Von Neumann algebra description of inflationary cosmology

Min-Seok Seoa

aDepartment of Physics Education, Korea National University of Education,
Cheongju 28173, Republic of Korea

We study the von Neumann algebra description of the inflationary quasi-de
Sitter (dS) space. Unlike perfect dS space, quasi-dS space allows the nonzero
energy flux across the horizon, which can be identified with the expectation
value of the static time translation generator. Moreover, as a dS isometry
associated with the static time translation is spontaneously broken, the
fluctuation in time is accumulated, which induces the fluctuation in the
energy flux. When the inflationary period is given by $(\epsilon_{H}H)^{-1}$
where $\epsilon_{H}$ is the slow-roll parameter measuring the increasing rate
of the Hubble radius, both the energy flux and its fluctuation diverge in the
$G\to 0$ limit. Taking the fluctuation in the energy flux and that in the
observer’s energy into account, we argue that the inflationary quasi-dS space
is described by Type II∞ algebra. As the entropy is not bounded from above,
this is different from Type II1 description of perfect dS space in which the
entropy is maximized by the maximal entanglement. We also show that our result
is consistent with the observation that the von Neumann entropy for the
density matrix reflecting the fluctuations above is interpreted as the
generalized entropy.

##  1 Introduction

Whereas the spacetime geometry close to de Sitter (dS) well describes the
primordial inflation and the current accelerating expansion of the universe,
understanding its quantum nature is challenging. Studies on the quantum field
theory in the dS background tell us that a static observer in dS space is
surrounded by the (cosmological) horizon of radius $r_{H}=H^{-1}$ having
thermodynamic properties characterized by the Gibbons-Hawking temperature
$\beta^{-1}=H/(2\pi)$ and the entropy $S_{\rm GH}=A/(4G)=\pi M_{\rm
Pl}^{2}/H^{2}$, where $A$ is the horizon area given by $4\pi r_{H}^{2}$ and
$M_{\rm Pl}^{2}$ is defined as $G^{-1}$ [1]. This is quite similar to the
black hole as seen from far outside the horizon but the geometric structure of
dS space different from that of the black hole gives rise to several
ambiguities. That is, unlike the black hole horizon, the boundary of the
physically well defined compact object, the cosmological horizon in dS space
is observer dependent. Moreover, it is not clear that the dS entropy given by
the finite number counts the number of degrees of freedom of the region beyond
the horizon which is not compact.

Such ambiguities are expected to be fixed by the more complete description of
the thermodynamic behavior in quantum gravity. As an attempt to find it, it
was recently proposed that the entanglement needs to be described in terms of
the algebra of local observables, rather than the tensor product of two
Hilbert spaces defined on two separated regions (for reviews, see, e.g., [2,
3]). Here the algebra of local observables ${\cal A}$ called the von Neumann
algebra is required to satisfy the following conditions (see Section II. F of
[2]):

  * •

Any element $a$ in ${\cal A}$ is bounded: this means that the eigenvalues
$\alpha$ of $a$ are bounded, i.e., $|\alpha|<\infty$. Then given the vector
$|\psi\rangle$ in the Hilbert space consisting of the normalized vectors,
$a|\psi\rangle$ is also normalizable, so belongs to the Hilbert space. More
concretely, expanding $|\psi\rangle$ in terms of the eigenvectors
$|\alpha\rangle$ of $a$ as
$|\psi\rangle=\sum_{\alpha}c_{\alpha}|\alpha\rangle$ with
$\langle\psi|\psi\rangle=\sum_{\alpha}|c_{\alpha}|^{2}<\infty$, the norm of
$a|\psi\rangle=\sum_{\alpha}\alpha c_{\alpha}|\alpha\rangle$ given by
$\sum_{\alpha}|\alpha|^{2}|c_{\alpha}|^{2}$ is also finite provided
$|\alpha|<\infty$.

  * •

The algebra is closed under the Hermitian conjugation : Typically, ${\cal A}$
consists of the localized operators called the ‘smeared fields’, which is
written in the form of $\phi_{f}=\int d^{D}xf(x)\phi(x)$. Here $f(x)$ is
supported in the small region in which ${\cal A}$ is defined. Then its complex
conjugation $\phi_{f}^{\dagger}(x)=\int d^{D}xf(x)^{*}\phi^{\dagger}(x)$ is
also well defined and belongs to ${\cal A}$ as well.

  * •

The algebra is closed under the weak limit : if we consider the sequence of
operators $a_{1},a_{2},\cdots\in{\cal A}$ then any matrix element
$\langle\psi|a_{n}|\chi\rangle$ converges to $\langle\psi|a|\chi\rangle$ for
some $a\in{\cal A}$ in the $n\to\infty$ limit. From this, we can identify the
observables with the finite (but the large number of) observations within the
controllable error.

When the multiple of the identity is the only allowed center, a subset of
${\cal A}$ consisting of operators that commute with all elements in ${\cal
A}$, the von Neumann algebra is called a factor. Then the Hilbert space is
constructed by acting the local observables on the cyclic and separating state
$|\Psi\rangle$. Here the state $|\Psi\rangle$ is said to be cyclic for ${\cal
A}$ if the states $a|\Psi\rangle$ for $a\in{\cal A}$ are dense in the Hilbert
space, i.e., only the zero vector is orthogonal to all states in the form of
$a|\Psi\rangle$. Meanwhile, $|\Psi\rangle$ is called separating if $a=0$ is
the only local observables in ${\cal A}$ satisfying $a|\Psi\rangle=0$.

When we merely consider the quantized fluctuation around the fixed background
in the $G\to 0$ (or equivalently, $M_{\rm Pl}\to\infty$) limit, the algebra
typically belongs to Type III, in which neither the pure state nor the entropy
is well defined [4, 5, 6, 7]. 111Formally, the state is called pure if the
function $F_{\Psi}(a)=\langle\Psi|a|\Psi\rangle$ cannot be written in the form
of $p_{1}F_{\Phi_{1}}(a)+p_{2}F_{\Phi_{2}}(a)$ with $p_{1,2}>0$, which means
that decoherence does not take place thus interference effects appear. When
the state is not pure, it is called mixed. Indeed, $F_{\Psi}(a)$ is used to
define the density matrix through the trace, the linear functional of
operators satisfying the commutative property and the positivity. If the
finite trace is not defined, the divergent entropy is not renormalized. For
more complete discussion, see reviews [2, 3]. By taking dynamical gravity into
account through the ${\cal O}(G)$ corrections and treating diffeomorphism
invariance as gauge redundancy or constraint, the algebra becomes Type II
factor, in which the entropy as a finite, renormalized quantity can be
defined. For the black hole, $H_{L/R}-M$, the deviation of the ADM Hamiltonian
of the left/right patch around the ADM mass $M=r_{s}/(2G)$ ($r_{s}$ is the
horizon radius) can take any real value in the $G\to 0$ limit as $M$ becomes
divergent. Then the black hole thermodynamics is described by Type II∞ algebra
[8, 9], in which only a subset of observables has the finite trace hence the
entropy is not bounded from above. In contrast, in dS space, there is no
boundary at infinity. Instead, the static patch is bounded by the horizon
which is in thermal equilibrium with the Gibbons-Hawking radiation, resulting
in the vanishing energy flux across the horizon. As we will see, this implies
the absence of the operator analogous to $H_{L/R}-M$ in the black hole, hence
the algebraic description of dS space is different from that of the black
hole. In [10], it was found that when the static observer has the positive
energy as a random variable, a system of dS space and the observer is well
described by Type II1 algebra, in which the trace of any bounded operator is
finite. As a result, the entropy has an upper bound, which is saturated for
the maximally entangled state.

Meanwhile, in the inflationary era, $H$ is no longer a constant but a slowly
varying function of the flat time coordinate $t$. Then some of dS isometries
are slightly broken and the spacetime geometry is given by quasi-dS space. In
this case, the deviation of the background from perfect dS space is
parametrized by the slow-roll parameter $\epsilon_{H}$. When the inflation is
driven by the vacuum energy of the inflaton $\phi(t)$, a homogeneously
evolving scalar field, $\epsilon_{H}$ is proportional to $\dot{\phi}^{2}$ :

$\begin{split}\epsilon_{H}\equiv\dot{r_{H}}=-\frac{\dot{H}}{H^{2}}=\frac{4\pi\dot{\phi^{2}}}{M_{\rm
Pl}^{2}H^{2}}=\frac{4\pi G\dot{\phi^{2}}}{H^{2}},\end{split}$ (1)

where dot denotes the derivative with respect to $t$. In perfect dS case,
equations of motion are solved by the constant $H$ and $\dot{\phi}=0$, giving
$\epsilon_{H}=0$. On the other hand, even if $\dot{\phi}^{2}$ does not vanish,
we can suppress $\epsilon_{H}$ close to zero by taking the $G\to 0$ limit,
which we will focus on throughout this work. In any case, as $\epsilon_{H}\to
0$, the broken dS isometries are restored, implying the existence of the
approximate timelike Killing vector associated with the static time coordinate
$t_{s}$. At the same time, the time scale $(\epsilon_{H}H)^{-1}$ after which
$H$ is no longer approximated as a constant becomes infinity. We will
explicitly show that when we take this time scale to be the inflationary
period, the energy flux across the horizon and its fluctuation become
divergent in the $G\to 0$ limit. Moreover, the energy flux across the horizon
is interpreted as the expectation value of the static time translation
generator, and its fluctuation is driven by the fluctuation in time, hence
that in the value of $H$ at the end of inflation. Then we find that unlike
perfect dS space, the inflationary quasi-dS space is described by Type II∞
algebra rather than Type II1 algebra.

The organization of this article is as follows. In Section 2, we describe how
the change in the horizon area induced by the slow-roll gives rise to the
nonzero energy flux across the horizon, which is identified with the
expectation value of the static time translation generator. In Section 3.1, we
observe the modification of the von Neumann algebra description of the
inflationary quasi-dS space from that of perfect dS space when we take the
nonzero energy flux across the horizon into account. After claiming that
quasi-dS space is well described by Type II∞ algebra, we provide the
expression for the von Neumann entropy of the static patch in Section 3.2. In
Section 3.3, we relate this with the change in the horizon area considered in
Section 2 to complete our argument. Then we conclude with a brief comment
about the possibility that the inflationary quasi-dS space is described by
Type II1 algebra. This can happen when the inflationary period is much shorter
than $(\epsilon_{H}H)^{-1}$ as recently conjectured in the swampland program.
In Appendix A, we summarize various coordinates on dS space which are used
throughout the discussion. In Appendix B, details of the density matrix
considered in Section 3.2 are given.

##  2 Horizon dynamics of quasi-dS space

Figure 1: Penrose diagram of dS space. Over the whole period of inflation, a
static observer travelling along the timelike trajectory $\gamma$ is causally
accessible to the static patch $R$, the region bounded by the future horizon
${\cal H}^{+}$ and the past horizon ${\cal H}^{-}$. Two horizons intersect at
the bifurcation cut $b$. At a time close to the end of inflation, the static
observer is causally connected to the region denoted by $\infty$, the right of
the future horizon cut $b_{\infty}$. In the inflationary era, the universe
covers the region $R\cup T$, the past boundary ${\cal H}^{-}\cup{\cal
H}^{-\prime}$ of which corresponds to the initial singularity. The complement
static patch is denoted by $L$. The increasing directions of Kruskal-Szekeres
coordinates $U$ and $V$ are also depicted.

In this section, we estimate the energy flux across the future horizon ${\cal
H}^{+}$ during the inflationary period, which relates the change in the
horizon area to the static time translation generator. We assume that there is
no energy flux across the initial singularity, ${\cal H}^{-}\cup{\cal
H}^{-\prime}$ in Figure 1, the past boundary of the region $T\cup R$ covered
by the flat coordinates. Since the past horizon ${\cal H}^{-}$ belongs to the
initial singularity, we do not consider the energy flux across ${\cal H}^{-}$.

The energy-momentum tensor of the inflaton field $\phi$ with the canonical
kinetic term is given by

$\begin{split}T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\Big{(}\frac{1}{2}(\nabla\phi)^{2}+V(\phi)\Big{)}g_{\mu\nu}.\end{split}$
(2)

Since spacetime during inflation is homogeneous and isotropic at large scale,
we expect that the equations of motion are solved by $\phi(t)$ which depends
only on the flat time coordinate $t$ (the metric of dS space in the flat
coordinates $(t,r)$ can be found in (A.65)). Since $\dot{\phi}$ becomes zero
in the perfect dS limit ($H=$constant), it measures the deviation of the
background from dS space, which is evident from (1), i.e.,
$\epsilon_{H}\propto\dot{\phi}^{2}$. So far as $\epsilon_{H}$ is very tiny,
one can find the approximate dS isometries, which allow an approximate
timelike Killing vector along the direction of the static time coordinate
$t_{s}$, $k^{a}=(\partial_{t_{s}})^{a}=(\partial_{t}-Hr\partial_{r})^{a}$ (the
metric of dS space in the static coordinates $(t_{s},r_{s})$ can be found in
(A.61)). The component of the energy-momentum tensor associated with the
$t_{s}$ direction is written as

$\begin{split}T_{t_{s}t_{s}}=\dot{\phi}^{2}+(1-H^{2}r_{s}^{2})\Big{(}-\frac{1}{2}\dot{\phi}^{2}+V(\phi)\Big{)},\end{split}$
(3)

hence $T_{t_{s}t_{s}}=\dot{\phi}^{2}$ on the horizon $r_{s}=r_{H}=H^{-1}$.
Furthermore, the relation $\partial_{r_{s}}\phi=(\partial t/\partial
r_{s})\dot{\phi}=-[(Hr_{s})/(1-H^{2}r_{s}^{2})]\dot{\phi}$ (see (A.67)) gives

$\begin{split}T_{t_{s}r_{s}}=-\frac{Hr_{s}}{1-H^{2}r_{s}^{2}}\dot{\phi}^{2}.\end{split}$
(4)

For a more straightforward interpretation of this, we consider
$T_{t_{s}r_{*}}$ by converting $r_{s}$ into the tortoise coordinate $r_{*}$
defined in (A.62). From this, we can define ‘luminosity’, the energy flux
across the surface of constant $r_{s}$ by ${\cal L}=-4\pi
r_{s}^{2}T_{t_{s}r_{*}}$ [11]. Since

$\begin{split}T_{t_{s}r_{*}}=-Hr_{s}\dot{\phi}^{2}\end{split}$ (5)

becomes $T_{t_{s}r_{*}}=-\dot{\phi}^{2}$ on the horizon, the luminosity on the
horizon is given by ${\cal L}=4\pi H^{-2}\dot{\phi}^{2}=\epsilon_{H}M_{\rm
Pl}^{2}$. Meanwhile, the Kruskal-Szekeres coordinates $U$ and $V$, in terms of
which the metric is written as (A.64), are natural affine parameters on ${\cal
H}^{+}$ and ${\cal H}^{-}$, respectively. The energy-momentum tensor
components in the Eddington-Finkelstein coordinates ($t,r_{*}$) and those in
the Kruskal-Szekeres coordinates ($U,V$) are related as

$\begin{split}&T_{t_{s}t_{s}}=H^{2}U^{2}T_{UU}+H^{2}V^{2}T_{VV}-2H^{2}UVT_{UV},\\\
&T_{t_{s}r_{*}}=-H^{2}U^{2}T_{UU}+H^{2}V^{2}T_{VV}.\end{split}$ (6)

Then the simple relations

$\begin{split}T_{UU}=\frac{T_{t_{s}t_{s}}}{H^{2}U^{2}}=-\frac{T_{t_{s}r_{*}}}{H^{2}U^{2}}=\frac{\dot{\phi}^{2}}{H^{2}U^{2}}\end{split}$
(7)

are satisfied on ${\cal H}^{+}$ ($V=0$).

The energy-momentum tensor components on ${\cal H}^{+}$ are used to find the
first law of thermodynamics, which relates the energy flux across the horizon
to the change in the horizon area [12, 13]. In the perfect dS limit, we can
use the timelike Killing vector
$k^{a}=(\partial_{t_{s}})^{a}=H(U\partial_{U}-V\partial_{V})$ to find the
conserved current

$\begin{split}J^{a}=-T^{a}_{~{}b}k^{b}.\end{split}$ (8)

Since $V=0=$(constant) on ${\cal H}^{+}$, relations $k^{a}=HU\partial_{U}$ and
$dV=0$ thus $g^{ab}(dV)_{a}(dV)_{b}=0$ are satisfied, implying that the vector
$(\partial_{U})^{a}$ which is proportional to $g^{ab}(dV)_{b}$ is normal as
well as tangential to ${\cal H}^{+}$. Then ${\cal H}^{+}$ corresponds to the
Killing horizon and the energy flux across ${\cal H}^{+}$ is given by

$\begin{split}\Delta E&=-\int_{{\cal H}^{+}}d\Sigma_{a}J^{a}=\int_{{\cal
H}^{+}}d\Omega dU\sqrt{\gamma}T_{Ut_{s}}=\int_{{\cal H}^{+}}d\Omega
dU\sqrt{\gamma}(HU)T_{UU},\end{split}$ (9)

where $d\Sigma_{a}$ is the volume element on ${\cal H}^{+}$ and
$\sqrt{\gamma}=r_{H}^{2}=H^{-2}$. As can be inferred from the relation
$k^{a}=HU\partial_{U}$ on ${\cal H}^{+}$ and $T_{Ut_{s}}$ in the integrand,
$\Delta E$ is interpreted as the static time translation generator on ${\cal
H}^{+}$. More precisely, consider the semiclassical approximation in which the
$\hbar\to 0$ limit is taken and the quantum fluctuations around the mean
values of the operators are sufficiently small. 222But still, the solution can
be fluctuated by the statistical uncertainty or the ‘classical’ fluctuation.
This is generated by the decoherence, the loss of the interference effect
through the interaction of the system with the environment, the region the
observer is ignorant of. See discussion in the paragraph containing (20) and
references therein. Denoting the quantum state of the inflationary universe by
$|\Phi\rangle$ and the operator generating the static time translation on the
‘boundary’ ${\cal H}^{+}$ of the static patch $R$ by $H_{R}$, the solutions to
the equations of motion ${\phi}(t)$ and $T_{\mu\nu}$ can be regarded as
$\langle\Phi|\phi|\Phi\rangle$ and $\langle\Phi|T_{\mu\nu}|\Phi\rangle$,
respectively. Then $\Delta E$ is identified with
$\langle\Phi|H_{R}|\Phi\rangle$. Indeed, the horizon is regarded as a boundary
of the static patch. Whereas the bulk Hamiltonian as a constraint associated
with the static time translation vanishes which is evident in the ADM
formalism, the boundary Hamiltonian defined on the horizon is nonzero, and
plays the role of the generator of the static time translation. As we will
see, $\Delta E$ is proportional to the horizon area, and as argued in Section
2.4 and Section 3 of [9] (and references therein), it is canonically conjugate
to the ‘boost’ (static time translation) so identified with the static time
translation generator.

Meanwhile, since $r_{s}=r_{H}=H^{-1}$ is almost constant on ${\cal H}^{+}$,
the relations $U=H^{-1}e^{H(t_{s}-r_{*})}$ (see (A.63)) and
$t_{s}=t-\frac{1}{2H}\log\big{(}1-H^{2}r_{s}^{2}\big{)}$ (see (A.66)) indicate
that $dU=HUdt$ is satisfied on ${\cal H}^{+}$. Then from (7) one finds

$\begin{split}\langle\Phi|H_{R}|\Phi\rangle=\Delta E=\int
dt\frac{4\pi}{H^{2}}\dot{\phi}^{2}=\int dt\epsilon_{H}M_{\rm
Pl}^{2},\end{split}$ (10)

where the integrand is nothing more than the luminosity ${\cal L}$ and the
range of $t$ integration is taken to be the inflationary period during which
$H$ is almost constant. In addition, we assume
$|\dot{\epsilon_{H}}/(\epsilon_{H}H)|\ll 1$ such that $\epsilon_{H}$ does not
vary much during the inflationary period. Since the value of $H$ considerably
deviates from the initial value after $\Delta t={\cal
O}(1)\times(\epsilon_{H}H)^{-1}$, we may take
$t\in(-c(2\epsilon_{H}H)^{-1},c(2\epsilon_{H}H)^{-1})$ with $c$ being some
constant smaller than $1$ (not to spoil the perturbative expansion with
respect to $\epsilon_{H}$), which becomes $t\in(-\infty,+\infty)$ in the
perfect dS limit $\epsilon_{H}\to 0$. Then $\langle\Phi|H_{R}|\Phi\rangle$ is
estimated as

$\begin{split}\langle\Phi|H_{R}|\Phi\rangle\simeq\frac{c}{\epsilon_{H}H}\times\epsilon_{H}M_{\rm
Pl}^{2}=c\frac{M_{\rm Pl}^{2}}{H},\end{split}$ (11)

i.e., $M_{\rm Pl}^{2}/H$ up to ${\cal O}(1)$ (but smaller than $1$)
coefficient, showing that $\langle\Phi|H_{R}|\Phi\rangle$ is insensitive to
$\epsilon_{H}$, or equivalently, $\dot{\phi}^{2}$ at leading order.

The backreaction of the energy flux across the horizon leads to the
deformation of the geometry parametrized by expansion, shear, and rotation.
When the background is close to dS space, the horizon can be approximated as a
Killing horizon, where all the three parameters vanish at leading order. Then
the Raychaudhuri equation for the expansion $\Theta=A^{-1}dA/dU$ which
describes the change in the horizon area $A=4\pi r_{H}^{2}=4\pi H^{-2}$ is
approximated as

$\begin{split}\frac{d\Theta}{dU}\simeq-\frac{8\pi}{M_{\rm
Pl}^{2}}T_{ab}(\partial_{U})^{a}(\partial_{U})^{b}=-\frac{8\pi}{M_{\rm
Pl}^{2}}T_{UU},\end{split}$ (12)

from which we can replace $T_{UU}$ by $-[M_{\rm Pl}^{2}/(8\pi)]d\Theta/dU$.
Putting this into $\langle\Phi|H_{R}|\Phi\rangle$, we obtain

$\begin{split}\langle\Phi|H_{R}|\Phi\rangle&=-\frac{M_{\rm Pl}^{2}}{8\pi}\int
d\Omega dU\sqrt{\gamma}(HU)\frac{d\Theta}{dU}\\\ &=-\frac{M_{\rm
Pl}^{2}}{8\pi}\int
d\Omega\Big{[}\sqrt{\gamma}HU\Theta\Big{|}_{U=0}^{U\simeq\infty}-\int
dU\Big{(}\sqrt{\gamma}H+HU\frac{d\sqrt{\gamma}}{dU}+\sqrt{\gamma}U\frac{dH}{dU}\Big{)}\Theta\Big{]}.\end{split}$
(13)

Noting that $\partial_{U}=(HU)^{-1}\partial_{t_{s}}$ and $dt_{s}=dt$ on ${\cal
H}^{+}$, one finds that $dH/dU=-\epsilon_{H}H/U$,
$d\sqrt{\gamma}/dU=2\epsilon_{H}/(H^{2}U)$ and $\Theta=2\epsilon_{H}/U$. Then
the last two terms in (13) are ${\cal O}(\epsilon_{H}^{2}\Delta t)$. For the
first surface term, since $\sqrt{\gamma}HU\Theta$ is ${\cal O}(\epsilon_{H})$,
the variation of $\sqrt{\gamma}HU\Theta$ over ${\cal H}^{+}$ is ${\cal
O}(\epsilon_{H}^{2}\Delta t)$. Therefore, the second term in (13) gives the
leading contribution to $\langle\Phi|H_{R}|\Phi\rangle$ of ${\cal
O}(\epsilon_{H}\Delta t)$ :

$\begin{split}\langle\Phi|H_{R}|\Phi\rangle&=\frac{M_{\rm Pl}^{2}}{8\pi}\int
dUH\int d\Omega\sqrt{\gamma}\Theta+{\cal O}(\epsilon_{H}^{2}\Delta
t)=\frac{M_{\rm Pl}^{2}}{8\pi}\int dUH\frac{dA}{dU}+{\cal
O}(\epsilon_{H}^{2}\Delta t)\\\ &=\frac{M_{\rm Pl}^{2}}{8\pi}\int
dtH\frac{dA}{dt}+{\cal O}(\epsilon_{H}^{2}\Delta t)=\int
dt\frac{H}{2\pi}\frac{d}{dt}\Big{(}\frac{A}{4G}\Big{)}+{\cal
O}(\epsilon_{H}^{2}\Delta t).\end{split}$ (14)

Since the Gibbons-Hawking temperature is given by $H/(2\pi)$, the integrand
can be written in the form of $T\Delta S$, which is consistent with the first
law of thermodynamics. We also note that the explicit calculation of the
integrand reproduces (10). To see the physical meaning of
$\langle\Phi|H_{R}|\Phi\rangle=\Delta E$ more clear, we recall that the static
time translation is a diffeomorphism, the gauge invariance of gravity, hence
it acts as a constraint on the dynamics of quantum gravity. As a result, just
like the ADM mass of the black hole, the associated charge gets contribution
from the surface integral on the boundary (${\cal H}^{+}$ for dS space) only,
which is given by $\beta^{-1}S=M_{\rm Pl}^{2}/(2H)=r_{H}/(2G)$ [14]. This is
supported by the fact that the energy inside the horizon in the perfect dS
limit is estimated as $M_{\rm Pl}^{2}/(2H)$, which is obtained by multiplying
the energy density during inflation $T_{tt}\simeq[3/(8\pi)]M_{\rm
Pl}^{2}H^{2}$ by the volume inside the horizon $(4\pi/3)H^{-3}$. 333The radius
of the horizon in the flat coordinates $H^{-1}e^{-Ht}$ gives
$V=(4\pi/3)e^{3Ht}\times(H^{-1}e^{-Ht})^{3}$ where the factor $e^{3Ht}$ comes
from $\sqrt{-g}$ restricted to the spatial directions. If we regard the slow-
roll as the adiabatic process, $H$ slowly decreases in time, then the ‘ADM
mass’ just after the end of inflation $M_{\rm Pl}^{2}/(2H)+\Delta E$ is
identified with $M_{\rm Pl}^{2}/(2H_{f})$ where $H_{f}$ is the value of $H$ at
that time. As $H_{f}$ significantly deviates from the initial $H$, we expect
that $\Delta E$ is at most given by the same order as $M_{\rm Pl}^{2}/(2H)$.

It is remarkable that $\langle\Phi|H_{R}|\Phi\rangle$ becomes divergent in the
$G\to 0$, or equivalently, $M_{\rm Pl}^{2}\to\infty$ limit. Indeed, whereas
the perfect dS limit $\epsilon_{H}\to 0$ is trivially obtained by taking
$\dot{\phi}\to 0$, we can also reach the same limit by taking $G\to 0$ even if
$\dot{\phi}^{2}$ is kept finite, as can be noticed from (1). In this case, as
$\epsilon_{H}$ almost vanishes, the spacetime geometry can be well
approximated by the perfect dS space. Moreover, even though the effects of
$\epsilon_{H}$ become negligible, the ratio $\epsilon_{H}/S_{\rm
GH}=4\pi^{2}\dot{\phi}^{2}/H^{4}$, which is independent of $G$, can be kept
nonvanishing. This is quite similar to what was assumed to find the AdS/CFT
correspondence : the string coupling $g_{s}$ is taken to vanish to decouple
string interactions, but the ‘t Hooft coupling $g_{\rm YM}^{2}N\sim g_{s}N$ is
kept fixed [15]. Therefore, by keeping $\epsilon_{H}/S_{\rm GH}>1$, i.e.,
$\epsilon_{H}>H^{2}/M_{\rm Pl}^{2}$, we can still forbid eternal inflation,
and the formulation based on the semiclassical approximation is reliable
against large quantum fluctuations or the non-perturbative effects [16].
Indeed, recent swampland conjectures concerning the instability of dS space
claimed the lower bound on the potential slope [17, 18, 19, 20], which forbids
eternal inflation (see, e.g., [21, 22, 23, 23]). But at the same time, as
implied by the nonzero energy flux, the horizon is no longer in thermal
equilibrium with the Gibbons-Hawking radiation. Then the state $|\Phi\rangle$
breaks the dS isometry by allowing the nonzero
$\langle\Phi|H_{R}|\Phi\rangle$, instead of being annihilated by $H_{R}$, just
like the Unruh state describing the evaporating black hole [25] (see also [26,
27] for recent discussions). This can be contrasted with perfect dS space
($\dot{\phi}=0$), in which the horizon is in thermal equilibrium hence the
energy flux across the horizon vanishes. Then the quantum state $|\Psi\rangle$
for perfect dS space respects the dS isometry. This is called the Bunch-Davies
state [28, 29], the dS analogy of the Hartle-Hawking state of the black hole
[30].

Before moving onto the fluctuation, we comment on our assumption
$|\dot{\epsilon_{H}}/(\epsilon_{H}H)|\ll 1$. Since

$\begin{split}\frac{\dot{\epsilon_{H}}}{\epsilon_{H}H}=\frac{\ddot{H}}{H{\dot{H}}}-2\frac{\dot{H}}{H^{2}}=2(-\eta_{H}+\epsilon_{H}),\end{split}$
(15)

where $\eta_{H}\equiv-{\ddot{H}}/(2{\dot{H}}H)$ is another slow-roll
parameter, this assumption indicates that $\epsilon_{H}\simeq\eta_{H}$. On the
other hand, there is a priori no reason that $\epsilon_{H}$ and $\eta_{H}$ are
similar in size : we just require that these two parameters are sufficiently
smaller than $1$. To see the role of $\eta_{H}$, we consider the perturbative
expansion of $H$ around some pivotal value $H_{0}$, say, the initial value of
$H$,

$\begin{split}H(t)&=H_{0}+\dot{H}_{0}\Delta t+\frac{1}{2}\ddot{H}_{0}\Delta
t^{2}+\cdots\\\ &=H_{0}\big{(}1-\epsilon_{H}(H_{0}\Delta
t)+\eta_{H}\epsilon_{H}(H_{0}\Delta t)^{2}+\cdots\big{)},\end{split}$ (16)

which is valid for $\epsilon_{H},|\eta_{H}|<1$ and $H_{0}\Delta t<{\rm
min}(1/\epsilon_{H},1/|\eta_{H}|)$. In particular, $\Delta t$ is at most
$c\times{\rm min}(1/\epsilon_{H},1/|\eta_{H}|)$ with $c$ being a constant
smaller than $1$, which can be employed as the inflationary period. Denoting
the value of $H(t)$ at the end of inflation, i.e., $H_{0}\Delta t\simeq
c\times{\rm min}(1/\epsilon_{H},1/|\eta_{H}|)$, by $H_{f}$, and using the
definition of $\epsilon_{H}\equiv-{\dot{H}}/H^{2}$, one finds that $\Delta E$
can be written as

$\begin{split}\Delta E&=\int dt\epsilon_{H}M_{\rm
Pl}^{2}=-\int\frac{dH}{H^{2}}M_{\rm Pl}^{2}=M_{\rm
Pl}^{2}\Big{(}\frac{1}{H_{f}}-\frac{1}{H_{0}}\Big{)}\\\ &=\epsilon_{H}\Delta
t+(\epsilon_{H}^{2}-\epsilon_{H}\eta_{H})H_{0}\Delta t^{2}+\cdots,\end{split}$
(17)

which is positive. Therefore, if $\epsilon_{H}\gtrsim|\eta_{H}|$, we take
$H\Delta t\simeq c/\epsilon_{H}$, which leads to

$\begin{split}\Delta E=\frac{M_{\rm
Pl}^{2}}{H_{0}}\Big{(}c+(\epsilon_{H}^{2}-\epsilon_{H}\eta_{H})c^{2}+\cdots\Big{)},\end{split}$
(18)

that is, with appropriately chosen $c<1$, $\Delta E\sim cM_{\rm Pl}^{2}/H$, as
we considered so far. In contrast, if $\epsilon_{H}<|\eta_{H}|$, we take
$H\Delta t\simeq c/|\eta_{H}|$, giving

$\begin{split}\Delta E=\frac{M_{\rm
Pl}^{2}}{H_{0}}\Big{(}c\frac{\epsilon_{H}}{|\eta_{H}|}+\Big{(}\frac{\epsilon_{H}^{2}}{\eta_{H}^{2}}-\frac{\epsilon_{H}}{\eta_{H}}\Big{)}c^{2}+\cdots\Big{)},\end{split}$
(19)

which is smaller than $M_{\rm Pl}^{2}/H$. This bound is saturated when
$\epsilon_{H}$ becomes close to $\eta_{H}$. This shows that our estimation
$\Delta E\sim M_{\rm Pl}^{2}/H$ up to a constant smaller than $1$ in fact
corresponds to the maximal value of the increment in the ADM mass. Treating
$\epsilon_{H}$ and $\eta_{H}$ as independent parameters, one may regard the
ratio $\epsilon_{H}/|\eta_{H}|$ to be another factor smaller than $1$
multiplied to $M_{\rm Pl}^{2}/H$.

Our discussion so far is made in terms of the solutions to the classical
equations of motion, which are regarded as the expectation values of the
operators with respect to $|\Phi\rangle$. On the other hand, as a dS isometry
associated with the static time translation is spontaneously broken by the
quasi-dS background, the quantum fluctuation in $\phi$ combines with that in
the trace of the spatial metric [31, 32] (see also [33, 34]), forming the
gauge invariant operator which excites the curvature perturbation [35, 36]. As
the universe undergoes accelerated expansion, the wavelength of the curvature
perturbation is stretched beyond the horizon scale, after which the
fluctuation can be treated as a classical one through the open system
description called decoherence [37, 38, 39, 40, 41, 42]. In terms of the
standard cosmological perturbation theory, this can be explained by the fact
that the perturbation with wavelength larger than $H^{-1}$ no longer
oscillates and may be treated as a frozen distribution of a classical field
$\phi(t)$ [43]. This contributes to the accumulated uncertainty of the
classical trajectory $\phi(t)$ during $\Delta t$ given by [43] (see also [44,
45, 46])

$\begin{split}\langle\phi(t)^{2}\rangle\equiv\delta\phi^{2}=\Big{(}\frac{H}{2\pi}\Big{)}^{2}H\Delta
t.\end{split}$ (20)

Noting $\beta^{-1}=H/(2\pi)$, the accumulated uncertainty may be interpreted
as the thermal fluctuation, which induces the fluctuation in $H_{R}$ estimated
as

$\begin{split}\delta
H_{R}=\Big{(}\frac{\partial}{\partial\beta}\langle\Phi|H_{R}|\Phi\rangle\Big{)}^{1/2}\simeq\frac{M_{\rm
Pl}}{\sqrt{2\pi}},\end{split}$ (21)

where the multiplication by $c^{1/2}$ is implicit. We can reach the similar
conclusion in the following way. In perfect dS space, different constant $t$
(flat time) slices are physically equivalent due to the isometry associated
with the $t_{s}$ (static time) translation : the translation of $t$ can be
compensated by the scaling of $r$, leaving $r_{s}=re^{Ht}$ (see (A.66)) hence
the metric in the static coordinates (A.61) unchanged. In quasi-dS space,
however, the time dependent classical solution $\phi(t)$ plays the role of
‘clock’ distinguishing the specific constant $t$ slice from others. Then the
fluctuation in $\phi(t)$ given by (20) leads to the fluctuation in time
$\delta t$, 444This should not be confused with $\Delta t$, the time interval
without fluctuation during which the fluctuation in $\phi(t)$ is accumulated :
$t$ is the time appearing in the background geometry, at each instant of which
the fluctuation of $t$ given by $\delta t$ is accumulated by $\delta\phi$.
thus that in $H$ :

$\begin{split}|\delta H|=|\dot{H}\delta
t|=\Big{|}\frac{\dot{H}}{\dot{\phi}}\delta\phi\Big{|}=\sqrt{\frac{\epsilon_{H}}{\pi}}H\frac{H}{M_{\rm
Pl}}(H\Delta t)^{1/2}.\end{split}$ (22)

From this, we can estimate the fluctuation in $\langle\Phi|H_{R}|\Phi\rangle$
over the inflationary period $\Delta t=c(\epsilon_{H}H)^{-1}$ to obtain

$\begin{split}|\delta H_{R}|=c^{1/2}\frac{M_{\rm Pl}^{2}}{H^{2}}|\delta
H|\simeq\frac{M_{\rm Pl}}{\sqrt{\pi}},\end{split}$ (23)

which diverges in the $G\to 0$ limit.

Since $\langle\Phi|H_{R}|\Phi\rangle\sim{\cal O}(G^{-1})$ is divergent in the
$G\to 0$ limit, one may define the ‘renormalized’ static time translation
generator by $H^{\prime}_{R}=H_{R}-\langle\Phi|H_{R}|\Phi\rangle$. But still,
$\langle\Phi|(H^{\prime}_{R})^{2}|\Phi\rangle=(\delta H_{R})^{2}\sim{\cal
O}(G^{-1})$ is also divergent in the $G\to 0$ limit, so
$H^{\prime}_{R}|\Phi\rangle$ has a divergent norm and $H^{\prime}_{R}$ is not
well defined. In order for the static time translation generator to be well
defined, i.e., the expectation values of both the generator and its square to
be finite in the $G\to 0$ limit, we need to extend the spacetime to the
regions $L$ and $B$ in Figure 1 such that $|\Phi\rangle$ describes the whole
quasi-dS manifold covering $R\cup T\cup L\cup B$, as suggested in Section 2 of
[8]. Then we can introduce $H_{L}$, the static time translation generator in
the complementary static patch $L$. Since $L$ is just a copy of $R$, the
energy flux across the future horizon ${\cal H}^{+\prime}$ is the same in size
as that across ${\cal H}^{+}$. But the static time in $L$ flows in the
opposite direction to that in $R$, so $\Delta E$ through ${\cal H}^{+}$ (say,
flowing from $R$ to $T$) has an opposite sign to that through ${\cal
H}^{+\prime}$ (say, flowing from $L$ to $B$). From this, we expect that
$\Delta E$ on ${\cal H}^{+\prime}$ is identified with
$-\langle\Phi|H_{L}|\Phi\rangle$ and the sum of $\Delta E$ on ${\cal H}^{+}$
and that on ${\cal H}^{+\prime}$ vanishes, giving
$\langle\Phi|H_{R}|\Phi\rangle=\langle\Phi|H_{L}|\Phi\rangle$. Now let us
define the total static time translation generator by
$H_{0}=H_{R}-H_{L}=H^{\prime}_{R}-H^{\prime}_{L}$. This acts on the
thermofield double state $|{\rm
TFD}\rangle=\sum_{n}c_{n}|E_{n}\rangle_{R}|E_{n}\rangle_{L}$ describing the
entanglement between states living on ${\cal H}^{+}$ and ${\cal H}^{+\prime}$.
Since both $H_{0}$ and $(H_{0})^{2}$ annihilate the thermofield double state
($H_{0}|{\rm
TFD}\rangle=\sum_{n}c_{n}(E_{n}-E_{n})|E_{n}\rangle_{R}|E_{n}\rangle_{L}=0$),
their expectation values are finite, hence $H_{0}$ is well defined.

But the fact that $H^{\prime}_{L}$ and $H^{\prime}_{R}$ are not well defined
indicates that a factorization of the Hilbert space into the Hilbert spaces
defined on $R$ and $L$ is not well defined in the $G\to 0$ limit. We can
compare our results, $\langle\Phi|H_{R}|\Phi\rangle\sim{\cal O}(G^{-1})$ and
$\delta H_{R}\sim{\cal O}(G^{-1/2})$, with the boundary Hamiltonian of the
eternal AdS black hole, which describes the ${\cal N}=4$ super Yang-Mills
theory [8]. In the large $N$ limit, Hamiltonians in the left and right
boundaries $H_{L}$ and $H_{R}$ have thermal expectation values of ${\cal
O}(N^{2})\sim{\cal O}(G^{-1})$ and $H^{\prime}_{L/R}=H_{L/R}-\langle
H_{L/R}\rangle$ satisfy $\langle{H^{\prime}_{L/R}}^{2}\rangle\sim{\cal
O}(N^{2})$ hence $\delta H_{L/R}\sim{\cal O}(N)\sim{\cal O}(G^{-1/2})$,
showing the same behaviors as $\langle\Phi|H_{R}|\Phi\rangle$ and $\delta
H_{R}$, respectively.

##  3 von Neumann algebra for inflationary quasi-dS space

###  3.1 von Neumann algebra for quasi-dS space

In order to find the von Neumann algebra description of quasi-dS space, we
first consider Type II1 algebra for dS space discussed in [10] and see how it
is modified by the nonzero energy flux across the horizon we obtained in
Section 2. Since a static observer can access the static patch $R$ only, the
quantum description of (quasi-)dS space as seen by the static observer is made
in terms of the local observables on $R$. Moreover, in the dS limit, the
static patch is invariant under the subgroup of the dS isometry consists of
the static time translation and rotation hence operators on the static patch
are required to be invariant under the subgroup. However, as pointed out in
[10], the only operators that commute with the static time translation
generator are those proportional to the identity. In order to resolve this
issue, [10] suggested that the nontrivial operators can be considered by
taking the Hamiltonian of the static observer into account in addition, such
that the total Hamiltonian is given by $H_{0}+\widehat{q}$, where $H_{0}$ is
the static time translation generator and $\widehat{q}$ is the observer
Hamiltonian. To see the meaning of $\widehat{q}$, we note that the observer
detects the thermal radiation using, for example, the apparatus consisting of
the large number of atoms. In this case, the energy eigenvalues $q$ are almost
continuous, forming the band structure such that we can detect the thermal
radiation with any frequency by observing the transition between the energy
levels of the apparatus. Since the energy levels of the apparatus described
above is also bounded from below, it is reasonable to assume that the
eigenvalues $q$ are nonnegative and $\widehat{q}$ acts on
$L^{2}(\mathbb{R}^{+})$, the Hilbert space of the square integrable function
of $q$. Moreover, we can define the time measured by the observer’s clock
along the observer’s worldline, which corresponds to the eigenvalue of the
operator $\widehat{q}$ conjugate to $\widehat{p}$.

For quasi-dS space, by taking the inflationary period to be
$(\epsilon_{H}H)^{-1}$, the nonzero energy flux across the horizon
$\langle\Phi|H_{R}|\Phi\rangle$ becomes divergent in the $G\to 0$ limit. As
discussed in Section 2, $H^{\prime}_{R}=H_{R}-\langle\Phi|H_{R}|\Phi\rangle$,
the renormalized static time translation generator (restricted to the static
patch $R$) is not well defined since
$\langle\Phi|(H^{\prime}_{R})^{2}|\Phi\rangle\sim{\cal O}(G^{-1})$ also
diverges in the $G\to 0$ limit. The similar problem also arises in the
boundary Hamiltonian of the AdS black hole, in which the issue is circumvented
by considering $H^{\prime}_{R}/N$ since ${\cal O}(N)$ is equivalent to ${\cal
O}(G^{-1/2})$ in the large $N$ limit [8]. Motivated by this, we may define
$H^{\prime}_{R}/N$ where $N$ is now the dimensionless parameter of ${\cal
O}(G^{-1/2})$, say, $M_{\rm Pl}/H$. Then following [8], $H^{\prime}_{R}/N$ can
be expressed as

$\begin{split}\frac{H^{\prime}_{R}}{N}=U+\frac{H_{0}}{N},\end{split}$ (24)

where $U$ is an operator which commutes with any observables on $R$. Whereas
$U$ is in fact $H^{\prime}_{L}/N$, it is also identified with
$H^{\prime}_{R}/N$ in the $G\to 0$ limit, in which $H_{0}/N\to 0$ and
$[a,H^{\prime}_{R}/N]=(i/N)\partial a/\partial t_{s}\to 0$ for any operator
$a$ on $R$. Moreover, the divergence of
$\langle\Phi|H_{R}|\Phi\rangle/N\sim{\cal O}(G^{-1/2})$ in the $G\to 0$ limit
indicates that the lower bound on $U$ is $-\infty$ thus the eigenvalue of $U$
can take any real number in $(-\infty,+\infty)$ and the Hilbert space relevant
to $U$ is given by $L^{2}(\mathbb{R})$. In the following discussion, we will
take the factor $1/N$ to be implicit for convenience. Indeed, as pointed out
in [8], if we are interested in the ordinary functions of $N$, we may work
with $H^{\prime}_{R}$ instead of $H^{\prime}_{R}/N$. 555Our discussion is
based on the canonical ensemble in which the fluctuation $\delta
H_{R}\sim{\cal O}(G^{-1/2})$ is divergent in the $G\to 0$ limit. On the other
hand, we don’t need to divide $H_{R}$ by $N$ in the microcanonical ensemble as
the fluctuation in $H_{R}$ is restricted to be ${\cal O}(1)$ [9]. The reason
we do not consider the microcanonical ensemble is that the divergent
fluctuation in $H_{R}$ is induced by the fluctuation in the curvature
perturbation and so far as we know there is no physical reason to restrict the
fluctuation to be ${\cal O}(1)$.  Defining $X\equiv NU=H^{\prime}_{L}$,
$H^{\prime}_{R}$ is written as $X+H_{0}$ with $X$ acting on the Hilbert space
$L^{2}(\mathbb{R})$ and the (renormalized) Hamiltonian restricted to the
static patch $R$ is given by $\widehat{H}=H_{0}+X+\widehat{q}$.

Now we can construct Type III algebra ${\cal A}_{R}\otimes
B(\mathbb{R})\otimes B(\mathbb{R}^{+})$, where ${\cal A}_{R}$ is the algebra
of the observables on $R$ and $B(\mathbb{R}^{(+)})$ is the algebra of bounded
operators acting on $\mathbb{R}^{(+)}$. This is converted into Type II algebra
by imposing the diffeomorphism invariance as a gauge constraint. Focusing on
an (approximate) isometry of the static time translation, the algebra of the
observables on $R$ is given by an invariant subalgebra with respect to
$\widehat{H}$,

$\begin{split}\widehat{\cal A}_{R}=\big{(}{\cal A}_{R}\otimes
B(\mathbb{R})\otimes B(\mathbb{R}^{+})\big{)}^{\widehat{H}}.\end{split}$ (25)

The elements of $\widehat{{\cal A}}_{R}$ can be explicitly written by
introducing an operator $\widehat{p}$ conjugate to $\widehat{q}$ satisfying
$[\widehat{q},\widehat{p}]=i$, which is interpreted as a (fluctuating) time
measured by the static observer. By requiring $[\widehat{q},H_{0}]=0$,
$\widehat{q}$ belongs to $\widehat{{\cal A}}_{R}$. Moreover, for any
$a\in{\cal A}_{R}$, one finds that its gravitational dressing or the outer
automorphism,

$\begin{split}{a}^{\prime}=e^{i(H_{0}+X)\widehat{p}}ae^{-i(H_{0}+X)\widehat{p}}\end{split}$
(26)

satisfies $[\widehat{q},{a}^{\prime}]=-[H_{0}+X,{a}^{\prime}]$, or
equivalently, $[\widehat{H},{a}^{\prime}]=0$. Therefore, $\widehat{{\cal
A}}_{R}$ is generated by

$\begin{split}\\{{a}^{\prime}=e^{i(H_{0}+X)\widehat{p}}ae^{-i(H_{0}+X)\widehat{p}},\widehat{q}\\}.\end{split}$
(27)

By taking the conjugation by $e^{-i(H_{0}+X)\widehat{p}}$, one finds that it
is equivalent to $\\{a,\widehat{q}-(H_{0}+X)\\}$.

In order to implement the finite (renormaliazed) entropy, we need to define
‘trace’ in a sensible way. The trace here refers to, in a somewhat abstract
sense, a linear functional of operators satisfying ${\rm Tr}(ab)={\rm Tr}(ba)$
and ${\rm Tr}(a^{\dagger}a)>0$ for a nonzero $a$. This can be used to defined
the ‘renormalized’ (thus finite) entropy in dS space (for further discussion,
see [2] and [10], which is also reviewed in the paragraph containing (29)).
Just like the case of perfect dS space, the trace can be defined in terms of
the Bunch-Davies state $|\Psi\rangle$ which is invariant under the dS
isometries. 666 We note that since $H_{0}$ belongs to the isometry generators,
$H_{0}|\Psi\rangle=0$ is satisfied and $|\Psi\rangle$ does not distinguish
$\widehat{q}-H_{0}$ from $\widehat{q}$. From this and $[X,a]=0$, one finds
that for $a^{\prime}$ defined in (26), $\langle\Psi|a^{\prime}|\Psi\rangle$ is
identified with $\langle\Psi|a|\Psi\rangle$.  Then we define the trace of any
operator $\widehat{a}\in\widehat{\cal A}_{R}$ by

$\begin{split}{\rm Tr}(\widehat{a})=\int_{-\infty}^{\infty}\beta dxe^{\beta
x}\int_{0}^{\infty}\beta dqe^{-\beta
q}\langle\Psi|\widehat{a}|\Psi\rangle,\end{split}$ (28)

where $x$ is the eigenvalue of $X$.

For perfect dS space, since the divergent fluctuation in $X$ is not taken into
account, the integration over $x$ is absent and the trace of the identity is
not divergent but finite : ${\rm Tr}(1)=1$. To see the physical meaning of the
identity in this case, let us observe the trace of other operators in
$\widehat{{\cal A}}_{R}$ generated by $\\{a,\widehat{q}-H_{0}\\}$. For
$a\in{\cal A}_{R}$ which is independent of $\widehat{q}-H_{0}$,

$\begin{split}&{\rm Tr}(a)={\rm
Tr}(a1)=\Big{(}\beta\int_{0}^{\infty}dqe^{-\beta
q}\Big{)}\langle\Psi|a|\Psi\rangle=\langle\Psi|a|\Psi\rangle,\end{split}$ (29)

showing that ${\rm Tr}(a)$ is the expectation value of the local operator $a$
with respect to $|\Psi\rangle$. On the other hand, when the operator
$G\in\widehat{{\cal A}}_{R}$ is given by a function of $\widehat{q}-H_{0}$,

$\begin{split}{\rm Tr}(G(\widehat{q}-H_{0}))&={\rm
Tr}(G(\widehat{q}-H_{0})1)=\beta\int_{0}^{\infty}dqe^{-\beta
q}\langle\Psi|G(\widehat{q}-H_{0})|\Psi\rangle\\\
&=\beta\int_{0}^{\infty}dqe^{-\beta
q}\langle\Psi|G(q)|\Psi\rangle=\beta\int_{0}^{\infty}dqe^{-\beta
q}G(q),\end{split}$ (30)

from which one finds that for Tr$(G)$ to be an expectation value, $\beta
e^{-\beta q}$ is interpreted as the probability distribution of the
eigenvalues of $\widehat{q}$. Then it is reasonable to interpret the identity
as a density matrix describing the maximal entanglement : the entropy of the
system in the perfect dS background is maximized at $-{\rm Tr}(1\log 1)=0$. In
other words, among the states in the Hilbert space ${\cal H}\otimes
L^{2}(\mathbb{R}^{+})$ on which the algebra $\widehat{{\cal A}}_{R}$ acts,

$\begin{split}|\Psi_{\rm
max}\rangle=|\Psi\rangle\otimes\int_{0}^{\infty}dq\sqrt{\beta}e^{-\beta{q}/2}|q\rangle\end{split}$
(31)

gives the maximal entropy as the density matrix $\rho_{\rm max}=1$ is obtained
from $\langle\Psi_{\rm max}|\widehat{a}|\Psi_{\rm max}\rangle={\rm
Tr}(\rho_{\rm max}\widehat{a})$ [10]. This is a feature of Type II1 von
Neumann algebra.

In contrast, for quasi-dS space, the integration over $x$ in (28) leads to
Tr$(1)=\infty$, which means that the density matrix for the maximal
entanglement is not renormalized. Then $\widehat{{\cal A}}_{R}$ belongs to
Type II∞ von Neumann algebra, in which the trace is sensibly defined only for
the subset of the algebra.

###  3.2 Density matrix and entropy

Since the inflationary quasi-dS background slightly breaks the dS isometry,
the quantum state during inflation $|\Phi\rangle$ is no longer the Bunch-
Davies state $|\Psi\rangle$. In order to find the density matrix in this case,
we consider a state

$\begin{split}&|\widehat{\Phi}\rangle=|\Phi\rangle\otimes g(X)\otimes
f(\widehat{q})\end{split}$ (32)

in ${\cal H}\otimes L^{2}(\mathbb{R})\otimes L^{2}(\mathbb{R}^{+})$, where

$\begin{split}&g(X)=\int_{-\infty}^{\infty}g(x)|x\rangle,\quad\quad
f(\widehat{q})=\int_{0}^{\infty}f(q)|q\rangle.\end{split}$ (33)

Here $|g(x)|^{2}$ and $|f(q)|^{2}$ are interpreted as the probability
distributions of $x$ and $p$ with the normalizations
$\int_{-\infty}^{\infty}|g(x)|^{2}dx=1$ and $\int_{0}^{\infty}|f(q)|^{2}dq=1$,
which are assumed to be slowly varying functions of $x$ and $q$, respectively.
Then $\rho_{\widehat{\Phi}}$, the density matrix associated with
$|\widehat{\Phi}\rangle$ is defined as

$\begin{split}\langle\widehat{\Phi}|\widehat{a}|\widehat{\Phi}\rangle={\rm
Tr}(\rho_{\widehat{\Phi}}\widehat{a})=\int_{-\infty}^{\infty}\beta dxe^{\beta
x}\langle\Psi_{\rm max}|\rho_{\widehat{\Phi}}\widehat{a}|\Psi_{\rm
max}\rangle,\end{split}$ (34)

for any $\widehat{a}\in\widehat{\cal A}_{R}$.

In order to obtain $\rho_{\widehat{\Phi}}$, we need to convert the states
written in terms of $|\Psi\rangle$, i.e., the states constructed by acting the
operators in ${\widehat{\cal A}_{R}}$ on $|\Psi\rangle$, into those written in
terms of $|\Phi\rangle$. This is well described by the relative Tomita
operator $S_{\Phi|\Psi}$, an antilinear operator satisfying 777While we follow
the notations in [10], they are different from those in [2] : $S_{\Phi|\Psi}$
and $\Delta_{\Phi|\Psi}$ in [10] are $S_{\Psi|\Phi}$ and $\Delta_{\Psi|\Phi}$
in [2], respectively.

$\begin{split}S_{\Phi|\Psi}a|\Psi\rangle=a^{\dagger}|\Phi\rangle\end{split}$
(35)

for all $a\in{\cal A}_{R}$. From this, we define the relative modular operator

$\begin{split}\Delta_{\Phi|\Psi}=e^{-h_{\Phi|\Psi}}=S_{\Phi|\Psi}^{\dagger}S_{\Phi|\Psi},\end{split}$
(36)

which gives the relation

$\begin{split}\langle\Psi|\Delta_{\Psi|\Phi}a|\Psi\rangle&=\langle\Psi|S_{\Phi|\Psi}^{\dagger}S_{\Phi|\Psi}a|\Psi\rangle=\langle\Psi|S_{\Phi|\Psi}^{\dagger}a^{\dagger}|\Phi\rangle=\langle\Phi|a^{\dagger}|\Phi\rangle^{*}\\\
&=\langle\Phi|a|\Phi\rangle.\end{split}$ (37)

In the same way, we can also define the Tomita operators and the modular
operators $\Delta_{\Phi}=e^{-h_{\Phi}}=S_{\Phi}^{\dagger}S_{\Phi}$ and
$\Delta_{\Psi}=e^{-h_{\Psi}}=S_{\Psi}^{\dagger}S_{\Psi}$ satisfying

$\begin{split}S_{\Phi}a|\Phi\rangle=a^{\dagger}|\Phi\rangle,\quad
S_{\Psi}a|\Psi\rangle=a^{\dagger}|\Psi\rangle,\end{split}$ (38)

respectively. In order to find the physical meaning of these modular
operators, in addition to ${\cal A}_{R}$, the local algebra restricted to the
static patch $R$, we consider another local algebra ${\cal A}^{\prime}_{R}$, a
commutant of ${\cal A}_{R}$. When we extend the spacetime manifold to $L$ and
$B$, ${\cal A}^{\prime}_{R}$ can be given by the local algebra on the
complementary static patch $L$. Then it was shown that (see, e.g., Section IV.
A of [2]) the density matrix $\rho_{\Psi}$ for algebra ${\cal A}_{R}$ and
$\rho^{\prime}_{\Psi}$ for ${\cal A}^{\prime}_{R}$ associated with the state
$|\Psi\rangle$ satisfy

$\begin{split}\Delta_{\Psi}=\rho_{\Psi}\otimes{\rho^{\prime}_{\Psi}}^{-1}.\end{split}$
(39)

We can also find the similar relations
$\Delta_{\Phi}=\rho_{\Phi}\otimes{\rho^{\prime}_{\Phi}}^{-1}$,
$\Delta_{\Psi|\Phi}=\rho_{\Psi}\otimes{\rho^{\prime}_{\Phi}}^{-1}$, and
$\Delta_{\Phi|\Psi}=\rho_{\Phi}\otimes{\rho^{\prime}_{\Psi}}^{-1}$. Given the
static time translation generators $H_{R}$ for ${\cal A}_{R}$ and $H_{L}$ for
${\cal A}^{\prime}_{R}={\cal A}_{L}$, the density matrices can be written as

$\begin{split}\log\rho_{\Psi}=-\beta
H_{R}+C,\quad\log\rho^{\prime}_{\Psi}=-\beta H_{L}+C,\end{split}$ (40)

respectively, from which one finds that $h_{\Psi}=\beta(H_{R}-H_{L})$. Since
the static time in $R$ runs in the opposite direction to that in $L$,
$H_{R}-H_{L}$ is nothing more than the total static time translation generator
$H_{0}$. Thus, $h_{\Psi}$ is identified with $\beta H_{0}$. The explicit value
of $C$ in (40) can be obtained by taking the expectation value with respect to
$|\Psi\rangle$. From $\langle\Psi|H_{R}|\Psi\rangle=0$ (no energy flux across
the horizon) and $\langle\Psi|\log\rho_{\Psi}|\Psi\rangle={\rm
Tr}(\rho_{\Psi}\log\rho_{\Psi})=-S_{\rm bulk}(R)_{\Psi}$ where $R$ denotes the
bulk of the static patch $R$, we obtain

$\begin{split}\log\rho_{\Psi}=-\beta H_{R}-S_{\rm bulk}(R)_{\Psi}.\end{split}$
(41)

Comparing (34) and (37), it is reasonable to expect that
$\rho_{\widehat{\Phi}}$ contains $\Delta_{\Phi|\Psi}$, which converts the
expectation value with respect to $|\Psi\rangle$ into that with respect to
$|\Phi\rangle$. This indeed is supported by the relation
$\Delta_{\Phi|\Psi}=\rho_{\Phi}\otimes{\rho^{\prime}_{\Psi}}^{-1}$ and the
observation that $L\cup B$ is a copy of $R\cup T$ thus $\rho^{\prime}_{\Psi}$
is just given by $\rho_{\Psi}$. The rest part of $\rho_{\widehat{\Phi}}$
depends on the probability distribution of $x$ and $q$. Moreover, the fact
that the observer’s energy is described in a probabilistic way implies that
the observer’s time has an uncertainty. In [10] and [9], it was argued that
when the fluctuation in the observer’s time is bounded by
$\varepsilon\ll\beta$ and $g(x)$ as well as $f(q)$ is slowly varying over
$1/\varepsilon$, i.e., $|\Delta x|,|\Delta q|\sim 1/\varepsilon$, the density
matrix $\rho_{\widehat{\Phi}}$ is given by

$\begin{split}\rho_{\widehat{\Phi}}=\frac{1}{\beta^{2}}\big{|}g(X+h_{\Psi}/\beta)f(\widehat{q})\big{|}^{2}e^{\beta(-X+\widehat{q})}\Delta_{\Phi|\Psi}+{\cal
O}(\varepsilon).\end{split}$ (42)

The justification of (42) can be found in Appendix B. We note that in quasi-dS
space, the curvature perturbation induces the fluctuation in $\phi(t)$ hence
that in the flat time given by $\delta t=\delta\phi/\dot{\phi}(t)$. As can be
inferred from (20), these fluctuations are accumulated as time goes on, hence
negligibly small compared to the fluctuation in the static observer’s time
provided

$\begin{split}\delta
t=\frac{\delta\phi}{\dot{\phi}(t)}=\sqrt{\frac{1}{\pi\epsilon_{H}}}\frac{(H\Delta
t)^{1/2}}{M_{\rm Pl}}<\varepsilon\ll\beta=\frac{2\pi}{H}.\end{split}$ (43)

If the inequality is satisfied until $\Delta t\sim\beta$, i.e., several
$e$-folds, $\epsilon_{H}$ is required to be larger than $H^{2}/M_{\rm
Pl}^{2}$, which is known as the condition that the eternal inflation does not
take place. That is, if $\epsilon_{H}$ is too small, the field value of
$\phi(t)$ strongly fluctuates and $H$ may stay at some constant value for a
long time instead of decreasing in time through the slow-roll. In this case,
the semiclassical description we have considered is no longer valid. If the
inequality is satisfied until the end of inflation $\Delta
t\sim(\epsilon_{H}H)^{-1}$, we have the stronger bound $\epsilon_{H}>H/M_{\rm
Pl}$.

Then the von Neumann entropy of the static patch associated with the state
$|\widehat{\Phi}\rangle$, which will be identified with the generalized
entropy up to the addition of constant, is written as

$\begin{split}S(R)_{\widehat{\Phi}}&=-\langle\widehat{\Phi}|\log\rho_{\widehat{\Phi}}|\widehat{\Phi}\rangle\\\
&=\langle\widehat{\Phi}|h_{\Phi|\Psi}|\widehat{\Phi}\rangle-\langle\widehat{\Phi}|(\beta(\widehat{q}-X)+h_{\Phi})|\widehat{\Phi}\rangle\\\
&\quad-\int_{0}^{\infty}dq|f(q)|^{2}(\log|f(q)|^{2}-\log\beta)-\int_{-\infty}^{\infty}dx|g(H_{R})|^{2}(\log|g(H_{R})|^{2}-\log\beta),\end{split}$
(44)

where in the second term $\langle\Phi|h_{\Phi}|\Phi\rangle=0$ which is
obtained from $S_{\Phi}|\Phi\rangle=|\Phi\rangle$ is added and $H_{R}$
indicates $X+h_{\Psi}/\beta=X+H_{0}$. From
$\Delta_{\Psi}=\rho_{\Psi}\otimes{\rho^{\prime}_{\Psi}}^{-1}$ and
$\Delta_{\Psi|\Phi}=\rho_{\Psi}\otimes{\rho^{\prime}_{\Phi}}^{-1}$, one finds
$\Delta_{\Phi|\Psi}^{is}\Delta_{\Psi}^{-is}=\Delta_{\Phi}^{is}\Delta_{\Psi|\Phi}^{-is}$
(this quantity is called the Connes cocycle), which leads to
$h_{\Phi|\Psi}-h_{\Phi}=h_{\Psi}-h_{\Psi|\Phi}$. Then $S(R)_{\widehat{\Phi}}$
is rewritten as

$\begin{split}S(R)_{\widehat{\Phi}}=&-\langle\widehat{\Phi}|h_{\Psi|\Phi}|\widehat{\Phi}\rangle-\langle\widehat{\Phi}|(\beta(\widehat{q}-X)-h_{\Psi})|\widehat{\Phi}\rangle\\\
&-\int_{0}^{\infty}dq|f(q)|^{2}(\log|f(q)|^{2}-\log\beta)-\int_{-\infty}^{\infty}dx|g(H_{R})|^{2}(\log|g(H_{R})|^{2}-\log\beta).\end{split}$
(45)

Let us first consider the second line. The first integral is interpreted as
the entropy of the static observer, which is evident from the fact that
$|f(q)|^{2}$ is the probability distribution of $\widehat{q}$ eigenvalues, the
observer’s energy. As for the last term, we recall that the fluctuation in
$H_{R}$ originates from the fluctuation in $\phi(t)$, or equivalently, the
flat time $t$. Since $H$ also varies depending on $t$, the value of $H$ at the
end of inflation also fluctuates, the probability distribution of which is
described by $|g(H_{R})|^{2}$. These two integrals in the second line are not
explicitly relevant to the excitations of matter in the static patch, and can
be identified with $S_{\rm bulk}(\infty)_{\widehat{\Phi}}$, the bulk entropy
associated with $|\widehat{\Phi}\rangle$ at the end of inflation. This is
because at late time, the wavelength of almost all the excitations will be
stretched beyond the horizon so the static observer does not find any
excitation except for that of the observer state inside the horizon. We can
compare it with the static observer in the Bunch-Davies state $|\Psi\rangle$.
As $|\Psi\rangle$ is invariant under the static time translation generated by
$H_{0}$, the bulk entropy will be constant in time, i.e., $S_{\rm
bulk}(R)_{\widehat{\Psi}}=S_{\rm bulk}(\infty)_{\widehat{\Psi}}$, and it
measures the entropy of empty dS space without any excitation except for that
of the observer state. For the state during inflation $|\Phi\rangle$, in
contrast, the background does not respect the isometry generated by $H_{0}$
any longer. Then the spontaneous breaking of the isometry by the background
gives rise to the curvature perturbation which does not appear in perfect dS
space. But the background is still close to dS space and the wavelength of
these excitations would be stretched as the universe undergoes accelerated
expansion. Then just like the perfect dS background, almost all the
excitations cross the horizon after several $e$-folds and only the
fluctuations of the observer’s energy and the value of $H$ contribute to the
entropy. Then $S_{\rm bulk}(\infty)_{\widehat{\Phi}}$ can be identified with
the sum of $S_{\rm bulk}(\infty)_{\widehat{\Psi}}=S_{\rm
bulk}(R)_{\widehat{\Psi}}$ and the last integral in (45). The same argument
leads to $S_{\rm bulk}(\infty)_{{\Phi}}\simeq S_{\rm
bulk}(\infty)_{{\Psi}}=S_{\rm bulk}(R)_{{\Psi}}$.

We also note that the last two integrals in (45) reflects two different ways
to give rise to the uncertainty of the horizon area. First, since the horizon
is deformed by the backreaction of observer’s energy $q$, the horizon area
fluctuates as $q$ fluctuates [47, 48, 10]. Second, as we remarked earlier, the
probability distribution $|g(H_{R})|^{2}$ is induced by the fluctuation in
time, which leads to the fluctuation in the horizon radius, thus that in the
horizon area at late time.

Summarizing the discussion so far, $S(R)_{\widehat{\Phi}}$ can be written as

$\begin{split}S(R)_{\widehat{\Phi}}=-\langle\widehat{\Phi}|h_{\Psi|\Phi}|\widehat{\Phi}\rangle-\langle\widehat{\Phi}|(\beta(\widehat{q}-X)-h_{\Psi})|\widehat{\Phi}\rangle+S_{\rm
bulk}(\infty)_{\widehat{\Phi}}.\end{split}$ (46)

As we will see in the next section, whereas the first term is identified with
the negative of the change in the generalized entropy, the term
$-\langle\widehat{\Phi}|(\beta(\widehat{q}-X)-h_{\Psi})|\widehat{\Phi}\rangle=\langle\widehat{\Phi}|(H_{R}-\beta\widehat{q})|\widehat{\Phi}\rangle$
can be interpreted as the deformation of the horizon area from the initial
value. This leads to the relation $S(R)_{\widehat{\Phi}}=S_{\rm
gen}(R)_{\widehat{\Phi}}+$(constant).

###  3.3 Horizon dynamics and entropy in quasi-dS space

We now focus on the first term of the RHS in (46). It was argued in [9] that
this term is identified with a negative of the relative entropy, $-S_{\rm
rel}(\Phi||\Psi)$. Indeed, this is the case if we consider the expectation
value of $-h_{\Psi|\Phi}$ with respect to $|\Phi\rangle$,

$\begin{split}-\langle\Phi|h_{\Psi|\Phi}|\Phi\rangle&=\langle\Phi|\log(\Delta_{\Psi|\Phi})|\Phi\rangle=\langle\Phi|(\log(\rho_{\Psi})-\log(\rho_{\Phi}))|\Phi\rangle\\\
&={\rm Tr}(\rho_{\Phi}\log\rho_{\Psi})-{\rm
Tr}(\rho_{\Phi}\log\rho_{\Phi})=-S_{\rm rel}(\Phi||\Psi).\end{split}$ (47)

Moreover, the relative entropy can be rewritten as [49, 9]

$\begin{split}S_{\rm rel}({\Phi}||{\Psi})=S_{\rm gen}(\infty)_{{\Phi}}-S_{\rm
gen}(R)_{{\Phi}},\end{split}$ (48)

i.e., the difference between the generalized entropies at initial ($U=0)$ and
late ($U=\infty)$ times given by

$\begin{split}S_{\rm gen}(R)=\frac{A(b)}{4G}+S_{\rm bulk}(R),\quad S_{\rm
gen}(\infty)=\frac{A(b_{\infty})}{4G}+S_{\rm bulk}(\infty),\end{split}$ (49)

respectively, where $b$ and $b_{\infty}$ indicate the horizon cuts at initial
and late times as depicted in Figure 1. This comes from the observation that
$H_{R}$, the static time translation generator restricted to the static patch
$R$ (recall that $H_{0}=h_{\Psi}/\beta=H_{R}-H_{L}$) satisfies

$\begin{split}\langle{\Phi}|\beta
H_{R}|{\Phi}\rangle=\frac{A(b_{\infty})}{4G}-\frac{A(b)}{4G}.\end{split}$ (50)

Then from $S_{\rm bulk}(\infty)_{\Phi}\simeq S_{\rm
bulk}(\infty)_{\Psi}=S_{\rm bulk}(R)_{\Psi}$ we obtain

$\begin{split}S_{\rm gen}(\infty)_{\Phi}-S_{\rm
gen}(R)_{\Phi}&=\frac{A(b_{\infty})}{4G}-\frac{A(b)}{4G}+S_{\rm
bulk}(\infty)_{\Phi}-S_{\rm bulk}(R)_{\Phi}\\\
&\simeq\frac{A(b_{\infty})}{4G}-\frac{A(b)}{4G}+S_{\rm bulk}(R)_{\Psi}-S_{\rm
bulk}(R)_{\Phi}.\end{split}$ (51)

From (41), the change in the horizon area can be written as

$\begin{split}\frac{A(b_{\infty})}{4G}-\frac{A(b)}{4G}&=\beta\langle\Phi|H_{R}|\Phi\rangle=-\langle\Phi|\log\rho_{\Psi}|\Phi\rangle-
S_{\rm bulk}(R)_{\Psi},\end{split}$ (52)

from which the above expression is rewritten as

$\begin{split}S_{\rm gen}(\infty)_{\Phi}-S_{\rm
gen}(R)_{\Phi}&=-\langle\Phi|\log\rho_{\Psi}|\Phi\rangle-S_{\rm
bulk}(R)_{\Phi}\\\
&=-\langle\Phi|\log\rho_{\Psi}|\Phi\rangle+\langle\Phi|\log\rho_{\Phi}|\Phi\rangle=S_{\rm
rel}(\Phi||\Psi),\end{split}$ (53)

which confirms (48).

Meanwhile, when we replace $|\Phi\rangle$ by $|\widehat{\Phi}\rangle$, the
first term in (46) is given by

$\begin{split}-\langle\widehat{\Phi}|h_{\Psi|\Phi}|\widehat{\Phi}\rangle&=\langle\widehat{\Phi}|(\log\rho_{\Psi}-\log\rho_{\Phi})|\widehat{\Phi}\rangle\\\
&=-\langle\widehat{\Phi}|\beta H_{R}|\widehat{\Phi}\rangle-S_{\rm
bulk}(R)_{\Psi}-\langle\widehat{\Phi}|\log\rho_{\Phi}|\widehat{\Phi}\rangle,\end{split}$
(54)

where again the expression (41) for $\log\rho_{\Psi}$ is used for the first
two terms in the second line. To proceed, we note that the relation (50) holds
even if $|\Phi\rangle$ is replaced by $|\widehat{\Phi}\rangle$, with the
explicit values of $A(b_{\infty})$ and $A(b)$ are changed reflecting the
backreaction of the observer. Moreover, since $\log\rho_{\Phi}$ contains the
probability distribution of the matter excitations in the bulk, it is tempting
to relate the last term
$-\langle\widehat{\Phi}|\log\rho_{\Phi}|\widehat{\Phi}\rangle=-{\rm
Tr}(\rho_{\widehat{\Phi}}\log\rho_{\Phi})$ to the bulk entropy. However, it
cannot be identified with $S_{\rm bulk}(R)_{\widehat{\Phi}}$ as the dynamics
of the bulk in the state $|\widehat{\Phi}\rangle$ is also affected by the
fluctuations in $q$ and $x$, which are not reflected in $\log\rho_{\Phi}$.
Since these fluctuations remain until the end of inflation when all the matter
excitations cross the horizon, if we conjecture that the difference $S_{\rm
bulk}(R)_{\widehat{\Phi}}-(-{\rm Tr}(\rho_{\widehat{\Phi}}\log\rho_{\Phi}))$
which contains the effects of the fluctuations in $q$ and $x$ on the bulk
dynamics are time independent at leading order, it can be identified with
$S_{\rm bulk}(\infty)_{\widehat{\Phi}}-S_{\rm bulk}(\infty)_{{\Phi}}$. Then we
obtain

$\begin{split}-\langle\widehat{\Phi}|h_{\Psi|\Phi}|\widehat{\Phi}\rangle&=\Big{(}-\frac{A(b_{\infty})}{4G}+\frac{A(b)}{4G}\Big{)}-S_{\rm
bulk}(R)_{\Psi}+\big{(}S_{\rm bulk}(R)_{\widehat{\Phi}}-S_{\rm
bulk}(\infty)_{\widehat{\Phi}}+S_{\rm bulk}(\infty)_{{\Phi}}\big{)}\\\
&=-\Big{(}\frac{A(b_{\infty})}{4G}+S_{\rm
bulk}(\infty)_{\widehat{\Phi}}\Big{)}+\Big{(}\frac{A(b)}{4G}+S_{\rm
bulk}(R)_{\widehat{\Phi}}\Big{)}\\\ &=-\big{(}S_{\rm
gen}(\infty)_{\widehat{\Phi}}-S_{\rm
gen}(R)_{\widehat{\Phi}}\big{)},\end{split}$ (55)

where in the second line we use the relation $S_{\rm
bulk}(\infty)_{{\Phi}}\simeq S_{\rm bulk}(\infty)_{{\Psi}}=S_{\rm
bulk}(R)_{{\Psi}}$.

Now we investigate the change in the horizon area more explicitly. When the
observer’s energy $q$ in perfect dS space is concentrated in the tiny region,
the Schwarzschild-de Sitter black hole can be created and the metric is
modified as 888There has been a conjecture that in the absence of an observer
collecting information, quantum gravity forbids the production of the black
hole through the fluctuation [50]. For discussions on how this conjecture
applies to the dS background, see, e,g, [51, 52, 53, 54].

$\begin{split}&ds^{2}=-f(r_{s})dt_{s}^{2}+\frac{1}{f(r_{s})}dr_{s}^{2}+r_{s}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),\\\ &f(r_{s})=1-\frac{2Gq}{r_{s}}-H^{2}r_{s}^{2}.\end{split}$ (56)

If $q$ is small enough, say, $q\lesssim H$ or $q/\Delta E\sim H^{2}/M_{\rm
Pl}^{2}\ll 1$, the linear expansion in $q$ is valid such that the cosmological
horizon and the black hole radii are approximated as $H^{-1}-Gq$ and $2Gq$,
respectively. For quasi-dS space, the radius of the cosmological horizon
deformed by the slow-roll as well as the backreaction of the observer’s energy
is given by $r_{H}=H^{-1}+\epsilon_{H}\Delta t-Gq$. Here $H$ is the initial
value of the Hubble parameter, and the Hubble parameter during the
inflationary period can be approximated by this constant value. Then the
(cosmological) horizon area at initial time is estimated as

$\begin{split}\frac{A(b)}{4G}\simeq\frac{\pi}{G}(H^{-1}-Gq)^{2}\simeq\frac{\pi}{GH^{2}}-\beta
q,\end{split}$ (57)

and that at late time, namely, just after the inflationary period $\Delta
t\simeq(\epsilon_{H}H)^{-1}$, is approximated as

$\begin{split}\frac{A(b_{\infty})}{4G}\simeq\frac{\pi}{G}(H_{f}^{-1}-Gq)^{2}\simeq\frac{\pi}{GH_{f}^{2}}-\beta_{f}q,\end{split}$
(58)

where $H_{f}^{-1}=H^{-1}+\epsilon_{H}\Delta t$ is the value of the Hubble
radius at the end of inflation and $\beta_{f}=(2\pi)/H_{f}$. As we have seen
in Section 2, the change in the horizon area leads to the energy flux across
the horizon as $\langle\widehat{\Phi}|H_{R}|\widehat{\Phi}\rangle=\int
dt\beta^{-1}(d/dt)[A/(4G)]$, or symbolically, $\langle\widehat{\Phi}|\beta
H_{R}|\widehat{\Phi}\rangle=[A(b_{\infty})-A(b)]/4G$. Since
$X+H_{0}=X+h_{\Psi}/\beta=H_{R}$, (46) is rewritten as

$\begin{split}S(R)_{\widehat{\Phi}}&=-(S_{\rm
gen}(\infty)_{\widehat{\Phi}}-S_{\rm
gen}(b)_{\widehat{\Phi}})-\langle\widehat{\Phi}|\beta\widehat{q}|\widehat{\Phi}\rangle+\Big{(}\frac{A(b_{\infty})}{4G}-\frac{A(b)}{4G}\Big{)}+S_{\rm
bulk}(\infty)_{\widehat{\Phi}}\\\ &=S_{\rm
gen}(b)_{\widehat{\Phi}}-\langle\widehat{\Phi}|\beta\widehat{q}|\widehat{\Phi}\rangle-\frac{A(b)}{4G}.\end{split}$
(59)

We also note that the state $|\widehat{\Phi}\rangle$ contains the probability
distribution of the observer’s energy $q$, in which $-\beta q$ in (57) is
modified to $-\langle\widehat{\Phi}|\beta\widehat{q}|\widehat{\Phi}\rangle$.
Therefore, we arrive at

$\begin{split}S(R)_{\widehat{\Phi}}=S_{\rm
gen}(b)_{\widehat{\Phi}}-\frac{\pi}{GH^{2}},\end{split}$ (60)

and since $H$ is a constant, $S(R)_{\widehat{\Phi}}$ is identified with
$S_{\rm gen}(b)_{\widehat{\Phi}}$ up to the addition of a constant.

##  4 Conclusion

Throughout this article, we have investigated how Type II1 von Neumann algebra
description of perfect dS space is modified in the inflationary quasi-dS
space. Unlike perfect dS space, quasi-dS space allows the nonvanishing energy
flux across the horizon, which is identified with the expectation value of the
static time translation generator. In the evaluation of the energy flux, we
assume the inflationary period to be $(\epsilon_{H}H)^{-1}$, which is natural
in the sense that after this time scale $H$ deviates significantly from the
initial value hence it is no longer approximated as a constant. Then both the
energy flux and its fluctuation diverge in the $G\to 0$ limit. Here the
fluctuation originates from the breaking of the dS isometry associated with
the static time translation, which induces the uncertainty of time, and also
the fluctuation in the value of $H$ at the end of inflation. As a result, the
inflationary quasi-dS space can be described by Type II∞ algebra. This is
different from Type II1 algebra for perfect dS space : since the horizon
radius fluctuates by the uncertainty of the observer’s energy only, the
entropy of any quantum state cannot exceed that of empty dS space in the
Bunch-Davies state. In contrast, in Type II∞ algebra for quasi-dS space, due
to the divergent fluctuation of the energy flux, the trace hence the entropy
is not well defined for the identity describing the maximal entanglement of
the Bunch-Davies state, and there is no upper bound on the entropy.

On the other hand, there has been a claim called the ‘dS swampland conjecture’
that dS space is unstable in quantum gravity, which is supported by the
distance conjecture and the covariant entropy bound [17, 18, 19, 20].
Estimation based on the conjecture suggests the much shorter inflationary
period given by $(\epsilon_{H}^{-1/2}H^{-1})\log(M_{\rm Pl}/H)$, after which
$\epsilon_{H}$ becomes ${\cal O}(1)$ and the background geometry is no longer
close to dS space [55, 56]. Even shorter inflationary period
$H^{-1}\log(M_{\rm Pl}/H)$ was conjectured under the name of ‘trans-Planckian
censorship conjecture’, which forbids the horizon crossing of the trans-
Planckian modes [57] (see also [58, 59, 60, 61]). In these cases, the energy
flux across the horizon, or $\langle\Phi|H_{R}|\Phi\rangle$ is given by
$\epsilon_{H}^{1/2}[1/(GH)]\log(M_{\rm Pl}/H)$ and
$\epsilon_{H}[1/(GH)]\log(M_{\rm Pl}/H)$, respectively. Since
$\epsilon_{H}\sim{\cal O}(G)$, the former still diverges but the latter is of
${\cal O}(1)$ in the $G\to 0$ limit. In both cases, the fluctuation $\delta
H_{R}$ becomes vanishing in the $G\to 0$ limit as the values in two cases are
estimated as $(\epsilon_{H}/\pi)^{1/2}\log(M_{\rm Pl}/H)[\log(M_{\rm
Pl}/H)+1]$ and $(\epsilon_{H}^{3}/\pi)^{1/2}\log(M_{\rm Pl}/H)[\log(M_{\rm
Pl}/H)+1]$, respectively. Hence, the renormalized operator
$H^{\prime}_{R}=H_{R}-\langle\Phi|H_{R}|\Phi\rangle$ is well defined. Then in
this limit, we do not need to consider the probability distribution $g(X)$
reflecting the fluctuation in time, and the von Neumann algebra can be defined
in the same way as that in the pure dS space with observer, i.e., Type II1
algebra. This shows that in addition to the nonzero energy flux across the
horizon, the inflationary period plays the crucial role in determining the
appropriate von Neumann algebra description of spacetime during the inflation.

##  Appendix A Coordinate systems on de Sitter space

We list several coordinate descriptions of dS space which are useful in our
discussion. For more complete reviews, see, e.g., [62, 63]. A natural
description of dS space as seen by a static observer surrounded by the horizon
is the static coordinates, in which the metric is written as

$\begin{split}ds^{2}=-(1-H^{2}r_{s}^{2})dt_{s}^{2}+\frac{dr_{s}^{2}}{1-H^{2}r_{s}^{2}}+r_{s}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).\end{split}$ (A.61)

From this, one immediately finds that the timelike Killing vector is just
given by $k^{a}=(\partial_{t_{s}})^{a}$ and the horizon is located at
$r_{s}=H^{-1}$. In order to see the causal structure, it is convenient to
introduce the tortoise coordinate,

$\begin{split}dr_{*}=\frac{dr_{s}}{1-H^{2}r_{s}^{2}},\quad
r_{*}=\frac{1}{2H}\log\Big{(}\frac{1+Hr_{s}}{1-Hr_{s}}\Big{)},\end{split}$
(A.62)

such that the $(t_{s},r_{*})$ part of the metric is written in the conformally
flat form. Then one can define the Eddington-Finkelstein coordinates
$u=t_{s}-r_{*}$ and $v=t_{s}+r_{*}$ as lightcone coordinates. For the
extension to the region beyond the horizon, the Kruskal-Szekeres coordinates
are useful. In the static patch, they are given by

$\begin{split}&U=\frac{1}{H}e^{Hu}=\frac{1}{H}e^{Ht_{s}}\sqrt{\frac{1-Hr_{s}}{1+Hr_{s}}},\\\
&V=-\frac{1}{H}e^{-Hv}=-\frac{1}{H}e^{-Ht_{s}}\sqrt{\frac{1-Hr_{s}}{1+Hr_{s}}},\end{split}$
(A.63)

in terms of which the metric is written as

$\begin{split}ds^{2}=-\frac{4}{(1-H^{2}UV)^{2}}dUdV+\frac{(1+H^{2}UV)^{2}}{H^{2}(1-H^{2}UV)^{2}}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).\end{split}$ (A.64)

Using (A.63), the timelike Killing vector in dS is rewritten as
$k^{a}=(\partial_{t_{s}})^{a}=H(U\partial_{U}-V\partial_{V})^{a}$. The future
(past) horizon is a null hypersurface satisfying $V=0$ ($U=0$), which is
normal to $k^{a}=HU\partial_{U}$ ($k^{a}=-HV\partial_{V}$) or
$k_{a}=-2HU(dV)_{a}$ ($k_{a}=2HV(dU)_{a}$). Thus, $U$ ($V$) is a natural
canonical affine parameter on the future (past) horizon.

Meanwhile, in order to describe the inflationary cosmology, the flat
coordinates $(t,r)$ in terms of which the metric is written as

$\begin{split}ds^{2}=-dt^{2}+e^{2Ht}\big{[}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\big{]}\end{split}$ (A.65)

are useful. They are related to the static coordinates by

$\begin{split}t_{s}=t-\frac{1}{2H}\log\big{(}1-H^{2}r^{2}e^{2Ht}\big{)},\quad
r_{s}=re^{Ht},\end{split}$ (A.66)

which give

$\begin{split}&\frac{\partial t_{s}}{\partial
t}=\frac{1}{1-H^{2}r_{s}^{2}},\quad\frac{\partial r_{s}}{\partial
t}=Hr_{s},\quad\frac{\partial t_{s}}{\partial
r}=\frac{e^{Ht}Hr_{s}}{1-H^{2}r_{s}^{2}},\quad\frac{\partial r_{s}}{\partial
r}=e^{Ht},\\\ &\frac{\partial t}{\partial t_{s}}=1,\quad\frac{\partial
r}{\partial t_{s}}=-Hr,\quad\frac{\partial t}{\partial
r_{s}}=-\frac{Hr_{s}}{1-H^{2}r_{s}^{2}},\quad\frac{\partial r}{\partial
r_{s}}=\frac{e^{-Ht}}{1-H^{2}r_{s}^{2}}.\end{split}$ (A.67)

Then the timelike Killing vector is written as
$k^{a}=(\partial_{t_{s}})^{a}=(\partial_{t}-Hr\partial_{r})^{a}$.

##  Appendix B Derivation of the density matrix

Here we briefly sketch how (42), the expression for the density matrix
$\rho_{\widehat{\Phi}}$ associated with $|\widehat{\Phi}\rangle$ is obtained,
following [10]. The relation (37),
$\langle\Psi|\Delta_{\Phi|\Psi}a|\Psi\rangle=\langle\Phi|a|\Phi\rangle$
indicates that the ${\cal A}_{R}$ part of $\rho_{\widehat{\Phi}}$ is given by
$\Delta_{\Phi|\Psi}$. Meanwhile, from the facts that the algebra
$\widehat{{\cal A}}_{R}$ is generated by $\\{a,\widehat{q}-(H_{0}+X)\\}$ where
$H_{0}=\beta h_{\Psi}$ and that
$h_{\Psi}-h_{\Phi|\Psi}=h_{\Phi}-h_{\Psi|\Phi}$ belongs to ${\cal A}_{R}$, one
finds that the combination

$\begin{split}e^{\beta(-X+\widehat{q})}\Delta_{\Phi|\Psi}=e^{-\beta
X+\beta(\widehat{q}-h_{\Psi}/\beta)+(h_{\Psi}-h_{\Phi|\Psi})}\end{split}$
(B.68)

is well factorized into ${\cal A}_{R}\otimes B(\mathbb{R})\otimes
B(\mathbb{R}^{+})$. This motivates us to consider an ansatz

$\begin{split}\rho_{\widehat{\Phi}}=\frac{1}{\beta^{2}}g(X)^{*}f(\widehat{q}-h_{\Psi}/\beta)^{*}e^{\beta\widehat{q}}\Delta_{\Phi|\Psi}f(\widehat{q}-h_{\Psi}/\beta)g(X)+{\cal
O}(\varepsilon).\end{split}$ (B.69)

Since we assume that both $f(\widehat{q}-h_{\Psi}/\beta)$ and $g(X)$ are
slowly varying such that they are taken to be almost constant over
$|q-h_{\Psi}/\beta|<1/\varepsilon$ and $|x|<1/\varepsilon$, respectively,
their commutators with any other operators are expected to be suppressed by
${\cal O}(\varepsilon)$. Then $\rho_{\widehat{\Phi}}$ can be written in the
form of (42).

To see $\rho_{\widehat{\Phi}}$ satisfies (34), we consider the operator

$\begin{split}\widehat{a}=ae^{-iu(\beta(\widehat{q}-X)-h_{\Psi})},\end{split}$
(B.70)

which does not vanish for $\beta|u|<\varepsilon$ as it avoids the strong
oscillation. Imposing $e^{iuh_{\Psi}}=1+{\cal O}(\varepsilon)$, its
expectation value with respect to $|\widehat{\Phi}\rangle=|\Phi\rangle\otimes
g(X)\otimes f(\widehat{q})$ is given by

$\begin{split}\langle\widehat{\Phi}|\widehat{a}|\widehat{\Phi}\rangle&=\int_{-\infty}^{\infty}dx|g(x)|^{2}\int_{0}^{\infty}dq|f(q)|^{2}\langle\Phi|ae^{-iu(\beta({q}-x)-h_{\Psi})}|\Phi\rangle\\\
&=\int_{-\infty}^{\infty}dx|g(x)|^{2}\int_{0}^{\infty}dq|f(q)|^{2}e^{-iu\beta({q}-x)}\langle\Phi|a|\Phi\rangle+{\cal
O}(\varepsilon).\end{split}$ (B.71)

Here $\langle\Phi|a|\Phi\rangle$ can be replaced by
$\langle\Psi|\Delta_{\Phi|\Psi}a|\Psi\rangle$. Ignoring ${\cal
O}(\varepsilon)$ terms and using $h_{\Psi}|\Psi\rangle=0$, it becomes

$\begin{split}\langle\widehat{\Phi}|\widehat{a}|\widehat{\Phi}\rangle&=\int_{-\infty}^{\infty}dx\int_{0}^{\infty}dq\langle\Psi|\big{|}g(x+h_{\Psi}/\beta)\big{|}^{2}\big{|}f(q)\big{|}^{2}e^{-iu(\beta({q}-x)-h_{\Psi})}\Delta_{\Phi|\Psi}a|\Psi\rangle\\\
&=\int_{-\infty}^{\infty}dx\int_{0}^{\infty}dq\langle\Psi|\big{|}g(x+h_{\Psi}/\beta)\big{|}^{2}\big{|}f(q)\big{|}^{2}\Delta_{\Phi|\Psi}\widehat{a}|\Psi\rangle\\\
&=\int_{-\infty}^{\infty}\beta dxe^{\beta x}\int_{0}^{\infty}\beta dqe^{-\beta
q}\langle\Psi|\frac{1}{\beta^{2}}\big{|}g(x+h_{\Psi}/\beta)f(q)\big{|}^{2}e^{\beta(-x+q)}\Delta_{\Phi|\Psi}\widehat{a}|\Psi\rangle.\end{split}$
(B.72)

Matching this with $\int\beta dxe^{\beta x}\langle\Psi_{\rm
max}|\rho_{\widehat{\Phi}}\widehat{a}|\Psi_{\rm max}\rangle$, we find that
${\rho}_{\widehat{\Phi}}$ is written as (42).

### Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (2021R1A4A5031460).

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