rm L}^{\prime}$ and ${\cal S}_{\rm
R}^{\prime}$ on the apparent horizons of the pode and the antipode. The light
red shaded region is the causal diamond in the exterior region E, and the dark
blue region is the entanglement wedge of the single left-screen system. The
biggest entanglement wedge is obtained when both screens lie at $M$ (Figure
(c)). It coincides with the full causal diamond of any slice in the
entanglement wedge of any of the equivalent screen configurations.

In fact, as shown in Figure 11(b), the full entanglement wedge of ${\cal
S}_{{\rm L}}$ is bigger and contains the entanglement wedge of the parent
screen ${\cal S}_{{\rm L}}^{\prime}$. Since there is unitary evolution among
the Cauchy slices in the entanglement wedge of ${\cal S}_{{\rm L}}$, the state
on them can be determined from the state on a Cauchy slice in the entanglement
wedge of the parent screen ${\cal S}_{{\rm L}}^{\prime}$ (since the latter
lies in the entanglement wedge of ${\cal S}_{{\rm L}}$).

The contrary is also true. The state on slices in the entanglement wedge of
the parent screen can be determined from the state on the Cauchy slices in the
entanglement wedge of the “daughter” screen. Therefore, the extra degrees of
freedom that are integrated out in order to obtain the effective holographic
theory on the “daughter” screen are not really needed for the reconstruction
of the bulk state on these slices.

It may seem contradictory that the entanglement wedge of the “parent” screen
is contained in the entanglement wedge of a “daughter” screen, as it seems to
imply that a greater bulk region can be reconstructed from the “daughter”
screen ${\cal S}_{{\rm L}}$ with fewer degrees of freedom. Recall however that
the entanglement wedge of a screen does not contain all Cauchy slices that are
unitarily related (and hence reconstructible from each other), but only those
passing through the screen.151515This ensures that all points of the Cauchy
slice are spacelike (or lightlike) separated from the screen, and the local
bulk fields at these points are independent from the fields on the screen. In
particular, the causal diamond of a Cauchy slice in the entanglement wedge is
in general bigger than the entanglement wedge itself. By examining Figure
11(b), we can easily infer that all Cauchy slices in the entanglement wedge of
${\cal S}_{{\rm L}}$ lie in the full causal diamond of the Cauchy slices
$\hat{\Sigma}_{{\rm L}}^{\prime}\cup\Delta_{\rm L}\cup\Delta_{{\cal S}_{{\rm
L}}M}$, the latter spanning the entanglement wedge of ${\cal S}_{{\rm
L}}^{\prime}$. In fact, the full causal diamond coincides with the
entanglement wedge of the minimal extremal sphere $M$ – see Figure 11(c). So
the state on Cauchy slices in the entanglement wedge of ${\cal S}_{{\rm L}}$
can be reconstructed from the state on $\hat{\Sigma}_{{\rm
L}}^{\prime}\cup\Delta_{\rm L}\cup\Delta_{{\cal S}_{{\rm L}}M}$.

In particular, consider the bulk slice $\hat{\Sigma}_{\rm L}\cup\Delta_{{\cal
S}_{{\rm L}}M}$ (lying in the entanglement wedge of ${\cal S}_{{\rm L}}$) and
the bulk subsystem associated with this slice. The holographic theories on
${\cal S}_{{\rm L}}\cup{\cal S}_{{\rm R}}$ and ${\cal S}_{{\rm
L}}^{\prime}\cup{\cal S}_{{\rm R}}^{\prime}$ lead to the same prediction
concerning the fine-grained entropy of this slice to leading order. Since the
first corrected entropy requires adding the semiclassical entropy on slices in
the entanglement wedge of ${\cal S}_{{\rm L}}$ and $\hat{\Sigma}_{{\rm
L}}^{\prime}\cup\Delta_{\rm L}\cup\Delta_{{\cal S}_{{\rm L}}M}$ lies in this
entanglement wedge, we see that the equivalence holds at the level of quantum
corrections.

Notice that among the equivalent choices is the choice where both screens are
placed at $M$ (the top vertex of the exterior causal diamond of the parent
screens). The entanglement wedge of the left screen at $M$ now coincides with
the full causal diamond of its Cauchy slices. Furthermore, this is the minimal
configuration with the smallest possible number of degrees of freedom that are
necessary to reconstruct the dual bulk subsystems (associated with Cauchy
slices in the left causal diamond of $M$). The density matrix in this minimal
dual holographic theory is maximally entangled since the von Neumann entropy
is of the same order as the area of the screen in Planck units – the latter
sets the number of degrees of freedom in the holographic theory. For the rest
of the configurations, one uses a larger density matrix, but as we emphasized
before, this should have many zero eigenvalues, with the non-trivial block of
non-zero eigenvalues corresponding to the subspace associated with the minimal
extremal area sphere $M$, reflecting the fact that the additional number of
degrees of freedom do not participate in the entanglement pattern with the
right screen.

The situation can be generalized in the case when the exterior causal diamond
of the parent screens acquires a fifth edge, along past lightlike infinity for
the bouncing cases ($\gamma<-1$), or the Big Crunch singularity for the Big
Bang/Big Crunch cosmologies ($\gamma>1$). For the bouncing cases, the
generalization is straightforward since the relevant minimal extremal surface
is the top vertex of the causal diamond. For the singular cosmologies
$\gamma>1$, note that the nesting property continues to hold. All subsequent
causal diamonds share the same fifth edge along the Big Crunch singularity and
are contained in the previous ones. Hence, the disentanglement phenomenon
occurs for all equivalent configurations simultaneously, and moreover the
subsequent entanglement wedge structures resulting from quantum corrections
will continue to yield identical predictions for the entropy of bulk
subsystems associated with the screens.

###  4.6 Discussion

The entropies (4.28) and (4.36) are plotted in Figure 12,

$|\gamma|\pi/2$$|\gamma|\pi$$0$$\frac{\pi}{2G}$$\eta_{\rm L}$${\cal S}_{{\rm
L}}$$w=1/2$$w=1/4$$w=-1/2$$w=-2/3$$w=-1$ Figure 12: Entropy of a single screen
subsystem as a function of the conformal time, for $n=2$ and various values of
$w$. The two screens are placed on the apparent horizons and taken to lie at
equal conformal times. The entropy is rescaled with respect to $|\gamma|\pi$.

for three-dimensional cosmological spacetimes ($n=2$), and for several values
of $w$ in the range $[-1,1]$. For simplicity, we consider the case where the
screen trajectories coincide with the apparent horizons, and take the screen
conformal times to be equal, $\eta_{\rm L}=\eta_{\rm R}$.

To summarize, the entanglement entropy between the two screens can be directly
related to the geometry of an effective geometrical bridge in the exterior
region. The effective geometrical bridge is the union of the two lightlike
$\hat{{\cal C}}_{\rm E}$’s, which connect the screens with the minimal
extremal homologous surface $M$, respectively. It corresponds to the limiting
Cauchy slice on the boundary of the exterior causal diamond, which contains
$M$. This is indeed the slice with the smallest bottleneck. In the Big
Bang/Big Crunch cases, the bottleneck pinches and closes off, precisely when
the two screen subsystems effectively disentangle. Indeed, when the leading
classical entanglement entropy between the screens vanishes, the area of $M$
vanishes since this lies either at the Big Bang or the Big Crunch singularity.
The presence of a complete geometrical bridge between the screen subsystems is
a manifestation of the ER=EPR paradigm in a cosmological, time-dependent
setting [16]. This is depicted in Figure 13,

(a) For $(x^{+}_{\rm R}+x^{-}_{\rm L})/2<\gamma\pi$, $M$ has a non-vanishing
area.

(b) When $(x^{+}_{\rm R}+x^{-}_{\rm L})/2=\gamma\pi$, $M$ hits the Big Crunch
singularity: its area vanishes and the bridge pinches and closes off.

(c) The bridge remains disconnected for the rest of the cosmological
evolution, for $(x^{+}_{\rm R}+x^{-}_{\rm L})/2>\gamma\pi$.

Figure 13: Topology of the effective bridge in the exterior region connecting
the screens ${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$ via the minimal extremal
surface $M$, depicted at different times in the contracting phase of a Big
Bang/Big Crunch cosmology. The Cauchy slice $\hat{\Sigma}_{\rm E}$ is the
union of the blue and red shaded regions. It corresponds to the limiting
Cauchy slice consisting of the two upper edges of the exterior causal
rectangle in the Penrose diagram.

in the case where the two screens lie in the contracting phase of a Big
Bang/Big Crunch cosmology.

Note that for the $\gamma<-1$ and $\gamma>1$ cases, there is a certain period
of time of finite duration, where the two screens are in causal contact and
can exchange information. However, the screens are always spacelike separated,
comprising two sets of independent degrees of freedom with possible
interactions among them.

The minimal extremal surface associated with a single screen subsystem in the
exterior region E is lightlike separated with respect to the screens ${\cal
S}_{{\rm L}}$ and ${\cal S}_{{\rm R}}$. As a result, the entanglement wedge of
${\cal S}_{{\rm L}}$ in the exterior has an infinitesimally small extent. This
supports the fact that a single screen does not suffice to encode the full
bulk spacetime.

Let us conclude this section by commenting on the difference between maximal
coarse grained entropy and the fine grained entropy associated with a single
screen. The first one provides a measure to the number of degrees of freedom
on the screen. Since it is a maximal entropy, counting the underlying degrees
of freedom, it is a property of the system itself rather than the state of the
system at a given time. The second entropy quantifies the entanglement between
the two screen subsystems. In general, these two entropies are not equal. The
coarse grained entropy is always greater or equal to the entanglement entropy.
The latter, being a fine grained entropy is not in general additive and it
does not obey a 2nd thermodynamical law.

The maximal coarse grained entropy of a single screen subsystem is given by
the area of the screen in Planck units. Contrary to the case of de Sitter
space, for which the number of degrees of freedom on the screens remains
constant as the screens evolve along the cosmological horizons, for all other
cases of closed FRW cosmologies, the number of degrees of freedom of the dual
holographic theory on the screens is not constant. As we already remarked,
this suggests that the evolution of the holographic dual to a generic closed
FRW Universe is not unitary, a very interesting possibility that has been
studied recently in [80, 83].

The entanglement entropy between the two screens has been studied in this
section. We find that the entanglement entropy to leading order is less than
the maximal coarse grained entropy (given by the area divided by $4G\hbar$),
except for the cases where both screens coincide at the same point in the
contracting phase of the cosmology. This highlights the particularity of the
de Sitter case, for which, as in the examples of eternal AdS black holes, the
entanglement entropy between the screens is given by the area of the horizon
divided by $4G\hbar$.

  1. 1.

In the Big Bang/Big Crunch cases, the area of the screens tends to zero as we
approach the Big Bang or the Crunch singularity. The classical entanglement
entropy vanishes well before, and the quantum corrections dominate. As the
Universe contracts the number of fundamental holographic degrees of freedom
decreases. The entanglement entropy between the screens should remain bounded
by the smaller (quantum) area between the screens, as expected. Near the Big
Bang/Big Crunch singularities, quantum corrections to the geometry as well as
to the generalized entropy become important.

  2. 2.

In the bouncing cases, the number of holographic degrees of freedom goes to
infinity as we approach the future or past null infinity, but the entanglement
entropy between the screens saturates a finite bound. In other words, the
entanglement entropy grows without ever becoming infinite, despite the fact
that extra degrees of freedom keep being added on the two screens.

The particularity of the de Sitter and the eternal black hole in AdS is due to
the fact that the holographic dual to these spacetimes is expected to be in a
thermofield double state. Indeed, taking a partial trace over the degrees of
freedom of one of the two copies leads to a thermal or maximally entangled
density matrix for the other copy. The corresponding von Neumann entropy is
given by the area of the horizon in Planck units. In the de Sitter case, when
the two screens coincide at the bifurcate horizon, their state is dual to the
Bunch-Davies vacuum state, which can be expressed as a thermofield double
state over degrees of freedom associated with the two static patches [84].
However, we saw that in the general closed FRW spacetimes, the patches
associated with the pode and antipode are not always causally disconnected.
The states of the single screen systems are thus expected to be more
complicated.

##  5 Big Bang/Big Crunch cosmologies with
$1/(n-1)\leq\gamma<1$

Let us discuss the holographic proposal for Big Bang/Big Crunch cosmologies
with $1/(n-1)\leq\gamma<1$. This class includes the moduli dominated cases
with $w=1$ and the radiation dominated cases for $n>3$. Recall from Section 2
that the Penrose diagrams are wider than tall, and the causal patches
associated with comoving observers at the pode and antipode, which are bounded
by cosmological horizons, are causally disconnected. No light signal from one
patch will reach the other before the Big Crunch time at $\eta=\gamma\pi$. The
apparent horizons are spacelike and lie in the region exterior of the causal
patches. See Figure 14. The new feature of these examples is that future
directed left-moving light rays emanating from the pode apparent horizon can
terminate at the Big Crunch singularity without entering the causal patch of
the pode. So we cannot in general apply Bousso’s spacelike projection theorem
to bound the entropy on spacelike slices between the pode and the apparent
horizon, without invoking new physics to describe the Big Crunch singularity.
Instead, following our rules, we place the screens at the boundaries of the
causal patches, namely the cosmological horizons. Bousso’s spacelike
projection theorem can be applied in order to encode holographically the
interior regions. Notice that as the screens evolve, they remain out of causal
contact and their trajectories never intersect, as shown in Figure 14.

${\cal S}_{\rm L}$${\cal S}_{\rm R}$ Figure 14: FRW cosmologies for
$1/(n-1)\leq\gamma<1$. The two screens ${\cal S}_{\rm L}$ and ${\cal S}_{\rm
R}$ are placed at the cosmological horizons (depicted by the dark dashed
lines). The apparent horizons are the dark diagonal lines.

So, in contrast with the cases $|\gamma|\geq 1$, the screen trajectories never
intersect. The distance between the screens starts small near the Big Bang
singularity, grows to a maximum value of order $a_{0}$ and then decreases in
the contracting phase. However, the screens remain out of causal contact for
the entire cosmological evolution. In particular, no light signal from one
screen will ever reach the other. The bulk picture suggests that the two
screens cannot be in a pure state. Indeed, the region between the cosmological
horizons remains inaccessible to the observers at the pode and antipode at all
times. Particles created at the Big Bang singularity in the past may reach the
Big Crunch singularity without entering the pode and antipode causal patches
and without crossing the screen trajectories along the cosmological horizons.
In order to describe the exterior region, we would need additional observers
following trajectories in the exterior region and the corresponding
holographic screens. Therefore, we do not have evidence to believe that the
degrees of freedom in the exterior region can be encoded holographically on a
two-screen system at the cosmological horizons. It seems plausible that the
bilayer prescription, which requires both exterior and interior contributions,
is not applicable in these cosmological cases. Indeed, applying this proposal
blindly for these cases leads to contradictory results. Rather, the
holographic entanglement entropy prescription could involve contributions from
the two interior regions only.161616It would be interesting to derive this
result directly from a bulk replica path integral and contrast with the
$|\gamma|\geq 1$ cases.

For the single screen subsystem, the dual homologous surface in region L is
the empty surface, implying that the leading classical geometrical
contributions to the entanglement entropy vanish. The entanglement entropy
should arise from semiclassical entropy contributions associated with bulk
field degrees of freedom on Cauchy slices connecting the pode and the screen.
Indeed, we can imagine entangled spin pairs, which start in the early past and
then separate further, entering the pode and antipode causal patches,
respectively. As a result, there can be non-trivial quantum entanglement