Transcript Entropy Bounds and Holography Constraints for

Cosmic Holography
Bin Wang(王 斌)
Department of Physics,
Fudan University
Entropy of our universe
2.7K cosmic microwave background (CMB)
S 0 = 10
90
The universe started in a low-entropy state and has
not yet reached its maximal attainable entropy.
Questions:
1. Which is this maximal possible value of entropy?
2. Why has it not already been reached after so many billion
years of cosmic evolution?
Bekenstein Entropy Bound (BEB)
For Isolated Objects

Isolated physical system of energyE and size R
S £ S BEB = ER / h

(J.D. Bekenstein, PRD23(1981)287)
Charged system with energy E , radius R and charge e
S £ (ER / h )(1 - e 2 / (2ER ))
(Bekenstein and Mayo, PRD61(2000)024022; S. Hod,
PRD61(2000)024023; B. Linet, GRG31(1999)1609)

Rotating system
S £ ER / h(1 - s 2 / (E 2R 2 ))1/ 2
(S. Hod, PRD61(2000)024012; B. Wang and E. Abdalla,
PRD62(2000)044030)
•
Charged rotating system
S £ (ER / h )[(1 - s 2 / (E 2R 2 )1/ 2 - e 2 / (2ER )]
(W. Qiu, B. Wang, R-K Su and E. Abdalla, PRD 64 (2001) 027503 )
+
“Entropy bounds for isolated system depend neither
On background spacetime nor on spacetime
dimensions.”
Universal
Holographic Entropy Bound (HEB)
Holographic Principle
Entropy cannot exceed one unit per Planckian area of its
boundary surface
S £ S HEB = Alp- 2
(Hooft, gr-qc/9310026; L. Susskind,
J. Math. Phys. 36(1995)6337)
AdS/CFT Correspondence
“Real conceptual change in our thinking about
Gravity.”
(Witten, Science 285(1999)512)
Comparison of BEB and HEB
Isolated System
S BEB = ER / h = GER / (G h ) = R s R lp- 2 R s = GE
S HEB = A lp- 2 = R 2lp- 2
For R > R s , S BEB < S HEB
For R = R s , S BEB = S HEB
Cosmological Consideration
Cosmological entropy S : R 3
S HEB : R 2 HEB violated for large R
S BEB : ER : R 4 BEB too loose for large R
Two bounds cannot naively be used in cosmology.
Both of them need revision in a cosmological context.
Problem: In a general cosmological setting,
Natural Boundary?
Particle Horizon
BEB (J.D. Bekenstein, Inter. J. Theor. Phys. 28(1989)967)
FS(W. Fischler and L. Susskind, hep-th/9806039)
Comparison of BEB and FS-HEB

Around the Plank time, they appear to be saturated,
which could justify the initial “low” entropy value.

S BEB = ER / h = Md p / h = r d p4 / h = (Hd p )2 d p2 / lp2 =
(Hd p )2 S FS - HEB
S FS - HEB < S BEB
+
Questions still exit

For collapsing universe, FS entropy bound fails

For universes with negative cosmological constants,
FS bound fails
(W. Fischler and L. Susskind, hep-th/9806039; N. Kaloper and
A. Linde, PRD60(1999)103509; B. Wang and E. Abdalla,
PLB471(2000)346)
Hubble Entropy Bound (HB)
H
- 1
H - 1 - Hubble radius
n H - number of Hubble-size regions within the volume V
S H = H - 2lp- 2- maximum entropy of each Hubble-size region
S (V ) < S HB = n H S H = (V / H - 3 )H - 2lp- 2 = VHlp- 2
Relation among HB, FS, BEB
1/ 2 1/ 2
S HB = S BEB
S SF
A possible relation between the FS, HB and a generalized
second law of thermodynamics (GSL) has been discussed.
(R. Brustein, PRL84(2000)2072; B. Wang and E. Abdalla,
PLB466(1999)122, PLB471(2000)346)
Validity of BEB and HB
S BEB = 2p ER / n , limited self-gravity, R < H - 1, HR < 1
S HB = (n - 1)HV / (4G ), strong self-gravity, R > H - 1, HR > 1
Friedmann equation
H2 =
16pGE
- 1 2 (1)
R
n (n - 1)V
For
S BH = (n - 1)
V
4GR
(2)
Holographic Bekenstein-Hawking entropy of a universe-sized
black hole
HR > 1 : S BEB > S BH
HR < 1 : S BEB < S BH
(3)
(4)
Relation among S BEB , S HB , S BH
2
2
2
2
S HB
+ (S BEB
- S BH
) = S BEB
(5)
Substituting S BH = 2p E BH R / n , S BEB = 2p ER / n
The relation can be written as
S HB
2p R
=
n
E BH (2E - E BH ) (6)
This is very similar to the 2D Cardy formula
S = 2p c(L0 - c / 24) / 6 (7)
At the turning point between the limited self-gravity and
strong self-gravity
 BEB and HB have been unified
 Friedmann equation corresponds to the generalized Cardy
formula
(E. Verlinde, hep-th/0008140; B. Wang, E. Abdalla,
PLB503(2001)394)
Inhomogeneous Cosmologies
Pietronero’s (1987) case that luminous large-scale matter
distribution follows a fractal pattern has started a sharp
controversy in the literature.
 CfA1 redshift survey (de Lapparent, Geller & Huchra, 1986,
1988) was the first to reveal structures such as filaments
and voids on scales where a random distribution of matter
was expected.
 Do observations of large-scale galaxy distribution support
or dismiss a fractal pattern? How inhomogeneous is
matter distribution?
 Relativistic aspects of cosmological models. Any relativistic
effect on observations?
(L. Pietronero and col.)

The real reason, though, for our adherence here to
the Cosmological Principle is not that it is surely
correct, but rather, that it allows us to make use of
the extremely limited data provided to cosmology by
observational astronomy. …
If the data will not fit into this framework, we shall
be able to conclude that either the Cosmological
Principle or the Principle of Equivalence is wrong.
Nothing could be more interesting.
Weinberg, 1972
Large Scale Structure
Description of the Inhomogeneous Universe
Relativistic Model
Let us start with the inhomogeneous spherically symmetric
metric as:
ds 2 = - dt 2 + R 2 (r , t )[dr 2 / f 2 (r ) + r 2dw2 ] (8)
where
where
d w2 = d q2 + sin 2 qd y 2 (9)
R (t , r ) = R (t )FR W + perturbation
(E. Abdalla, R. Mohayaee, PRD59(1999)084014)
Lemaitre-Tolman-Bondi model
In normalized comoving coordinates the metric of the
parabolic LTB model is
where hab
ds 2 = - dt 2 + R ¢2dr 2 + R 2 (d q2 + sin 2 qd y 2 )
(10)
a
b
2
2
2
2
%
= habdx dx + r (x )(d q + sin qd y )
= diag[- 1, R ¢2 ] , and
1
R = (9F )1/ 3 (t + b )2/ 3 (11)
2
Characteristics of the realistic model:
 Spherical symmetry
 Describing a fractal distribution of galaxies, not refering to
either initial of final moment
Our proposal of a holographic principle in
inhomogeneous cosmology
The entropy inside the apparent horizon can never exceed the
area of the apparent horizon in Plank units.
S £ A / lp2 = 4pbr%A2H
Defining Apparent Horizon
Ñ r%2 º hab¶ a r%¶ br%= 0 to the aerial radius, with the result
F (rA H ) = 3 [t + b (rA H ) ] (12)
r%
AH =
3
[t + b (rA H )] is the physical apparent horizon,
2
rA H denotes the proper apparent horizon.
Fractal behavior in parabolic models have been found by
Ribeiro (APJ1992). They are
ìï F
Model 1: íï b
ïî
ìF
Model 2: ïïí
ïïî b
Where a Î [10- 5 , 10- 4 ], p
= ar p
= b 0 + h0r q
(13)
= ar p
(14)
= ln (e b 0 + h1r )
and b 0 Î [0.5, 4 ], h1 Î [1000,1300 ]
q around 0.65 and h0 around 50 are required to obtain fractal
solutions.
Define the local entropy density
Standard big-bang cosmology: When a particle becomes
nonrelativistic and disappears, its entropy is transferred to
other relativistic particle species still present in the thermal
plasma.
Photons and neutrinos share the entropy of the universe.
Reasonable suppose: Entropy of the universe is mainly
produced before the dust-filled era.
+
First Law of thermodynamics, s = r / T = aT 4 / T .
Considering that in the expansion of the universe,
 The radiation always has the property of black body
 Conservation of the number density of the photon
We have:
h n0
hn
=
kT 0
kT
dt
1+ z =
dl
l
ædt
çç
èd l
, the expression for the redshift is
ö
÷
÷
÷
l =0ø
1
dr
¢
= R
dl
l
æ dr
ççR 0¢
çè d l
l0
We obtain the relation T R ¢ = T 0R 0¢ = Const .
ö
÷
÷
÷
ø
1
R¢
=
(15)
R 0¢
The local entropy density in the inhomogeneous case can
be expressed as
3
s (t , r ) = a (T 0R 0¢)
1
1
=
C
(16)
3
3
¢
¢
R (t , r )
R (t , r )
The total entropy measured in the comoving space inside the
apparent horizon is
S =
rA H
ò0
s (t , r )4p R ¢R 2dr . (17)
For homogeneous dust universe the local entropy density is
only a function of t proportional to a - 3 (t ) , the consistent
total entropy value
4p 3
S =
sr .
3 AH
(B. Wang, E. Abdalla and T. Osada, PRL85(2000)5507)
Fig. 1: Relation between S and A with different p at the
beginning of the dust-filled universe when t 0 = 0.97 ´ 10- 5.
We now face the question:

the holographic principle has to be challenged

it can be used to select a physically acceptable
model
We prefer the second, more constructive, alternative.
Fig. 2: Inhomogeneous models which can accommodate
reasonable entropy to meet the present observable value.
Fig. 3: Choosing parameters in order to meet the entropy
value in the present observable universe.
Conclusions

Possible to modelize highly inhomogeneous
structures

Entropy constraints can give valuable information
through the holographic principle
I. Upper bound on the
number of e-foldings from
holography
The number of e-foldings during inflation
Number of e-folds:


Horizon problem, flatness problem & entropy problem
Relate to the slow roll parameters and fluctuations prediction of
inflation
The number of e-foldings during inflation
The existence of an upper bound for the number of efoldings has been discussed.
In general it is model dependent. The bound has been
obtained in some very simple cosmological settings,
while it is still difficult to be obtained in nonstandard
models.
Using the holographic principle, the consideration of
physical details connected to the universe evolution can
be avoided. We have obtained the upper bound for the
number of e-foldings for a standard FRW universe as
well as non-standard cosmology based on the brane
inspired idea of Randall and Sundrum models.
Holographic Principle
Motivated by the well-known example of black hole
entropy, an influential holographic principle has put
forward, suggesting that microscopic degrees of
freedom that build up the gravitational dynamics actually
reside on the boundary of space-time.
This principle developed to the Maldacena's conjecture
on AdS/CFT correspondence and further very important
consequences, such as Witten's identification of the
entropy, energy and temperature of CFT at high
temperatures with the entropy, mass and Hawking
temperature of the AdS black hole.
Cosmic Holography
We thus seek at a description of the
powerful holographic principle in
cosmological settings, where its testing is
subtle.
The question of holography therein: for flat
and open FLRW universes the area of the
particle horizon should bound the entropy
on the backward-looking light cone.
Verlinde-Cardy formula
FLRW universe filled with CFT with a dual AdS description has
been done by Verlinde, revealing that when a universe-sized
black hole can be formed, an interesting and surprising
correspondence appears between entropy of CFT and
Friedmann equation governing the radiation dominated closed
FLRW universes.
Generalizing Verlinde's discussion to a broader class of
universes including a cosmological constant: matching of
Friedmann equation to Cardy formula holds for de Sitter closed
and AdS flat universes.
However for the remaining de Sitter and AdS universes, the
argument fails due to breaking down of the general philosophy
of the holographic principle. In high dimensions, various other
aspects of Verlinde's proposal have also been investigated in a
number of works.
Verlinde-Cardy formula in Brane Cosmology
Further light on the correspondence between Friedmann equation
and Cardy formula has been shed from a Randall-Sundrum.
CFT dominated universe as a co-dimension one brane with finetuned tension in a background of an AdS black hole, Savonije and
Verlinde found the correspondence between Friedmann equation
and Cardy formula for the entropy of CFT when the brane crosses
the black hole horizon.
Confirmed by studying a brane-universe filled with radiation and
stiff-matter, quantum-induced brane worlds and radially infalling
brane.
The discovered relation between Friedmann equation and Cardy
formula for the entropy shed significant light on the meaning of the
holographic principle in a cosmological setting.
The general proof for this correspondence for all CFTs is still difficult
at the moment.
The number of e-foldings from holography
Our motivation here is the use of the
correspondence between the CFT gas and the
Friedmann equation establishing an upper
bound for the number of e-foldings during
inflation.
Recently, Banks and Fischler have considered
the problem of the number of e-foldings in a
universe displaying an asymptotic de Sitter
phase, as our own. As a result the number of efoldings is not larger than 65/85 depending on
the type of matter considered.
The number of e-foldings from holography
Here we reconsider the problem from the
point of view of the entropy content of the
Universe, and considering the
correspondence between the Friedmann
equation and Cardy formula in Brane
Universes.
Brane cosmology
Metric:
We consider a bulk metric defined by
and L is the curvature radius
of AdS spacetime. k takes the values 0, -1, +1
corresponding to flat, open and closed
geometrics, and
is the corresponding
metric on the unit three dimensional sections.
Brane cosmology
Black hole horizon:
The relation between the parameter m and the
Arnowitt-Deser-Misner (ADM) mass of the five
dimensional black hole M is
is the volume of the unit 3 sphere.
Brane cosmology
Metric on the brane:
Here, the location and the metric on the
boundary are time dependent. We can choose
the brane time such that
The metric on the brane is given by
CFT on the brane
The Conformal Field Theory lives on the brane,
which is the boundary of the AdS hole. The energy
for a CFT on a sphere with volume
is
given by
The density of the CFT energy
can be expressed as
Entropy
The entropy of the CFT on the brane is equal to
the Bekenstein-Hawking entropy of the AdS
black hole
The entropy density of the CFT on the brane is
Friedmann Equation
From the matching conditions we find now the
cosmological equations in the brane,
is the critical brane tension. Taking
the Friedmann Eq. reduces to the Friedmann equation
of CFT radiation dominated brane universe without
cosmological constant.
If
the brane-world is a de Sitter
universe or AdS universe, respectively.
Friedmann Equation
Using
the Friedmann equation can be written in the form
is the effective positive cosmological constant in four
dimensions.
Using
Friedmann equation becomes
which corresponds to the movement of a mechanical
nonrelativistic particle in a given potential.
Entropy Bound
For a closed universe there is a critical value for
which the solution extends to infinity (no big
crunch)
The entropy in such a universe can be rewriten as
at the end of inflation. We take to be the energy
density during inflation, that is,
Upper bound on the number of e-foldings
Scale factor at the exit of inflation leads to the
value
, where
corresponds to the
apparent horizon during inflation, and we obtain
We get
where we used the usual values
Brane corrections to the Friedmann equation
Let us consider now very high energy brane corrections
to the Friedmann equation. From the Darmois-Israel
conditions we find
where l is the brane tension and in the very high energy
limit the term dominates.
Within the high-energy regime, the expansion laws
corresponding to matter and radiation domination are
slower than in the standard cosmology.
Slower expansion rates lead to a larger value of the
number of e-foldings. However, the full calculation has
not been obtained due to the lack of knowledge of this
high-energy regime.
CFT energy density and entropy density relation
The energy density of the CFT and the entropy
density are related as follows,
Substitute in the Friedmann equation as before,
leading to a bound for the entropy, as well as a
bound for the scale factor,
Upper bound on the number of e-foldings
The era when the quadratic energy density is important.
The brane tension is required to be bounded by
and then the number of e-foldings is
where
is taken.
The number of e-foldings obtained is bigger than the
value in standard FRW cosmology, which is consistent
with the argument of Liddle et al.
Upper bound on the number of e-foldings
In summary:

we have derived the upper limit for the number of e-foldings based
upon the arguments relating Friedmann equation and Cardy formula.

For the standard FRW universe our result is in good agreement with
Literatures.

For the brane inspired cosmology in four dimensions we obtained a
larger bound. Considering such a high energy context, the
expansion laws are slower than in the standard cosmology, and our
result can again be considered to be consistent with the known
argument.

The interesting point here is that using the holographic point of view,
we can avoid a complicated physics during the universe evolution
and give a reasonable value for the upper bound of the number of efoldings.
II. WMAP constraint on
P-term inflationary
model
Supersymmetric inflationary model
Besides the standard model, supersymmetry has been
considered both as a blessing and as a curse for
inflationary model building.
 It is a blessing, primarily because it allows one to have
very flat potential, as well as to fine-tune any parameters
at the tree level. Moreover it seems more natural than
the non-symmetric theories.
 It is a curse, because during inflation one needs to
consider supergravity, where usually all scalar fields
have too big masses to support inflation.
 Exceptions:
The N=1 generic D-term inflation
The N=1 supersymmetric F-term inflation
avoids the general problem of inflation in supergravity.
P-term inflationary model
A new version of hybrid inflation, the ``P-term
inflation'' has been introduced in the context of
N=2 supersymmetry. [Kallosh & Linde]
It is intriguing that once one breaks N=2
supersymmetry and implements the P-term
inflation in N=1 supergravity, this scenario
simultaneously leads to a new class of
inflationary models, which interpolates between
D-term and F-term models.
P-term inflationary model
The effective potential in units $M_p=1$ is
A general P-term inflation model has 0<f<1 with the
special case f=0 corresponding to the D-term inflation,
while f=1 corresponds to the F-term inflation.
Above, s_e is the bifurcation point indicating the end of
inflation.
The second term in the potential is due to the one-loop
correction and the third term to the supergravity
correction.
The inflationary space
In a single field slow-roll inflation model with a
potential V(s) the amplitude of curvature
perturbation is given by
The spectral index is defined by
The logarithmic derivative of the spectral index
is
The WMAP results
WMAP result favors purely adiabatic fluctuations
with a remarkable feature that the spectral index
runs from n>1 on a large scale to n<1 on a small
scale. More specifically on the scale
k=
It is of interest to investigate whether the P-term
inflation can accommodate these observational
result.
Number of e-folds
From the potential form we learnt that inflation consists of
two long stages, one of them is determined by the oneloop effect and the other is determined by the
supergravity corrections.
The total duration of inflation can be estimated by
Where N_k is supposed to be a reasonable number of efoldings.
Thus we require
.
For the F-term inflation f=1 and N_k=60, g<0.15, which
is exactly the argument given by Linde and Riotto (97)
Number of e-folds

The number of e-foldings during inflation
Slow-roll parameters
For the P-term inflation
The end of inflation is determined by
Slow-roll parameters

Behavior of slow-roll parameters
The value of $s_{end}$ obtained from $\epsilon=1$ for small $s$
($s_{end}<s_0$) is the real end point of inflation.
Comparison with WMAP result
Strategy:
Express s (s=s_k) as a function of s_{end} and N_k for
different values of f and g. N_k is the number of efoldings between the time the scales of interest leave
the horizon and the end of inflation. Inserting such an s
into slow-roll parameters and we can obtain the spectral
index and its logarithmic derivative.
Comparison with WMAP result
Dependence of the spectral index and its
logarithmic derivative on f for different values of
g when the number of e-foldings are 60 and 70,
respectively.
Comparison with WMAP result

There is a threshold value of g(min) to force the spectral
index n to meet the minimum observational result 1.04.

With the increase of N, the threshold value g(min) can be
smaller. However due to the existence of the upper
bound of the number of e-foldings, this threshold value
cannot be reduced arbitrarily.

For fixed f, both the values of the spectral index and its
logarithmic derivative increase with the increase of g. For
fixed g, they increase with f as well.

We cannot enforce both spectral index and its running to
meet the WMAP observational result at the same time
for the common range of f and g.
Comparison with WMAP result

For fixed g, the dependence of the spectral index and its
logarithmic derivative on f for different values of the
number of e-foldings is shown
with the increase of the number of e-foldings
both n and its running increase.
Comparison with WMAP result


The spectral index and its logarithmic derivative
depend on the number of e-foldings for different fixed
values of g and f.
n decreases from n>1 to n<1 as k increases (N )
 Again, n and its running cannot comply with
observation at the same time.
WMAP constraint on P-term inflationary model
In summary:

The P-term inflation model with a running parameter 0<f<1 displays
a richer physics.

In addition to the upper bound on g determined by a reasonable
number of e-foldings to solve the horizon problem as required by the
inflation, the observational data of the spectral index together with
the upper limit of the number of e-foldings puts the lower bound on
the choice of g.

Obtaining a logarithmic derivative spectral index such that n>1 on
large scales while n<1 on small scale for P-term inflation. (Not for D-,
F-term inflations)

It is not possible to accommodate both observational ranges of n
and its running at the same time.

The larger values of the logarithmic derivative of the spectral index
can be around -0.011 for values of f and g keeping the spectral
index within the WMAP range.