Holographic dynamics as a way to solve the basic cosmological problems
We review recent results on the cosmological models based on the holographic principle which were proposed to explain the most of the problems occurring in the Standard Cosmological Model. It is shown that these models naturally solve the cosmological constant problem and coincidence problem. Well-d...
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Bolotin, Yu.L. Lemets, O.A. Yerokhin, D.A. Zazunov, L.G. 2016-10-12T11:41:14Z 2016-10-12T11:41:14Z 2012 Holographic dynamics as a way to solve the basic cosmological problems / Yu.L. Bolotin, O.A. Lemets, D.A. Yerokhin, L.G. Zazunov // Вопросы атомной науки и техники. — 2012. — №. — С. 157-162. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 98.80.-k https://nasplib.isofts.kiev.ua/handle/123456789/107055 We review recent results on the cosmological models based on the holographic principle which were proposed to explain the most of the problems occurring in the Standard Cosmological Model. It is shown that these models naturally solve the cosmological constant problem and coincidence problem. Well-documented cosmic acceleration at the present time was analyzed in the light of holographic dark energy. In particular, we showed that in the model of the universe consisting of dark matter interacting with a scalar field on the agegraphic background can explain the transient acceleration. We also study the impact of ideas of the physics of entangled states on these cosmological models. Entanglement entropy of the universe gives holographic dark energy with the equation of state consistent with the current observation data. Рассмотрены космологические модели, основанные на голографическом принципе, позволяющие облегчить решение таких космологических проблем, как проблема космологической постоянной и проблема совпадений, а также объяснить ускоренное расширение Вселенной. В частности, мы показали, что модель Вселенной, состоящая из тёмной материи, взаимодействующая со скалярным полем, на фоне голографической тёмной энергии может объяснить переходное ускорение. В работе также рассмотрено влияние идей физики запутанных состояний на космологические модели. Розглянуто космологічні моделі, засновані на голографічному принципі, які можуть не тільки пояснити прискорене розширення Всесвіту, але й дають можливість вирішити проблему космологічної постійної та проблему збігів. Зокрема, ми показали, що модель Всесвіту, заповненого темною матерією, яка взаємодіє зі скалярним полем, на фоні голографічної темної енергії може пояснити перехідне прискорення. В роботі також розглянуто вплив ідей фізики заплутаних станів на космологічні моделі. The work was supported in part by the Joint DFFD-RFBR Grant # F40.2/040 en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section C. Theory of Elementary Particles. Cosmology Holographic dynamics as a way to solve the basic cosmological problems Голографическая динамика как способ решения основных космологических проблем Голографічна динаміка як спосіб вирішення основних космологічних проблем Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Holographic dynamics as a way to solve the basic cosmological problems |
| spellingShingle |
Holographic dynamics as a way to solve the basic cosmological problems Bolotin, Yu.L. Lemets, O.A. Yerokhin, D.A. Zazunov, L.G. Section C. Theory of Elementary Particles. Cosmology |
| title_short |
Holographic dynamics as a way to solve the basic cosmological problems |
| title_full |
Holographic dynamics as a way to solve the basic cosmological problems |
| title_fullStr |
Holographic dynamics as a way to solve the basic cosmological problems |
| title_full_unstemmed |
Holographic dynamics as a way to solve the basic cosmological problems |
| title_sort |
holographic dynamics as a way to solve the basic cosmological problems |
| author |
Bolotin, Yu.L. Lemets, O.A. Yerokhin, D.A. Zazunov, L.G. |
| author_facet |
Bolotin, Yu.L. Lemets, O.A. Yerokhin, D.A. Zazunov, L.G. |
| topic |
Section C. Theory of Elementary Particles. Cosmology |
| topic_facet |
Section C. Theory of Elementary Particles. Cosmology |
| publishDate |
2012 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Голографическая динамика как способ решения основных космологических проблем Голографічна динаміка як спосіб вирішення основних космологічних проблем |
| description |
We review recent results on the cosmological models based on the holographic principle which were proposed to explain the most of the problems occurring in the Standard Cosmological Model. It is shown that these models naturally solve the cosmological constant problem and coincidence problem. Well-documented cosmic acceleration at the present time was analyzed in the light of holographic dark energy. In particular, we showed that in the model of the universe consisting of dark matter interacting with a scalar field on the agegraphic background can explain the transient acceleration. We also study the impact of ideas of the physics of entangled states on these cosmological models. Entanglement entropy of the universe gives holographic dark energy with the equation of state consistent with the current observation data.
Рассмотрены космологические модели, основанные на голографическом принципе, позволяющие облегчить решение таких космологических проблем, как проблема космологической постоянной и проблема совпадений, а также объяснить ускоренное расширение Вселенной. В частности, мы показали, что модель Вселенной, состоящая из тёмной материи, взаимодействующая со скалярным полем, на фоне голографической тёмной энергии может объяснить переходное ускорение. В работе также рассмотрено влияние идей физики запутанных состояний на космологические модели.
Розглянуто космологічні моделі, засновані на голографічному принципі, які можуть не тільки пояснити прискорене розширення Всесвіту, але й дають можливість вирішити проблему космологічної постійної та проблему збігів. Зокрема, ми показали, що модель Всесвіту, заповненого темною матерією, яка взаємодіє зі скалярним полем, на фоні голографічної темної енергії може пояснити перехідне прискорення. В роботі також розглянуто вплив ідей фізики заплутаних станів на космологічні моделі.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/107055 |
| citation_txt |
Holographic dynamics as a way to solve the basic cosmological problems / Yu.L. Bolotin, O.A. Lemets, D.A. Yerokhin, L.G. Zazunov // Вопросы атомной науки и техники. — 2012. — №. — С. 157-162. — Бібліогр.: 11 назв. — англ. |
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| fulltext |
HOLOGRAPHIC DYNAMICS AS A WAY TO SOLVE THE
BASIC COSMOLOGICAL PROBLEMS
Yu.L. Bolotin ∗, O.A. Lemets, D.A. Yerokhin, L.G. Zazunov
National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received October 31, 2011)
We review recent results on the cosmological models based on the holographic principle which were proposed to
explain the most of the problems occurring in the Standard Cosmological Model. It is shown that these models
naturally solve the cosmological constant problem and coincidence problem. Well-documented cosmic acceleration at
the present time was analyzed in the light of holographic dark energy. In particular, we showed that in the model of
the universe consisting of dark matter interacting with a scalar field on the agegraphic background can explain the
transient acceleration. We also study the impact of ideas of the physics of entangled states on these cosmological
models. Entanglement entropy of the universe gives holographic dark energy with the equation of state consistent
with the current observation data.
PACS: 98.80.-k
1. INTRODUCTION
One of the most promising ideas that emerged in the-
oretical physics during the last decade was the Holo-
graphic Principle proposed by ’t Hooft and Susskind
[1–4]; it appears to be a new guiding paradigm for
the true understanding of quantum gravity theories.
Basically, the Holographic Principle states that the
fundamental degrees of freedom of a physical system
are bound by its surface area in Planck units.
It concerns the number of degrees of freedom in
nature and states that the entropy of matter sys-
tems is drastically reduced compared to conventional
quantum field theory. This claim is supported by the
covariant entropy bound which is valid in a rather
general class of spacetime geometries. The notion
of holography is well developed in certain models
and backgrounds, in particular in the context of the
AdS/CFT correspondence. A more general formula-
tion is lacking, however, and the ultimate role of the
holographic principle in fundamental physics remains
to be identified.
The holographic principle is composed of the two
main statements:
1) The number of possible states of a region of space
is the same as that of a system of binary degree of
freedom distributed on the boundary of the region,
i.e. physics inside a causal horizon can be described
completely by physics on the horizon;
2) The number of such degrees of freedom N is not
indefinitely large but is bounded by the area A of the
region (on causal horizon) in Planck units:
N ≤ Ac3
Gh̄
. (1)
Therefore, the holography says that in a quan-
tum theory of gravity we should be able to describe
physics in some region of space by a theory with at
most one degree of freedom per unit Planck area. No-
tice that the number of degrees of freedom N would
then increase with the area and not with the volume
as we are normally used to. Of course, for all physi-
cal systems that we normally encounter the number
of degrees of freedom is much smaller than the area,
since the Planck length is so small. It is called “holog-
raphy” because it would be analogous to a hologram
which can store a three dimensional image in a two
dimensional surface.
2. COSMOLOGICAL CONSTANT AND
HOLOGRAPHIC PRINCIPLE
Why cosmological constant observed today is so much
smaller than the Planck scale? This is one of the most
important problems in modern physics. In history,
Einstein first introduced the cosmological constant in
his famous field equation to achieve a static universe
in 1917.
2.1. The basic problems of the standard cos-
mological model
Recent observations of supernovæ CMB anisotropies
and large scale structure point to the presence of a flat
universe with a dark energy component. Understand-
ing the origin of dark energy is one of the most im-
portant challenges facing cosmology and theoretical
physics. One aspect of the problem is to understand
what is the role of zero-point vacuum fluctuations.
In particle physics, the cosmological constant nat-
urally arises as an energy density of the vacuum,
∗Corresponding author E-mail address: ybolotin@gmail.com
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 157-162.
157
which is evaluated by the sum of zero-point energies
of quantum fields with mass m as follows:
ρΛ =
1
2
∫ Λ
0
4πk2dk
(2π)3
√
k2 + m2 ≈ Λ4
16π2
, (2)
where Λ � m is the UV cutoff. Usually the quan-
tum field theory is considered to be valid just below
the Planck scale: Mp ∼ 1018 GeV, where we used de-
duced Planck mass M−2
p = 8πG for convenience. If
we pick up Λ = Mp, we find that the energy density
of the vacuum in this case is estimated as 1070 GeV4,
which is about 10120 orders of magnitude larger than
the observation value 10−47 GeV4. This problem is
called the cosmological constant problem [5].
Another related but distinct difficulty with
ΛCDM is the so-called “why now?” or coincidence
problem. Why the densities of dark energy and dark
matter are comparable today? While a cosmological
constant is by definition time-independent, the mat-
ter energy density is diluted as 1/a3 as the universe
expands. Thus, despite evolution of a over many or-
ders of magnitude, we appear to live in an era during
which the two energy densities are roughly the same.
In other words, if Λ is tuned to give ΩΛ ∼ ΩM to-
day, then for essentially all of the previous history
of the universe, the cosmological constant was negli-
gible in the dynamics of the Hubble expansion, and
for the indefinite future, the universe will undergo a
de Sitter-type expansion in which ΩΛ is near unity
and all other components are negligible. The present
epoch would then be a very special time in the history
of the universe, the only period when ΩM ∼ ΩΛ.
2.2. Cosmological constant: holographic view
We now turn to the question of whether some form
of a holographic bound may apply to a cosmological
theory in which no boundary conditions have been
enforced.
For an effective quantum field theory in a box of
size L with UV cutoff Λ the entropy S scales exten-
sively, S ∼ L3Λ3. However the specific thermody-
namics of black holes has led Bekenstein to postulate
that the maximum entropy in a box of volume L3 be-
haves non-extensively, growing only as the area of the
box. For any Λ, there is a sufficiently large volume
for which the entropy of an effective field theory will
exceed the Bekenstein limit. The Bekenstein entropy
bound may be satisfied in an effective field theory if
we limit the volume of the system according to
L3Λ3 ≤ SBH ≡ πL2M2
P , (3)
where SBH is the entropy of a black hole of ra-
dius L. Consequently the length L, which acts as
an IR cutoff, cannot be chosen independently of the
UV cutoff, and scales as Λ−3. An effective field the-
ory that can saturate (3) necessarily includes many
states with Schwarzschild radius much larger than the
box size.To avoid these difficulties an even stronger
constraint on the IR cutoff 1/L has been proposed
in [6] which excludes all states that lie within their
Schwarzschild radius. Since the maximum energy
density in the effective theory is Λ4, the constraint
on L is
L3Λ4 ≤ LM2
P . (4)
Here the IR cutoff scales like Λ−2. This bound is far
more restrictive than (3). While saturating this in-
equality by choosing the largest L it gives rise to a
holographic energy density:
ρΛ = 3c2M2
pL−2, (5)
where c is a dimensionless constant. Then the key
issue is what possible physical scale one can choose
as the cutoff L constrained by the fact of the current
acceleration of the universe.
Applying the relation (5) to the universe as a
whole it is naturally to identify the IR-scale in the
simplest case with the Hubble radius H−1. Then for
the upper bound of the energy density one finds:
ρΛ ∼ L−2M2
Pl ∼ H2M2
Pl. (6)
We will below denote its density as ρ
DE
. Accounting
that MPl � 1.2 × 1019 GeV, H0 � 1.6 × 10−42 GeV
one finds:
ρDE � 3 × 10−47 GeV4. (7)
So, this value is comparable to the observed dark
energy density ρobs ∼ 10−46 GeV4. Therefore, the
holographic dynamics is free from the cosmological
constant problem.
The coincidences problem can also be solved
within the framework of holographic cosmology. Set-
ting L = H−1 in the equation (5) and working
with the equality (i.e., assuming that the holographic
bound is saturated) give ρDE = 3 c2M2
P H2. Let us
consider the flat universe consisting of nonrelativis-
tic matter and holographic dark energy. The Fried-
mann equation in this case take the following form
3M2
P H2 = ρ
DE
+ ρm and can be rewritten in a very
neat form
ρm = 3
(
1 − c2
)
M2
P H2. (8)
Now, the argument runs as follows. The energy
density ρm varies as H2, which coincides with the
dependence of ρ
DE
on H. The energy density of cold
matter is known to scale as ρm ∝ a−3. So, theirs
ratio is constant and has the form
ρm
ρDE
=
1 − c2
c2
. (9)
Therefore the holographic dynamics is free from the
cosmic coincidences problem also. If ρ
DE
∝ H2, then
dynamical behavior of holographic dark energy coin-
cides with that of normal matter, thus the accelerated
expansion is impossible.
In order to produce the accelerated expansion of
the universe in frames of holographic dark energy
model we will try to use the IR-cutoff spatial scale
different from the Hubble radius. The first thing that
comes to mind is a consideration as the cutoff value
of the cosmological particle or event horizon.
158
The particle horizon is given by
Rh = a
∫ t
0
dt
a
= a
∫ a
0
da
Ha2
. (10)
Substituting in (5) expression for RH and using
the equation of covariant conservation, it is easy to
verify that expression for the equation of state para-
meter w = p/ρ takes the form w = − 1
3 + 2
3c > − 1
3 . So,
this IR-scale can not provide the accelerated expan-
sion of the universe. To get an accelerating expansion
of the universe, we supersede the particle horizon by
the future event horizon:
Rh = a
∫ ∞
t
dt
a
= a
∫ ∞
a
da
Ha2
.
This horizon is the boundary of the volume a fixed
observer may eventually observe. One is to formulate
a theory regarding a fixed observer within this hori-
zon. In this case, the equation of state parameter
acquires the form
w = −1
3
− 2
3c
. (11)
We obtain a component of energy behaving as
dark energy. If we take c = 1, its behavior is similar
to the cosmological constant. If c < 1 then w < −1, a
value achieved in the past only in the phantom model
in the traditional approach.
For the first impression the declared task is com-
pleted. The holographic dark energy with equation of
state parameter (11) from the one hand provides cor-
respondence between the observed density and the
theoretical estimate, and from the other it leads to
the state equation which is able to generate the accel-
erated expansion of the universe. However the holo-
graphic dark energy with IR-cutoff on the event hori-
zon still leaves unsolved problems connected with the
causality principle: according to the definition of the
event horizon the holographic dark energy dynamics
depends on future evolution of the scale factor. Such
dependence is hard to agree with the causality prin-
ciple. The former based on the fact that space-time
curvature is non-zero, so it can be associated with a
horizon, that is considered as a holographic screen.
The latter type of energy is so-called the agegraphic
dark energy. This kind of dark energy we study in
more detail.
2.3. Agegraphic dark energy
According to the definition the agegraphic dark en-
ergy is the holographic dark energy in the infrared
cutoff scale equal to the age of the universe. It is
remarkable that this kind of energy can be obtained
from independent and less radical conception.
The existence of quantum fluctuations in the met-
ric directly leads to the following conclusion, re-
lated to the problem of distance measurements in the
Minkowski space: the distance t 1 cannot be mea-
sured with precision exceeding the following:
δt = βt
2/3
Pl t1/3, (12)
where β is a factor of order of unity. This expression
is so-called Károlyházy uncertainty relation [7].
The Károlyházy relation (12) together with the
time-energy uncertainty relation enables one to esti-
mate a quantum energy density of the metric fluctua-
tions of Minkowski space-time [8]. With the relation
(12), a length scale t can be known with a maximal
precision δt determining a minimal detectable cell δ3
over a spatial region t3. Thus one is able to look at the
microstructure of space-time over a region t3 by view-
ing the region as the one consisting of cells δt3 ∼ t2pt.
Therefore such a cell δt3 is the minimal detectable
unit of space-time over a given length scale t and if
the age of the space-time is t, its existence due to the
time-energy uncertainty relation cannot be justified
with energy smaller than ∼ t−1 . Hence, as a result
of the relation (12), one can conclude that if the age
of the Minkowski space-time is t over a spatial re-
gion with linear size t (determining the maximal ob-
servable patch) there exists a minimal cell δt3, which
energy, due to the time-energy uncertainty relation,
cannot be smaller than [8]
Eδt3 ∼ t−1. (13)
It was also argued [8] that the energy density of met-
ric fluctuations of Minkowski spacetime is given by
ρq ∼ Eδt3
δt3
∼ 1
t2pt
2
∼ M2
p
t2
. (14)
In [8], it is noticed that the Károlyházy relation natu-
rally obeys the holographic black hole entropy bound.
It is worth noting that the form of energy density
Eq. (14) is similar to the one of holographic dark en-
ergy, i.e., ρΛ ∼ l−2
p l−2. The similarity between ρq and
ρΛ might reveal some universal features of quantum
gravity, although they arise from different sources. As
the most natural choice, the time scale t in Eq. (14)
is chosen to be the age of our Universe. Therefore,
we call it “agegraphic” dark energy [8]. The rela-
tion (14) allows to introduce an alternative model for
holographic dark energy, which uses the age of the
universe T for IR-cutoff scale. In such a model
ρq =
3n2M2
Pl
T 2
, (15)
where n is a free parameter of model, and the num-
ber coefficient 3 is introduced for convenience. So de-
fined energy density (15) with T ∼ H−1
0 , where H0 is
the current value of the Hubble parameter, leads to
the observed value of the dark energy density with
the coefficient n value of order of unity. Thus in
SCM, where H0 � 72 km sec−1Mpc−1, ΩDE � 0.73,
T � 13.7Gyr, one finds that n � 1.15. Suppose that
the universe is described by the Friedmann equation:
H2 =
1
3M2
Pl
(ρq + ρm) . (16)
1Recall that we use the system of units where the light speed equals c = h̄ = 1, so that LPl = tPl = M−1
Pl .
159
The state equation for the dark energy is
wq = −1 +
2
3n
√
Ωq. (17)
So such universe will be accelerated expanded, and
would be similar to ΛCDM. Thus the holographic
model for dark energy with IR-cutoff scale set to the
universe age, allows the following: 1) to obtain the
observed value of the dark energy density; 2) provide
the accelerated expansion regime on later stages of
the universe evolution; 3) resolve contradictions with
the causality principle.
Nevertheless we do not discard other possibilities,
to which the allusions of observations exist.
3. OBSERVATIONS CHALLENGE
A. Starobinsky and co-workers investigated [9] the
course of cosmic expansion in its recent past using the
Constitution SN Ia sample (which includes CfA data
at low redshifts), jointly with signatures of baryon
acoustic oscillations (BAO) in the galaxy distribution
and fluctuations in the cosmic microwave background
(CMB). Allowing the equation of state of dark en-
ergy (DE) to vary, they find that a coasting model of
the universe (q0 = 0) fits the data about as well as
ΛCDM. This effect, which is most clearly seen using
the recently introduced Om diagnostics, corresponds
to an increase of Om and q at redshifts z ≤ 0.3.
In geometrical terms, this suggests that cosmic ac-
celeration may have already peaked and that we are
currently witnessing its slowing down (Fig. 1).
Thus the main result of the analysis is the fol-
lowing: SCM is not unique though the simplest ex-
planation of the observational data, and the acceler-
ated expansion of the universe presently dominated
by dark energy is just a transient phenomenon.
Thus, the evolutional behavior of dark energy re-
constructed and the issue of whether the cosmic accel-
eration is slowing down or even speeding up is highly
dependent upon the SNIa data sets, the light curve
fitting method of the SNIa, and the parametrization
forms of the equation of state. In order to have a def-
inite answer, we must wait for data with more pre-
cision and search for the more reliable and efficient
methods to analyze these data.
Model with the holographic dark energy, as dis-
cussed above, in their original form, do not allow to
explain the nonmonotonic dependence of the cosmo-
logical parameters.
4. THE MODEL OF INTERACTING
DARK ENERGY WITH A TRANSIENT
ACCELERATION PHASE
Current literature usually considers the models where
the required dynamics of the universe is provided by
one or another, and always only one, type of dark
energy.
We consider the cosmological model which con-
tains both volume and surface terms. The role of for-
mer is played by homogeneous scalar field in exponen-
tial potential, which interacts with dark matter. The
boundary term responds to holographic dark energy
in form of (15). This scenario predicts a transient
accelerating phase.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.2 0.4 0.6 0.8 1 1.2 1.4
q
(z
)
z
1 sigma CL
best tfi
Fig. 1. The deceleration parameter dependence q(z)
reconstructed from independent observational data,
including the brightness curves for SN Ia, cosmic
microwave background temperature anisotropy and
baryon acoustic oscillations (BAO). The red solid
line shows the best fit on the confidence level 1σCL [9]
To describe the dynamical properties of the uni-
verse it is convenient to pass to dimensionless vari-
ables:
x = ϕ̇√
6MPlH
, y = 1
MPlH
√
V (ϕ)
3 ,
z = 1
MP lH
√
ρm
3 , u = 1
MP lH
√
ρq
3 .
(18)
The evolution of scalar field is described by the Klein-
Gordon equation, which in the case of interaction be-
tween the scalar field and matter takes the following
form:
ϕ̈ + 3Hϕ̇ +
dV
dϕ
= −Q
ϕ̇
. (19)
In the present section we consider the case when the
interaction parameter Q is a linear combination of
energy density for scalar field and dark energy:
Q = 3H(αρϕ + βρm), (20)
where α, β are constant parameters. It is for given
model, regardless the explicit form of the scalar field
potential V (ϕ).
As was mentioned above, here we consider
the simplest case of exponential potential V =
V0 exp
(√
2
3
μϕ
MPl
)
, where μ is constant. Taking into
account Eqs. (18), the system of equations describing
the dynamics of the universe in this model reads:
x′ = 3x
2
[
g(x, z, u) − α(x2+y2)+βz2
x2
]
− 3x − μy2,
y′ = 3y
2 g(x, z, u) + μxy,
z′ = 3z
2
[
g(x, z, u) + α(x2+y2)+βz2
z2
]
− 3
2z,
u′ = 3u
2 g(x, z, u) − u2
n ,
(21)
where
g(x, z, u) = 2x2+z2+
2
3n
u3, λ ≡ − 1
V
dV
dϕ
MPl. (22)
160
Next, we consider the simplest case in which this
model can obtain the regime of transient accelera-
tion, Q = 3Hαρϕ. We consider the case with the
interaction parameter of the form (20) with β = 0.
For example, we consider the case presented in the
Fig. 2. With these values of the parameters of in-
teraction, transient acceleration begins almost in the
present era. So, one of the deficiencies of original
agegraphic dark energy model, that is the inability
to explain the phenomenon of transient acceleration,
in this model can be solved.
5. ENTANGLEMENT ENTROPY AND
HOLOGRAPHY
In quantum information science, quantum entangle-
ment is a central concept and a precious resource al-
lowing various quantum information processing such
as quantum key distribution. The entanglement is a
quantum nonlocal correlation which can not be pre-
pared by local operations and classical communica-
tion. For pure states the entanglement entropy SEnt
is a good measure of entanglement. For a bipartite
system AB described by a full density matrix ρ
AB
the
von Neumann entropy SEnt is
SEnt = −Tr (ρ
A
ln ρ
A
), (23)
where ρ
A
is the reduced matrix obtained by “tracing
out” the degrees of freedom of system B (which is
quantum-correlated with A) and given by
ρ
A
≡ Tr
B
ρ
AB
. (24)
The Basic Conjectures of the Entanglement en-
tropy are: 1) Quantum entanglement of matter or the
vacuum in the universe increases like the entropy; 2)
There is a new kind of force — quantum entangle-
ment force associated with this tendency; 3) Gravity
and dark energy are types of the quantum entangle-
ment force associated with the increase of the en-
tanglement, similar to Verlinde’s [10] entropic force
linked with the increase of the entropy.
For an entanglement system in the flat spacetime,
we consider the three-dimensional spherical volume
V and its enclosed boundary Σ (Fig. 3). We assume
that this system with radius r and the cutoff scale
b is described by the local quantum field theory of a
free scalar field φ.
In general, the vacuum entanglement entropy of a
spherical region with a radius r with quantum fields
can be expressed in the form
Sent =
βr2
b2
, (25)
where β is an O(1) constant that depends on the na-
ture of the field (like n in the agegraphic dark energy)
and b is the UV-cutoff.
The entanglement energy is carried by the modes
around Σ, which implies that the cutoff scale b is
introduced only in the r direction through the con-
traction length b2/r2. We start by noting the sim-
ilarity between the entanglement system in the flat
spacetime and the stretched horizon formulation of
the Schwarzschild black hole.
q
0.2
0.0
0.4 0.6 0.8 1.0 1.2 1.4
0.2
0.4
0.6
0.0 0.2
Z
Fig. 2. Behavior of Ωϕ (dot line), Ωq (dash line)
and Ωm (solid line) as a function of N = ln a
for n = 3, α = 0.005 and μ = −5 (upper plot).
Evolution of deceleration parameter q(z) for this
model (lower plot)
r
M
m
Fig. 3. The space around a massive object with
mass M can be divided into two subspaces, the
inside and the outside of an imaginary spherical
surface with a radius r. The surface Σ has the
entanglement entropy Sent ∝ r2 and entanglement
energy Eent ≡ ∫
Σ TentdSent. If there is a test
particle with mass m, it feels an effective attrac-
tive force in the direction of increase of entanglement
So, SEnt has a form consistent with the holo-
graphic principle, although it is derived from quan-
tum field theory without using the principle. Thus,
from different and independent physical assumptions,
we come to equal physical consequence. We can use
both assumptions of this ideology with equal success
and equivalent effect. Why are we considering the
quantum entanglement as an essential concept for
gravity? First, there are interesting similarities be-
tween the holographic entropy and the entanglement
entropy of a given surface. Both are proportional
161
to its area in general and related to quantum non-
locality. Second, when there is a gravitational force,
there is always a Rindler horizon for some observers,
which acts as information barrier for the observers.
This can lead to ignorance of information beyond the
horizons, and the lost information can be described
by the entanglement entropy. The spacetime should
bend itself so that the increase of the entanglement
entropy compensate the lost information of matter.
Third, if we use the entanglement entropy of quan-
tum fields instead of thermal entropy of the holo-
graphic screen, we can understand the microstates of
the screen and explicitly calculate, in principle, rele-
vant physical quantities using the quantum field the-
ory in the curved spacetime. The microstates can be
thought of as just quantum fields on the surface or its
discretized oscillators. Finally, identifying the holo-
graphic entropy as the entanglement entropy could
explain why the derivations of the Einstein equation
is involved with entropy, the Planck constant h̄ and,
hence, quantum mechanics. All these facts indicate
that quantum mechanics and gravity has an intrinsic
connection, and the holographic principle itself has
something to do with quantum entanglement.
6. CONCLUSIONS
In this work we gave a brief overview of the appli-
cation of holographic principle for solving the basic
problems of the standard cosmological model. It is
shown how a model based on the holographic princi-
ple naturally solves the cosmological constant prob-
lem and the coincidence problem. Proposed modi-
fication of this model was capable of explaining the
possibility of nonlinear dynamics of the cosmological
parameters — the phenomenon of transient accelera-
tion. It is shown that there is a deep analogy between
the cosmological models with the holographic princi-
ple and models with quantum entanglement entropy.
Acknowledgements
The work was supported in part by the Joint DFFD-
RFBR Grant # F40.2/040.
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