Fischler-Susskind holographic cosmology revisited
Pablo Diaz∗ M. A. Per†, Antonio Segui‡
Departamento de Fisica Teorica
Universidad de Zaragoza. 50009-Zaragoza. Spain
Abstract
When Fischler and Susskind proposed a holographic prescription based on the
Particle Horizon, they found that spatially closed cosmological models do not verify
it due to the apparently unavoidable recontraction of the Particle Horizon area.
In this article, after a short review of their original work, we expose graphically
and analytically that spatially closed cosmological models can avoid this problem if
they expand fast enough. It has been also shown that the Holographic Principle is
saturated for a codimension one brane dominated Universe. The Fischler-Susskind
prescription is used to obtain the maximum number of degrees of freedom per Planck
volume at the Planck era compatible with the Holographic Principle.
∗e-mail: pablo@posta.unizar.es
†e-mail: maperb@gmail.com
‡e-mail: segui@posta.unizar.es
http://arxiv.org/abs/0704.1637v2
1 Introduction
One of the most promising ideas that emerged in theoretical physics during the last decade
was the Holographic Principle according to which a physical system can be described
uniquely by degrees of freedom living on its boundary [1, 2]. If the Holographic Principle
is indeed a primary principle of fundamental physics it should be verified when the entire
universe is considered as a physical system. That is, the physical information inside
any cosmological domain should be holographically codified on its boundary area. But
obviously, if an unlimited region of scale L is considered, its entropy content will scale like
volume L3 and its boundary area like L2; so inevitably the former will grow quicker than
the second and the holographic codification will be impossible for big size cosmological
domains. The origin of the Holographic Principle is related to black hole horizons; so,
it seems natural to relate it now to any kind of cosmological horizon. It is at this stage
when the causal relationship that gives rise to cosmological horizons should be taken into
account. William Fischler and Leonard Susskind proposed a cosmological holographic
prescription based on the particle horizon [3]
SPH ≤
. (1)
The entropy content inside the particle horizon of a cosmological observer cannot be
greater than one quarter of the horizon area in Planck units. Enforcing this condition
for the future of any cosmological model with constant ω = p/ρ (Friedmann-Robertson-
Walker models, FRW) spatially flat, Fischler and Susskind found the limit ω < 1. The
compatibility of this limit with the dominant energy condition seems to support the
Fischler-Susskind (FS) holographic prescription. In section 2, a detailed deduction of this
limit is shown. Moreover, the verification of the FS prescription in the past is enforced,
finding a limit for the entropy density in the Planck era.
On the other hand, in spatially closed cosmological models, the FS holographic pre-
scription yields to apparently unavoidable problems. Indeed, if the model has compact
homogeneous spatial sections, all of them of finite volume, then a physical system cannot
have an arbitrary big size at a given time. But for this kind of cosmological models the
boundary area does not grow uniformly when the size of a cosmological domain increases.
Graphically, it is shown that when the domain crosses the equator the boundary area
begins to decrease, going to zero when the domain reaches the antipodes and covers the
entire universe [3, 4]. Figure 1 show this behavior for spatial dimension n = 2.
Raphael Bousso proposed a different holographic prescription [4, 5] based on the evalua-
tion of the entropy content over certain null sections named light-sheets. This prescription
solves the problems associated to spatially closed cosmological models, but it also lacks
the simplicity of the FS prescription. The Bousso prescription will not be used here but
it can be shown that both prescriptions are closely related: Two of the light-sheets de-
fined by Bousso give rise to the past light cone of a cosmological observer1. According to
our previous work [6], the entropy content over the past light cone is proportional to the
entropy content over the particle horizon (defined over the homogeneous spatial section
of the observer), and for adiabatic expansion both will be exactly the same. In fact, the
1According to the Bousso’s nomenclature, every past light cone can be built with the light sheets (+-)
and (-+) associated to the maximum of that cosmological light cone, also called apparent horizon [4, 5].
Figure 1: Decrease of the area of a domain defined in a compact spatial section when its volume
increases and goes beyond one half of the total volume (further than the equator).
original FS prescription applies to the entropy content over the ingoing past directed null
section associated to a given spherical boundary; the key is that the verification for the
particle horizon (1) guarantees the verification for every spherical boundary. In conclu-
sion, the FS holographic prescription (1) also imposes a limit on the entropy content over
the past light cone, and then it may also be regarded covariant as well as the Bousso
prescription.
In section 3 of this paper general explicit solutions for the area and the volume of spherical
cosmological domains are obtained in spatially closed (n+1)-dimensional FRW models.
It is shown that, in fact, the boundary area of the particle horizon defined in recontract-
ing models (dominated by conventional matter) tends to zero; so, the FS holographic
prescription will be violated for this kind of models. But it is also shown that non-
recontracting models, that is, spatially closed (n+1)-dimensional FRW models dominated
by quintessence matter (bouncing models), do not necessarily present this problematical
behavior. These models present accelerated expansion, and particularly only the most
accelerated models avoid the collapse of the particle horizon. So, it is deduced that a
rapid enough cosmological expansion does not allow the particle horizon to evolve enough
over the hyperspheric spatial section to reach the antipodes, so the boundary area never
decreases. It will be shown that the sufficiently accelerated FRW model corresponds to
universes dominated by a codimension one brane gas; thus, such a fluid could saturate
the Holographic Principle.
Section 3 concludes with a discussion of our results in contrast with other related works.
Especially interesting are the recent works about holographic dark energy. The simplified
argument is that a holographic limit on the entropy of a cosmological domain could also
imply a limit of its energy content; thus, the Holographic Principle applied to cosmology
might illuminate the dark energy problem [7, 8]. It is argued how our results could improve
the compatibility between the particle horizon and the holographic dark energy. Finally,
section 4 exposes the basic conclusions of our work.
2 Fischler-Susskind holography in flat universes
We will consider (n+1)-dimensional cosmological models with constant parameter ω = p/ρ
(FRW models). Here we study the spatially flat case k = 0; the scale factor grows
according to the potential function
R(t) = R0
n(1+ω)
∝ t1−
α (2)
where subscript 0 refers to the value of a magnitude in an arbitrary reference time t0. For
later convenience we have defined
n(1 + ω)
n(1 + ω)− 2
n being the spatial dimension of the model. In this section, only conventional matter
dominated models –which are decelerated and verify α > 1– will be considered, and
quintessence dominated models –which are accelerated and verify α < 0– are left for the
next section. Table 1 summarizes these cases and gives the specific limiting values
acceleration ω-range α-range denomination
R̈ < 0
− 1 < ω ≤ +1 α ≥
> 0 conventional matter
R̈ = 0 ω =
− 1 α = ∞ curvature dominated
R̈ > 0 −1 ≤ ω <
− 1 α ≤ 0 quintessence matter
Table 1: Relation among the cosmological acceleration, the dynamically dominant matter and
the parameters of its equation of state ω and α. The ranges can be obtained from the spatially
flat case (2) but they are also valid for the positively (18) and negatively curved case. The
dominant energy condition |ω| ≤ 1 and the value ω = −1 related with a cosmological constant
(de Sitter universe) has been also included.
Given the scale factor, the particle horizon (named in [9] like future event horizon) for
decelerated FRW models can be obtained as [10, 11, 12]
DPH(t) = R(t)
R(t′)
= αt . (4)
Assuming adiabatic expansion, the entropy in a comoving volume must be constant; so,
the spatial entropy density scales like
s(t)R(t)n = s0R
0 = constant ⇒ s(t) = s0R
0 R(t)
−n. (5)
Now the entropy content inside the particle horizon can be computed
SPH(t) = s(t)VPH(t) = s0R
0 R(t)
−n ωn−1
DPH(t)
n , (6)
where ωn−1 is the area of the unit sphere. The FS holographic prescription [3] demands
that the above entropy content must not be greater than one quarter of the particle
horizon area (1). Then
SPH(t) = s(t)
DPH(t)
APH(t) =
ωn−1DPH(t)
n−1 , (7)
performing some cancelations and introducing (5) we arrive at
DPH(t) ≤
4s(t)
R(t)n . (8)
This inequality is the simplified form of the FS holographic prescription for spatially flat
cosmological models. Now, according to the FS work the inequality should be imposed
in the future of any FRW model. For this purpose, comparing the exponents of temporal
evolution is sufficient: the particle horizon evolves linearly (4) and the scale factor evolves
according to (2). Thus, we obtain a family of cosmological models which will verify the
FS holographic prescription in the future
n(1 + ω)
⇒ ω < 1 . (9)
This bound on the parameter of the equation of state coincides with the limit of Special
Relativity; the sound speed in a fluid given by v2 = δp/δρ must not be greater than
the speed of light. When ω = 1, the entropic limit could be also verified depending on
the numerical prefactors (see condition (11) below). So, according to this, the dominant
energy condition enables the verification of the FS holographic prescription2 in the future.
But the previous FS argument presents an objection that we will not obviate. If we
enforce that in the future the particle horizon area dominates over its entropy content,
being potential functions, it is unavoidable that in the past the entropy content dominates
over the horizon area. In other words, these mathematical functions intersects in a given
time, so that at any previous time the holographic codification will be impossible. This
intersection time depends on the numeric prefactors that we have previously left out.
Our proposal is the enforcement of the intersection time near the Planck time; thus,
the apparent violation of the holographic prescription will be restricted to the Planck era.
Imposing this limit we will obtain an interesting relation involving the numeric prefactors;
so, we have to enforce the simplified holographic relation (8) at the Planck time (tP l = 1).
Using (4) and (3) we reach
SPH(tP l) ≤
APH(tP l)
⇒ α <
4 sP l
⇒ sP l <
1 + ω
. (10)
The first idea about this result is that the verification of the Holographic Principle needs,
in general, not too high an entropy density; concretely, the FS prescription gives us a limit
2The reverse implication is not valid: the FS prescription allows temporal violations of the dominant
energy condition [13].
on the entropy density at the Planck time. This fact is usually skipped in the literature.
Perhaps it is assumed that an entropy density at the Planck time sP l of the same order
as one is not problematic. A second view at the previous result may take one to interpret
it as a restriction the Holographic Principle imposes on the complexity of our world: the
number of degrees of freedom per Planck volume at the Planck era must not be greater
than the previous value. Thus, taking n = 3 and assuming a radiation dominated universe
(ω = 1/3) at early times, we get sP l < 3/8. Note also that this result does not depend
on the final behavior of the model, in a way that is also valid for our universe which is
supposed to be dominated now by some kind of dark energy.
Restriction (10) is not trivial. If we consider a cosmological model dynamically dominated
by a fluid with ω very near to the limit
ωlim =
− 1 (α = ∞ ) , (11)
then, the entropy density required at Planck time (10) will be absurdly small. This is
because the models with fluid of matter driven by (11) do not present particle horizon
(R(t) ∝ t); near this limit the particle horizon becomes arbitrarily big, so the entropy
content –scaled with the volume– can hardly be codified on the horizon area. Moreover,
according to [14] the observational data are compatible with a universe very near the
linear evolution; so this case cannot be discarded.
Bousso [4], Kaloper and Linde [15] proposed an ad hoc solution based on a redefinition
of the particle horizon. They took integral (4) from the Planck time t = 1 instead of
t = 0 as the starting point. However, it is not a valid solution for accelerated models
(ω < ωlim ∼ α < 0); let us see the reason. According to the new prescription, the
redefined particle horizon D̃PH grows as the scale factor (2)
D̃PH(t) = R(t)
R(t′)
= α(t− t1−1/α) ∼ −α t1−1/α . (12)
So, computing the associated entropy content S̃PH –with the entropy density (5)– leads
to a function that approaches a constant value; it can be simplified taking the Planck
time as reference time
S̃PH(t) = s0R
0 R(t)
−n ωn−1
D̃PH(t)
n ⇒ lim
S̃PH(t) =
sP l|α|
n . (13)
This limit for the entropy content seems fairly unnatural because it is of the same order
as one.
3 Fischler-Susskind holography in closed universes
Let us focus on Robertson-Walker metrics with closed spatial sections (curvature param-
eter k = +1). The line element in conformal coordinates (η, χ) reads
ds2 = R2(η)
− dη2 + dχ2 + sin2(χ)dΩ2n−1
, (14)
where dΩn−1 is the metric of the (n-1)-dimensional unit sphere. The inner volume and
area of a spherical domain of coordinate radius χ can be obtained by integrating this
metric at a given cosmological time
A(η, χ) = ωn−1R(η)
n−1 sinn−1(χ) (15)
V (η, χ) = R(η)nωn−1
sinn−1(χ′) dχ′ . (16)
The entropy content inside this volume is obtained using the entropy density (5)
S(χ) = s0R
0 ωn−1
sinn−1(χ′) dχ′ , (17)
where scale factors R(t) have been cancelled; thus, the entropy content inside a comoving
volume is constant (adiabatic expansion). Note that S(χ) strictly grows with the confor-
mal size χ of the spherical domain; however boundary area A(η, χ) reaches a maximum
near the equator : for χ > π/2 the boundary area decreases, going to zero at the antipodes,
where χ → π (see Fig. 1). Similar problems appear when the cosmological model recon-
tracts to a Big Crunch, because every boundary area will shrink to zero. In both cases
holographic codification will be impossible. This problem will be reviewed in detail and
a solution based on the cosmological acceleration will be proposed in the next section.
3.1 Conventional matter dominated cosmological models
Fischler and Susskind applied the previous ideas to a FRW (3+1)-dimensional spatially
closed cosmological model, dynamically dominated by conventional matter [3]; the explicit
solution for the scale factor is
R(η) = Rm
. (18)
Here Rm is the maximum value of the scale factor on decelerated models (α > 1 for
conventional matter, see Table 1); it depends on the relation Ω between the energy density
of the model and the critical density
Rm ≡ R0
1− Ω−10
. (19)
Introducing this scale factor on (15), and computing (17) for the usual case n = 3, the
relation between the entropy content and the boundary area of a spherical domain of
coordinate size χ at the conformal time η is obtained
(η, χ) =
2χ− sin 2χ
(sin η
)2(α−1) sin2 χ
. (20)
It should also be kept in mind that the maximum domain accessible at a given time η is
the particle horizon; so this relation must be evaluated for χPH(η), the value that locates
the particle horizon for each η [10, 12]
χPH(η) = η − ηBB, (21)
where ηBB is the value of the conformal time assigned to the beginning of the universe
(usually the Big Bang). A quick observation of relation (20) shows that the denominator
goes to zero at χPH = π (antipodes) and also when the scale factor collapses in a Big
Crunch; for both cases the ratio SPH/APH diverges and so the holographic codification (1)
is impossible. All FRW spatially closed dynamically dominated by conventional matter
models (that is −1/3 < ω ≤ 1 for n = 3) will finally recollapse; so, these models will
violate the FS holographic prescription.
3.2 Quintessence dominated cosmological models
As seen in the last section, some scenarios can become problematic for the holographic
prescription. This section aims to expose an alternative solution for some of those trou-
bling cosmological models. The key point in what follows lies in the fact that not all
spatially closed cosmological models do recollapse; for example a positive cosmological
constant could avoid the recontraction and finally provide an accelerated expansion. The
same can be said for different mechanisms which drive acceleration. The present study
provides an example where the final accelerated expansion is driven by a negative pres-
sure fluid; this means considering FRW spatially closed (curvature parameter k = +1)
cosmological models dynamically dominated by quintessence matter, that is α < 0 (see
Table 1).
The explicit solution for this kind of models is (18) as well, but its behavior is very differ-
ent: a negative exponent for the scale factor prevents it from reaching the problematic zero
value and so these models are safe from recollapsing in a Big-Crunch and from presenting
a singular Big-Bang. Now, the scale factor take a minimum value at same η; firstly the
universe contracts, but after this minimum it undergoes an accelerated expansion for ever;
these are called bouncing models [16]. Bouncing models present the obvious advantage
of being free of singularities [17], and they also enjoy a renewed interest [18] due to the
observed cosmological acceleration [21] and especially in relation with brane-cosmology
[16]3. On the other hand bouncing cosmologies meets with many problems when trying
to reproduce the universe we observe; so the solution (18) must be only considered like
a toy model to study the final behavior of an spatially closed and finally accelerated cos-
mological model. Now, formula (19) gives the minimum value of scale factor Rm, and
according to it Rm tends to zero when the energy density tends to the critical density
(Ω → 1). For an almost flat bouncing cosmology, near the minimum on the scale fac-
tor Rm quantum gravity effects could dominate erasing every correlation coming from the
previous era4. So, in following calculations the beginning of the cosmological time is going
to be taken at the minimum on the scale factor (like a no-singular Big-Bang); according
to (18), this corresponds to a conformal time ηBB = π(1−α)/2. The coordinate distance
to the particle horizon (21) is then
χPH(η) = η − ηBB = η −
(1− α) . (22)
3However, our simplest bouncing models associated to the general solution (18) usually are not con-
sidered in the literature.
4George Gamow words refering to bouncing models: “from the physical point of view we must forget
entirely about the precollapse period” [19].
It was also obtained from (18) that the scale factor diverges for η∞ = π(1 − α). This
bounded value of the conformal time implies a bounded value for the coordinate size of
the particle horizon χPH(η∞) too.
As argued before, problems for the FS holographic prescription arise at χPH = π, i. e. the
value at which a refocusing of the particle horizon on the antipodes of the observer takes
place (the horizon area goes to zero). However, this scenario can be avoided by preventing
the conformal time from reaching the problematic value (see Fig. 2); such FRW spatially
closed models will never present any particle horizon recontraction
χPH∞ < π ⇔ η∞ − ηBB =
(1− α) < π ⇔ α > −1 . (23)
Quintessence models also verify α < 0; then the allowed range becomes 0 > α > −1 which
corresponds to very accelerated cosmological models.
This result can be physically interpreted as follows: For very accelerated spatially closed
cosmological models the growing rate of the scale factor is so high that it does not permit
null geodesics to develop even half a rotation over the spatial sections (see Fig. 3). So the
particle horizon, far from reaching the antipodal point, presents an eternally increasing
area. It also happens for the limiting case α = −1 (ω = −2/3 if n = 3) due to the diver-
gence of the scale factor. This can be summarized in the next statement: every spatially
closed quintessence model with α ≥ −1 has an eternally increasing particle horizon area.
The volume of the spatial sections for spatially closed cosmological models is always
finite, and so the entropy content will be; moreover the entropy content of the universe
for adiabatic expansion is constant. Then, in accordance with the previous result, the
relation SPH/APH remains finite and goes to zero (see Fig. 4); now, using (3) leads to the
conclusion that the FS holographic limit is also compatible with FRW spatially closed
models verifying
− 1 (n = 3, ω ≤ −
). (24)
D. Youm [22] applies the same argument to brane universes and arrives to similar con-
clusions. Note that the limiting value ω = 1
− 1 corresponds to a gas of co-dimension
one branes [23]; with this kind of matter the FS holographic limit could be saturated
depending on the numerical prefactors (like the value of the entropy density s0).
The FS prescription is neither violated in the past since entropy content SPH goes to zero
quicker than the particle horizon area APH as the beginning is approached, in a way that
the relation SPH/APH also goes to zero. This behavior may be checked by introducing
(22) in the general equation (20)
(χPH) = sm
χPH − sinχPH cosχPH
sin2 χPH
)2(1−α)
χPH ≪ π : ≃
sm χPH , (26)
where sm is the spatial entropy density at the beginning of the universe, which is chosen
as reference time (so s0 = sm and R0 = Rm). Fig. 4 shows function (25) for different
values of α(ω); there, the behavior that has been analytically deduced may be graphically
Figure 2: Penrose diagrams for spatially closed FRW universes dominated by quintessence
(spatial dimension n = 3); at the “Big-Bounce” the scale factor reaches a minimum but at the
“future infinite” diverges. Depending on the particle horizon behavior two very different cases
are shown:
• On the left the particle horizon reaches the antipodes χ = π; in this case the particle horizon
area firstly grows but later it surpasses the equator of the hyperspherical spatial section and
finally decreases and shrinks to zero (see Fig. 1) in a finite time. In this case the holographic
codification will be impossible.
• But on the right the model is more accelerated and so the scale factor diverges for a lower
value of the conformal time; so the diagram height is shorter and the particle horizon cannot
reach the antipodes. In this case the particle horizon area diverges (due to the divergence of the
scale factor at the future infinite) and the holographic codification is always possible.
The height of diagram ∆η discriminates both behaviors; so, the limit case is obviously ∆η = π;
then the limit value ω = −2/3 is obtained. For this limiting case the particle horizon reaches the
antipodes at the future infinite; the scale factor diverges, the particle horizon area also diverges
and, as a consequence, the holographic codification is allowed. So, the ω-range compatible
to the holographic codification on the particle horizon is −1 ≤ ω ≤ −2/3 which corresponds
to very accelerated spatially closed cosmological models. In general, a sufficient cosmological
acceleration do not permit the recontraction of the particle horizon at the antipodes and enables
the Fischler-Susskind holographic prescription.
Different Particle Horizon Behavior
accelerated FRW models k=+1 Hα<0: quintessence L
observer at the Big-Bounce
-expanding
particle
horizon
Figure 3: Polar representation of particle horizons for quintessence dominated (α < 0) spatially
closed FRW models. Future light cones are represented from the beginning η = ηBB (Big-
Bounce) for an observer at χ = 0. For α < −1 the particle horizon reconverges in the antipodes
(it reaches and surpasses value χ = π), so the particle horizon area shrinks to zero; this shrinkage
for a particular future light cone is also shown in the figure. However, for α ≥ −1 the particle
horizon does not reconverge since the cosmological acceleration does not allow it. The FS
holographic prescription would be verified in this case. A thick line has been used to show the
limit case α = −1 (ω = −2/3 if n = 3).
The accelerated growth of the closed spatial sections (3-spheres) is shown by concentric circles;
the smallest of them is considered the beginning of the universe, so all the particle horizons
(future light cones) arise from it. In this kind of representations the radial distance coincides
with the physical radius of the spatially closed model. So, in the figure, light cones do not
show the usual 45 degrees evolution. In fact, at the beginning, the future light cones are very
flattened since the scale factor of bouncing models evolves very slowly near the minimum which
is considered the beginning of time.
0.5 π π
Evolution of the Entropy - Area relation
α=-0.2
ω=-8 9
α=-1.6
ω=-3 5
ω=-2 3
Figure 4: Evolution of quotient SPH/APH depending on the coordinate distance χPH as the
particle horizon evolves and assuming sm = 1. Functions for different values of the parameter
α(ω) are shown. A thick line represents the limit case α = −1. For α < −1 (ω > −2/3 if
n = 3) the quotient diverges as the particle horizon reaches χPH = π (the particle horizon area
shrinks to zero at the antipodes of a fiducial observer). But for very accelerated models, α ≥ −1
(ω ≤ −2/3 if n = 3), the quotient is always finite which is a necessary condition for the FS
holographic prescription to be verified.
verified. Looking at maxima of the SPH/APH functions proves that, for non-problematic
cases (α ≥ −1), value 0.5 is an upper bound, so that
α ≥ −1 (n = 3, ω ≤ −2/3) ⇒
(η) < 0.5 sm . (27)
The maximum initial entropy density compatible with the FS entropic limit depends on
this bound and this turns out to be
sm ≤ 1/2 ⇒ SPH ≤
. (28)
This means that to impose not to have more than one degree of freedom for each two
Planck volumes is enough to ensure the verification of the FS prescription for spatially
closed and accelerated FRW models with α > −1.
3.3 A more realistic cosmological model
The previous results are based on a simple explicit solution for the scale factor (18) but its
beginning (the bounce) probably is far from the real evolution of our universe. Here the
opposite point of view is exposed: a two-fluid explicit, but not simple, solution mimics a
spatially closed cosmological model according to the observed behavior. The Friedmann
equations with curvature parameter k = +1 can be solved exactly for a universe initially
dominated by radiation plus a positive cosmological constant Λ that finally provides the
desired final acceleration5. The scale factor then evolves as
R(t) =
2− 2 cosh
, (29)
where Cγ is a constant related to the radiation density ργ 0 measured in an arbitrary
reference time:
ργ 0R
0 . (30)
Due to the initial deceleration (radiation dominated era) this model presents a genuine
particle horizon defined by the future light-cone from the Big-Bang. The evolution of this
light-front over the compact spatial sections is better described by the conformal angle
χPH(t) =
. (31)
Like in the previous section if this conformal angle reaches the value π for a finite time
this means that the particle horizon has covered all the spatial section, that is, it has
reached the antipodes. There the particle horizon area is zero and the FS holographic
prescription is not verified. But the proposed model is finally dominated by a positive
Λ that provides an extreme (exponential) cosmological acceleration that could prevent
the refocusing of the particle horizon. It can be checked that the conformal angle never
reaches the problematic value π when the parameters verify CγΛ > 1.2482 (in Planck
units).
Experimental measurements suggest that our universe is flat or almost flat; here the
second case is assumed, based on the value Ω = 1.02±0.02 from the combination of SDSS
and WMAP data [20]. The best fit of the scale factor (29) to the standard cosmological
parameters H0, t0 and ΩΛ takes place for CγΛ ∼ 700. Thus, the final acceleration of our
universe seems to be enough to avoid the refocusing of the particle horizon; particularly
it will tend to the asymptotic value χPH∞ ∼ 0.5 rad. The conclusion is that if our
universe is positively curved and its evolution is similar to (29) then it could verify the
FS holographic prescription far from saturation due to the ever increasing character of
the particle horizon area.
3.4 Discussion and related works
After the Fischler and Susskind exposition of the problematic application of the holo-
graphic principle for spatially closed models [3] and R. Easther and D. Lowe confirmed
these difficulties [24], several authors proposed feasible solutions. Kalyana Rama [25]
5For a small enough Λ the attractive character of the radiation always dominates and the universe
recollapses in a Big-Crunch. Like in the classical Lemâıtre’s model (initially dominated by pressureless
matter) there exists a critical value Λ
which provides a static but inestable model.
proposed a two-fluid cosmological model, and found that when one was of quintessence
type, the FS prescription would be verified under some additional conditions. N. Cruz
and S. Lepe [26] studied cosmological models with spatial dimension n = 2, and found
also that models with negative pressure could verify the FS prescription. There are some
alternative ways such as [13] which are worth quoting. All these authors analyzed math-
ematically the functional behavior of relation S/A; our work however claims to endorse
the mathematical work with a simple picture: ever expanding spatially closed cosmolog-
ical models could verify the FS holographic prescription, since, due to the cosmological
acceleration, future light cones could not reconverge into focal points and, so, the particle
horizon area would never shrink to zero.
As one can imagine, by virtue of the previous argument there are many spatially closed
cosmological models which fulfill the FS holographic prescription; ensuring a sufficiently
accelerated final era is enough. Examples other than quintessence concern spatially closed
models with conventional matter and a positive cosmological constant, the so-called oscil-
lating models of the second kind [27]. In fact, the late evolution of this family of models is
dominated by the cosmological constant which is compatible with ω = −1, and this value
verifies (24). Roughly speaking, an asymptotically exponential expansion will provide
acceleration enough to avoid the reconvergence of future light cones.
One more remark about observational result comes to support the study of quintessence
models. If the fundamental character of the Holographic Principle as a primary princi-
ple guiding the behavior of our universe is assumed, it looks reasonable to suppose the
saturation of the holographic limit. This is one of the arguments used by T. Banks and
W. Fischler [28, 29] to propose a holographic cosmology based on a an early universe,
spatially flat, dominated by a fluid with ω = 16. According to (9) this value saturates
the FS prescription for spatially flat FRW models, but it seems fairly incompatible with
observational results. However, for spatially closed FRW cosmological models, it has been
found that the saturation of the Holographic Principle is related to the value ω = −2/3
which is compatible with current observations (according to [30], ω < −0.76 at the 95%
confidence level). It is likely that the simplest bouncing model (18) does not describe our
universe correctly; however, as shown in this paper, the initial behavior of the universe
can enforce the evolution of the particle horizon (future light cone from the beginning)
to a saturated scenario compatible with the observed cosmological acceleration7. Thus,
the dark energy computation based on the Holographic Principle [7, 8] seems much more
plausible
ρDE ∼ s T ∼
SPH/VPH
APH/VPH
∼ D−2PH . (32)
Taking DPH ∼ 10Gy gives ρDE ∼ 10
−10 eV4 in agreement the measured value [31].
Finally, two recent conjectures concerning holography in spatially closed universes deserve
some comments. W. Zimdahl and D. Pavon [32] claim that dynamics of the holographic
dark energy in a spatially closed universe could solve the coincidence problem; however
the cosmological scale necessary for the definition of the holographic dark energy seems
to be incompatible with the particle horizon [7, 8, 33]. In a more recent paper F. Simpson
6Banks and Fischler propose a scenario where black holes of the maximum possible size –the size of
the particle horizon– coalesce saturating the holographic limit; this “fluid” evolves according to ω = 1.
7Work in progress.
[34] proposed an imaginative mechanism in which the non-monotonic evolution of the
particle horizon over a spatially closed universe controls the equation of state of the dark
energy. The abundant work in that line is still inconclusive but it seems to be a fairly
promising line of work.
4 Conclusions
It is usually believed that we live in a very complex and chaotic universe. The Holographic
Principle puts a bound for the complexity on our world arguing that a more complex
universe would undergo a gravitational collapse. So, one dare say that gravitational
interaction is responsible for the simplicity of our world. In this paper a measure of
the maximum complexity of the universe compatible with the FS prescription of the
Holographic Principle has been deduced. The maximum entropy density at the Planck
era under the assumption of a flat FRW universe (10) and a quintessence dominated
spatially closed FRW universe (28) has been computed as well.
One of the main points of this paper is to get over an extended prejudice which states that
the FS holographic prescription is, in general, incompatible with spatially closed cosmo-
logical models. Only two very particular solutions –[25] and [26]– solved the problem but
no physical arguments were given. It has been shown along this paper that cosmological
acceleration actually allows the verification of the FS prescription for a wide range of
spatially closed cosmological models.
Finally, let us take a further step, a step to a more clear suggestion. First let us assume
that the FS prescription is a correct method for the application of the Holographic Prin-
ciple in Cosmology, then if our universe is spatially closed (although almost flat) it should
be accelerated by virtue of the FS prescription. In this sense, the observed acceleration
[30] enforces the previous assumption. In fact, the experimental results are compatible
with k = 0 [31], but a very small positive curvature cannot be discarded [20, 30, 35, 36].
This reductionist use of the Holographic Principle is not usual in the literature. The most
common way is to search a valid prescription for every cosmological model and every sce-
nario (like the Bousso solution [4, 5]). However, the only possible world we have evidence
of is the one which is observed, and maybe it is so because the Holographic Principle does
not permit a different one.
Acknowledgements
We acknowledge R. Bousso criticism and suggestions. This work has been supported by
MCYT (Spain) under grant FPA 2003-02948.
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	Introduction
	Fischler-Susskind holography in flat universes
	Fischler-Susskind holography in closed universes
	Conventional matter dominated cosmological models
	Quintessence dominated cosmological models
	A more realistic cosmological model
	Discussion and related works
	Conclusions

  When Fischler and Susskind proposed a holographic prescription based on the
Particle Horizon, they found that spatially closed cosmological models do not
verify it due to the apparently unavoidable recontraction of the Particle
Horizon area. In this article, after a short review of their original work, we
expose graphically and analytically that spatially closed cosmological models
can avoid this problem if they expand fast enough. It has been also shown that
the Holographic Principle is saturated for a codimension one brane dominated
Universe. The Fischler-Susskind prescription is used to obtain the maximum
number of degrees of freedom per Planck volume at the Planck era compatible
with the Holographic Principle.

Introduction
One of the most promising ideas that emerged in theoretical physics during the last decade
was the Holographic Principle according to which a physical system can be described
uniquely by degrees of freedom living on its boundary [1, 2]. If the Holographic Principle
is indeed a primary principle of fundamental physics it should be verified when the entire
universe is considered as a physical system. That is, the physical information inside
any cosmological domain should be holographically codified on its boundary area. But
obviously, if an unlimited region of scale L is considered, its entropy content will scale like
volume L3 and its boundary area like L2; so inevitably the former will grow quicker than
the second and the holographic codification will be impossible for big size cosmological
domains. The origin of the Holographic Principle is related to black hole horizons; so,
it seems natural to relate it now to any kind of cosmological horizon. It is at this stage
when the causal relationship that gives rise to cosmological horizons should be taken into
account. William Fischler and Leonard Susskind proposed a cosmological holographic
prescription based on the particle horizon [3]
SPH ≤
. (1)
The entropy content inside the particle horizon of a cosmological observer cannot be
greater than one quarter of the horizon area in Planck units. Enforcing this condition
for the future of any cosmological model with constant ω = p/ρ (Friedmann-Robertson-
Walker models, FRW) spatially flat, Fischler and Susskind found the limit ω < 1. The
compatibility of this limit with the dominant energy condition seems to support the
Fischler-Susskind (FS) holographic prescription. In section 2, a detailed deduction of this
limit is shown. Moreover, the verification of the FS prescription in the past is enforced,
finding a limit for the entropy density in the Planck era.
On the other hand, in spatially closed cosmological models, the FS holographic pre-
scription yields to apparently unavoidable problems. Indeed, if the model has compact
homogeneous spatial sections, all of them of finite volume, then a physical system cannot
have an arbitrary big size at a given time. But for this kind of cosmological models the
boundary area does not grow uniformly when the size of a cosmological domain increases.
Graphically, it is shown that when the domain crosses the equator the boundary area
begins to decrease, going to zero when the domain reaches the antipodes and covers the
entire universe [3, 4]. Figure 1 show this behavior for spatial dimension n = 2.
Raphael Bousso proposed a different holographic prescription [4, 5] based on the evalua-
tion of the entropy content over certain null sections named light-sheets. This prescription
solves the problems associated to spatially closed cosmological models, but it also lacks
the simplicity of the FS prescription. The Bousso prescription will not be used here but
it can be shown that both prescriptions are closely related: Two of the light-sheets de-
fined by Bousso give rise to the past light cone of a cosmological observer1. According to
our previous work [6], the entropy content over the past light cone is proportional to the
entropy content over the particle horizon (defined over the homogeneous spatial section
of the observer), and for adiabatic expansion both will be exactly the same. In fact, the
1According to the Bousso’s nomenclature, every past light cone can be built with the light sheets (+-)
and (-+) associated to the maximum of that cosmological light cone, also called apparent horizon [4, 5].
Figure 1: Decrease of the area of a domain defined in a compact spatial section when its volume
increases and goes beyond one half of the total volume (further than the equator).
original FS prescription applies to the entropy content over the ingoing past directed null
section associated to a given spherical boundary; the key is that the verification for the
particle horizon (1) guarantees the verification for every spherical boundary. In conclu-
sion, the FS holographic prescription (1) also imposes a limit on the entropy content over
the past light cone, and then it may also be regarded covariant as well as the Bousso
prescription.
In section 3 of this paper general explicit solutions for the area and the volume of spherical
cosmological domains are obtained in spatially closed (n+1)-dimensional FRW models.
It is shown that, in fact, the boundary area of the particle horizon defined in recontract-
ing models (dominated by conventional matter) tends to zero; so, the FS holographic
prescription will be violated for this kind of models. But it is also shown that non-
recontracting models, that is, spatially closed (n+1)-dimensional FRW models dominated
by quintessence matter (bouncing models), do not necessarily present this problematical
behavior. These models present accelerated expansion, and particularly only the most
accelerated models avoid the collapse of the particle horizon. So, it is deduced that a
rapid enough cosmological expansion does not allow the particle horizon to evolve enough
over the hyperspheric spatial section to reach the antipodes, so the boundary area never
decreases. It will be shown that the sufficiently accelerated FRW model corresponds to
universes dominated by a codimension one brane gas; thus, such a fluid could saturate
the Holographic Principle.
Section 3 concludes with a discussion of our results in contrast with other related works.
Especially interesting are the recent works about holographic dark energy. The simplified
argument is that a holographic limit on the entropy of a cosmological domain could also
imply a limit of its energy content; thus, the Holographic Principle applied to cosmology
might illuminate the dark energy problem [7, 8]. It is argued how our results could improve
the compatibility between the particle horizon and the holographic dark energy. Finally,
section 4 exposes the basic conclusions of our work.
2 Fischler-Susskind holography in flat universes
We will consider (n+1)-dimensional cosmological models with constant parameter ω = p/ρ
(FRW models). Here we study the spatially flat case k = 0; the scale factor grows
according to the potential function
R(t) = R0
n(1+ω)
∝ t1−
α (2)
where subscript 0 refers to the value of a magnitude in an arbitrary reference time t0. For
later convenience we have defined
n(1 + ω)
n(1 + ω)− 2
n being the spatial dimension of the model. In this section, only conventional matter
dominated models –which are decelerated and verify α > 1– will be considered, and
quintessence dominated models –which are accelerated and verify α < 0– are left for the
next section. Table 1 summarizes these cases and gives the specific limiting values
acceleration ω-range α-range denomination
R̈ < 0
− 1 < ω ≤ +1 α ≥
> 0 conventional matter
R̈ = 0 ω =
− 1 α = ∞ curvature dominated
R̈ > 0 −1 ≤ ω <
− 1 α ≤ 0 quintessence matter
Table 1: Relation among the cosmological acceleration, the dynamically dominant matter and
the parameters of its equation of state ω and α. The ranges can be obtained from the spatially
flat case (2) but they are also valid for the positively (18) and negatively curved case. The
dominant energy condition |ω| ≤ 1 and the value ω = −1 related with a cosmological constant
(de Sitter universe) has been also included.
Given the scale factor, the particle horizon (named in [9] like future event horizon) for
decelerated FRW models can be obtained as [10, 11, 12]
DPH(t) = R(t)
R(t′)
= αt . (4)
Assuming adiabatic expansion, the entropy in a comoving volume must be constant; so,
the spatial entropy density scales like
s(t)R(t)n = s0R
0 = constant ⇒ s(t) = s0R
0 R(t)
−n. (5)
Now the entropy content inside the particle horizon can be computed
SPH(t) = s(t)VPH(t) = s0R
0 R(t)
−n ωn−1
DPH(t)
n , (6)
where ωn−1 is the area of the unit sphere. The FS holographic prescription [3] demands
that the above entropy content must not be greater than one quarter of the particle
horizon area (1). Then
SPH(t) = s(t)
DPH(t)
APH(t) =
ωn−1DPH(t)
n−1 , (7)
performing some cancelations and introducing (5) we arrive at
DPH(t) ≤
4s(t)
R(t)n . (8)
This inequality is the simplified form of the FS holographic prescription for spatially flat
cosmological models. Now, according to the FS work the inequality should be imposed
in the future of any FRW model. For this purpose, comparing the exponents of temporal
evolution is sufficient: the particle horizon evolves linearly (4) and the scale factor evolves
according to (2). Thus, we obtain a family of cosmological models which will verify the
FS holographic prescription in the future
n(1 + ω)
⇒ ω < 1 . (9)
This bound on the parameter of the equation of state coincides with the limit of Special
Relativity; the sound speed in a fluid given by v2 = δp/δρ must not be greater than
the speed of light. When ω = 1, the entropic limit could be also verified depending on
the numerical prefactors (see condition (11) below). So, according to this, the dominant
energy condition enables the verification of the FS holographic prescription2 in the future.
But the previous FS argument presents an objection that we will not obviate. If we
enforce that in the future the particle horizon area dominates over its entropy content,
being potential functions, it is unavoidable that in the past the entropy content dominates
over the horizon area. In other words, these mathematical functions intersects in a given
time, so that at any previous time the holographic codification will be impossible. This
intersection time depends on the numeric prefactors that we have previously left out.
Our proposal is the enforcement of the intersection time near the Planck time; thus,
the apparent violation of the holographic prescription will be restricted to the Planck era.
Imposing this limit we will obtain an interesting relation involving the numeric prefactors;
so, we have to enforce the simplified holographic relation (8) at the Planck time (tP l = 1).
Using (4) and (3) we reach
SPH(tP l) ≤
APH(tP l)
⇒ α <
4 sP l
⇒ sP l <
1 + ω
. (10)
The first idea about this result is that the verification of the Holographic Principle needs,
in general, not too high an entropy density; concretely, the FS prescription gives us a limit
2The reverse implication is not valid: the FS prescription allows temporal violations of the dominant
energy condition [13].
on the entropy density at the Planck time. This fact is usually skipped in the literature.
Perhaps it is assumed that an entropy density at the Planck time sP l of the same order
as one is not problematic. A second view at the previous result may take one to interpret
it as a restriction the Holographic Principle imposes on the complexity of our world: the
number of degrees of freedom per Planck volume at the Planck era must not be greater
than the previous value. Thus, taking n = 3 and assuming a radiation dominated universe
(ω = 1/3) at early times, we get sP l < 3/8. Note also that this result does not depend
on the final behavior of the model, in a way that is also valid for our universe which is
supposed to be dominated now by some kind of dark energy.
Restriction (10) is not trivial. If we consider a cosmological model dynamically dominated
by a fluid with ω very near to the limit
ωlim =
− 1 (α = ∞ ) , (11)
then, the entropy density required at Planck time (10) will be absurdly small. This is
because the models with fluid of matter driven by (11) do not present particle horizon
(R(t) ∝ t); near this limit the particle horizon becomes arbitrarily big, so the entropy
content –scaled with the volume– can hardly be codified on the horizon area. Moreover,
according to [14] the observational data are compatible with a universe very near the
linear evolution; so this case cannot be discarded.
Bousso [4], Kaloper and Linde [15] proposed an ad hoc solution based on a redefinition
of the particle horizon. They took integral (4) from the Planck time t = 1 instead of
t = 0 as the starting point. However, it is not a valid solution for accelerated models
(ω < ωlim ∼ α < 0); let us see the reason. According to the new prescription, the
redefined particle horizon D̃PH grows as the scale factor (2)
D̃PH(t) = R(t)
R(t′)
= α(t− t1−1/α) ∼ −α t1−1/α . (12)
So, computing the associated entropy content S̃PH –with the entropy density (5)– leads
to a function that approaches a constant value; it can be simplified taking the Planck
time as reference time
S̃PH(t) = s0R
0 R(t)
−n ωn−1
D̃PH(t)
n ⇒ lim
S̃PH(t) =
sP l|α|
n . (13)
This limit for the entropy content seems fairly unnatural because it is of the same order
as one.
3 Fischler-Susskind holography in closed universes
Let us focus on Robertson-Walker metrics with closed spatial sections (curvature param-
eter k = +1). The line element in conformal coordinates (η, χ) reads
ds2 = R2(η)
− dη2 + dχ2 + sin2(χ)dΩ2n−1
, (14)
where dΩn−1 is the metric of the (n-1)-dimensional unit sphere. The inner volume and
area of a spherical domain of coordinate radius χ can be obtained by integrating this
metric at a given cosmological time
A(η, χ) = ωn−1R(η)
n−1 sinn−1(χ) (15)
V (η, χ) = R(η)nωn−1
sinn−1(χ′) dχ′ . (16)
The entropy content inside this volume is obtained using the entropy density (5)
S(χ) = s0R
0 ωn−1
sinn−1(χ′) dχ′ , (17)
where scale factors R(t) have been cancelled; thus, the entropy content inside a comoving
volume is constant (adiabatic expansion). Note that S(χ) strictly grows with the confor-
mal size χ of the spherical domain; however boundary area A(η, χ) reaches a maximum
near the equator : for χ > π/2 the boundary area decreases, going to zero at the antipodes,
where χ → π (see Fig. 1). Similar problems appear when the cosmological model recon-
tracts to a Big Crunch, because every boundary area will shrink to zero. In both cases
holographic codification will be impossible. This problem will be reviewed in detail and
a solution based on the cosmological acceleration will be proposed in the next section.
3.1 Conventional matter dominated cosmological models
Fischler and Susskind applied the previous ideas to a FRW (3+1)-dimensional spatially
closed cosmological model, dynamically dominated by conventional matter [3]; the explicit
solution for the scale factor is
R(η) = Rm
. (18)
Here Rm is the maximum value of the scale factor on decelerated models (α > 1 for
conventional matter, see Table 1); it depends on the relation Ω between the energy density
of the model and the critical density
Rm ≡ R0
1− Ω−10
. (19)
Introducing this scale factor on (15), and computing (17) for the usual case n = 3, the
relation between the entropy content and the boundary area of a spherical domain of
coordinate size χ at the conformal time η is obtained
(η, χ) =
2χ− sin 2χ
(sin η
)2(α−1) sin2 χ
. (20)
It should also be kept in mind that the maximum domain accessible at a given time η is
the particle horizon; so this relation must be evaluated for χPH(η), the value that locates
the particle horizon for each η [10, 12]
χPH(η) = η − ηBB, (21)
where ηBB is the value of the conformal time assigned to the beginning of the universe
(usually the Big Bang). A quick observation of relation (20) shows that the denominator
goes to zero at χPH = π (antipodes) and also when the scale factor collapses in a Big
Crunch; for both cases the ratio SPH/APH diverges and so the holographic codification (1)
is impossible. All FRW spatially closed dynamically dominated by conventional matter
models (that is −1/3 < ω ≤ 1 for n = 3) will finally recollapse; so, these models will
violate the FS holographic prescription.
3.2 Quintessence dominated cosmological models
As seen in the last section, some scenarios can become problematic for the holographic
prescription. This section aims to expose an alternative solution for some of those trou-
bling cosmological models. The key point in what follows lies in the fact that not all
spatially closed cosmological models do recollapse; for example a positive cosmological
constant could avoid the recontraction and finally provide an accelerated expansion. The
same can be said for different mechanisms which drive acceleration. The present study
provides an example where the final accelerated expansion is driven by a negative pres-
sure fluid; this means considering FRW spatially closed (curvature parameter k = +1)
cosmological models dynamically dominated by quintessence matter, that is α < 0 (see
Table 1).
The explicit solution for this kind of models is (18) as well, but its behavior is very differ-
ent: a negative exponent for the scale factor prevents it from reaching the problematic zero
value and so these models are safe from recollapsing in a Big-Crunch and from presenting
a singular Big-Bang. Now, the scale factor take a minimum value at same η; firstly the
universe contracts, but after this minimum it undergoes an accelerated expansion for ever;
these are called bouncing models [16]. Bouncing models present the obvious advantage
of being free of singularities [17], and they also enjoy a renewed interest [18] due to the
observed cosmological acceleration [21] and especially in relation with brane-cosmology
[16]3. On the other hand bouncing cosmologies meets with many problems when trying
to reproduce the universe we observe; so the solution (18) must be only considered like
a toy model to study the final behavior of an spatially closed and finally accelerated cos-
mological model. Now, formula (19) gives the minimum value of scale factor Rm, and
according to it Rm tends to zero when the energy density tends to the critical density
(Ω → 1). For an almost flat bouncing cosmology, near the minimum on the scale fac-
tor Rm quantum gravity effects could dominate erasing every correlation coming from the
previous era4. So, in following calculations the beginning of the cosmological time is going
to be taken at the minimum on the scale factor (like a no-singular Big-Bang); according
to (18), this corresponds to a conformal time ηBB = π(1−α)/2. The coordinate distance
to the particle horizon (21) is then
χPH(η) = η − ηBB = η −
(1− α) . (22)
3However, our simplest bouncing models associated to the general solution (18) usually are not con-
sidered in the literature.
4George Gamow words refering to bouncing models: “from the physical point of view we must forget
entirely about the precollapse period” [19].
It was also obtained from (18) that the scale factor diverges for η∞ = π(1 − α). This
bounded value of the conformal time implies a bounded value for the coordinate size of
the particle horizon χPH(η∞) too.
As argued before, problems for the FS holographic prescription arise at χPH = π, i. e. the
value at which a refocusing of the particle horizon on the antipodes of the observer takes
place (the horizon area goes to zero). However, this scenario can be avoided by preventing
the conformal time from reaching the problematic value (see Fig. 2); such FRW spatially
closed models will never present any particle horizon recontraction
χPH∞ < π ⇔ η∞ − ηBB =
(1− α) < π ⇔ α > −1 . (23)
Quintessence models also verify α < 0; then the allowed range becomes 0 > α > −1 which
corresponds to very accelerated cosmological models.
This result can be physically interpreted as follows: For very accelerated spatially closed
cosmological models the growing rate of the scale factor is so high that it does not permit
null geodesics to develop even half a rotation over the spatial sections (see Fig. 3). So the
particle horizon, far from reaching the antipodal point, presents an eternally increasing
area. It also happens for the limiting case α = −1 (ω = −2/3 if n = 3) due to the diver-
gence of the scale factor. This can be summarized in the next statement: every spatially
closed quintessence model with α ≥ −1 has an eternally increasing particle horizon area.
The volume of the spatial sections for spatially closed cosmological models is always
finite, and so the entropy content will be; moreover the entropy content of the universe
for adiabatic expansion is constant. Then, in accordance with the previous result, the
relation SPH/APH remains finite and goes to zero (see Fig. 4); now, using (3) leads to the
conclusion that the FS holographic limit is also compatible with FRW spatially closed
models verifying
− 1 (n = 3, ω ≤ −
). (24)
D. Youm [22] applies the same argument to brane universes and arrives to similar con-
clusions. Note that the limiting value ω = 1
− 1 corresponds to a gas of co-dimension
one branes [23]; with this kind of matter the FS holographic limit could be saturated
depending on the numerical prefactors (like the value of the entropy density s0).
The FS prescription is neither violated in the past since entropy content SPH goes to zero
quicker than the particle horizon area APH as the beginning is approached, in a way that
the relation SPH/APH also goes to zero. This behavior may be checked by introducing
(22) in the general equation (20)
(χPH) = sm
χPH − sinχPH cosχPH
sin2 χPH
)2(1−α)
χPH ≪ π : ≃
sm χPH , (26)
where sm is the spatial entropy density at the beginning of the universe, which is chosen
as reference time (so s0 = sm and R0 = Rm). Fig. 4 shows function (25) for different
values of α(ω); there, the behavior that has been analytically deduced may be graphically
Figure 2: Penrose diagrams for spatially closed FRW universes dominated by quintessence
(spatial dimension n = 3); at the “Big-Bounce” the scale factor reaches a minimum but at the
“future infinite” diverges. Depending on the particle horizon behavior two very different cases
are shown:
• On the left the particle horizon reaches the antipodes χ = π; in this case the particle horizon
area firstly grows but later it surpasses the equator of the hyperspherical spatial section and
finally decreases and shrinks to zero (see Fig. 1) in a finite time. In this case the holographic
codification will be impossible.
• But on the right the model is more accelerated and so the scale factor diverges for a lower
value of the conformal time; so the diagram height is shorter and the particle horizon cannot
reach the antipodes. In this case the particle horizon area diverges (due to the divergence of the
scale factor at the future infinite) and the holographic codification is always possible.
The height of diagram ∆η discriminates both behaviors; so, the limit case is obviously ∆η = π;
then the limit value ω = −2/3 is obtained. For this limiting case the particle horizon reaches the
antipodes at the future infinite; the scale factor diverges, the particle horizon area also diverges
and, as a consequence, the holographic codification is allowed. So, the ω-range compatible
to the holographic codification on the particle horizon is −1 ≤ ω ≤ −2/3 which corresponds
to very accelerated spatially closed cosmological models. In general, a sufficient cosmological
acceleration do not permit the recontraction of the particle horizon at the antipodes and enables
the Fischler-Susskind holographic prescription.
Different Particle Horizon Behavior
accelerated FRW models k=+1 Hα<0: quintessence L
observer at the Big-Bounce
-expanding
particle
horizon
Figure 3: Polar representation of particle horizons for quintessence dominated (α < 0) spatially
closed FRW models. Future light cones are represented from the beginning η = ηBB (Big-
Bounce) for an observer at χ = 0. For α < −1 the particle horizon reconverges in the antipodes
(it reaches and surpasses value χ = π), so the particle horizon area shrinks to zero; this shrinkage
for a particular future light cone is also shown in the figure. However, for α ≥ −1 the particle
horizon does not reconverge since the cosmological acceleration does not allow it. The FS
holographic prescription would be verified in this case. A thick line has been used to show the
limit case α = −1 (ω = −2/3 if n = 3).
The accelerated growth of the closed spatial sections (3-spheres) is shown by concentric circles;
the smallest of them is considered the beginning of the universe, so all the particle horizons
(future light cones) arise from it. In this kind of representations the radial distance coincides
with the physical radius of the spatially closed model. So, in the figure, light cones do not
show the usual 45 degrees evolution. In fact, at the beginning, the future light cones are very
flattened since the scale factor of bouncing models evolves very slowly near the minimum which
is considered the beginning of time.
0.5 π π
Evolution of the Entropy - Area relation
α=-0.2
ω=-8 9
α=-1.6
ω=-3 5
ω=-2 3
Figure 4: Evolution of quotient SPH/APH depending on the coordinate distance χPH as the
particle horizon evolves and assuming sm = 1. Functions for different values of the parameter
α(ω) are shown. A thick line represents the limit case α = −1. For α < −1 (ω > −2/3 if
n = 3) the quotient diverges as the particle horizon reaches χPH = π (the particle horizon area
shrinks to zero at the antipodes of a fiducial observer). But for very accelerated models, α ≥ −1
(ω ≤ −2/3 if n = 3), the quotient is always finite which is a necessary condition for the FS
holographic prescription to be verified.
verified. Looking at maxima of the SPH/APH functions proves that, for non-problematic
cases (α ≥ −1), value 0.5 is an upper bound, so that
α ≥ −1 (n = 3, ω ≤ −2/3) ⇒
(η) < 0.5 sm . (27)
The maximum initial entropy density compatible with the FS entropic limit depends on
this bound and this turns out to be
sm ≤ 1/2 ⇒ SPH ≤
. (28)
This means that to impose not to have more than one degree of freedom for each two
Planck volumes is enough to ensure the verification of the FS prescription for spatially
closed and accelerated FRW models with α > −1.
3.3 A more realistic cosmological model
The previous results are based on a simple explicit solution for the scale factor (18) but its
beginning (the bounce) probably is far from the real evolution of our universe. Here the
opposite point of view is exposed: a two-fluid explicit, but not simple, solution mimics a
spatially closed cosmological model according to the observed behavior. The Friedmann
equations with curvature parameter k = +1 can be solved exactly for a universe initially
dominated by radiation plus a positive cosmological constant Λ that finally provides the
desired final acceleration5. The scale factor then evolves as
R(t) =
2− 2 cosh
, (29)
where Cγ is a constant related to the radiation density ργ 0 measured in an arbitrary
reference time:
ργ 0R
0 . (30)
Due to the initial deceleration (radiation dominated era) this model presents a genuine
particle horizon defined by the future light-cone from the Big-Bang. The evolution of this
light-front over the compact spatial sections is better described by the conformal angle
χPH(t) =
. (31)
Like in the previous section if this conformal angle reaches the value π for a finite time
this means that the particle horizon has covered all the spatial section, that is, it has
reached the antipodes. There the particle horizon area is zero and the FS holographic
prescription is not verified. But the proposed model is finally dominated by a positive
Λ that provides an extreme (exponential) cosmological acceleration that could prevent
the refocusing of the particle horizon. It can be checked that the conformal angle never
reaches the problematic value π when the parameters verify CγΛ > 1.2482 (in Planck
units).
Experimental measurements suggest that our universe is flat or almost flat; here the
second case is assumed, based on the value Ω = 1.02±0.02 from the combination of SDSS
and WMAP data [20]. The best fit of the scale factor (29) to the standard cosmological
parameters H0, t0 and ΩΛ takes place for CγΛ ∼ 700. Thus, the final acceleration of our
universe seems to be enough to avoid the refocusing of the particle horizon; particularly
it will tend to the asymptotic value χPH∞ ∼ 0.5 rad. The conclusion is that if our
universe is positively curved and its evolution is similar to (29) then it could verify the
FS holographic prescription far from saturation due to the ever increasing character of
the particle horizon area.
3.4 Discussion and related works
After the Fischler and Susskind exposition of the problematic application of the holo-
graphic principle for spatially closed models [3] and R. Easther and D. Lowe confirmed
these difficulties [24], several authors proposed feasible solutions. Kalyana Rama [25]
5For a small enough Λ the attractive character of the radiation always dominates and the universe
recollapses in a Big-Crunch. Like in the classical Lemâıtre’s model (initially dominated by pressureless
matter) there exists a critical value Λ
which provides a static but inestable model.
proposed a two-fluid cosmological model, and found that when one was of quintessence
type, the FS prescription would be verified under some additional conditions. N. Cruz
and S. Lepe [26] studied cosmological models with spatial dimension n = 2, and found
also that models with negative pressure could verify the FS prescription. There are some
alternative ways such as [13] which are worth quoting. All these authors analyzed math-
ematically the functional behavior of relation S/A; our work however claims to endorse
the mathematical work with a simple picture: ever expanding spatially closed cosmolog-
ical models could verify the FS holographic prescription, since, due to the cosmological
acceleration, future light cones could not reconverge into focal points and, so, the particle
horizon area would never shrink to zero.
As one can imagine, by virtue of the previous argument there are many spatially closed
cosmological models which fulfill the FS holographic prescription; ensuring a sufficiently
accelerated final era is enough. Examples other than quintessence concern spatially closed
models with conventional matter and a positive cosmological constant, the so-called oscil-
lating models of the second kind [27]. In fact, the late evolution of this family of models is
dominated by the cosmological constant which is compatible with ω = −1, and this value
verifies (24). Roughly speaking, an asymptotically exponential expansion will provide
acceleration enough to avoid the reconvergence of future light cones.
One more remark about observational result comes to support the study of quintessence
models. If the fundamental character of the Holographic Principle as a primary princi-
ple guiding the behavior of our universe is assumed, it looks reasonable to suppose the
saturation of the holographic limit. This is one of the arguments used by T. Banks and
W. Fischler [28, 29] to propose a holographic cosmology based on a an early universe,
spatially flat, dominated by a fluid with ω = 16. According to (9) this value saturates
the FS prescription for spatially flat FRW models, but it seems fairly incompatible with
observational results. However, for spatially closed FRW cosmological models, it has been
found that the saturation of the Holographic Principle is related to the value ω = −2/3
which is compatible with current observations (according to [30], ω < −0.76 at the 95%
confidence level). It is likely that the simplest bouncing model (18) does not describe our
universe correctly; however, as shown in this paper, the initial behavior of the universe
can enforce the evolution of the particle horizon (future light cone from the beginning)
to a saturated scenario compatible with the observed cosmological acceleration7. Thus,
the dark energy computation based on the Holographic Principle [7, 8] seems much more
plausible
ρDE ∼ s T ∼
SPH/VPH
APH/VPH
∼ D−2PH . (32)
Taking DPH ∼ 10Gy gives ρDE ∼ 10
−10 eV4 in agreement the measured value [31].
Finally, two recent conjectures concerning holography in spatially closed universes deserve
some comments. W. Zimdahl and D. Pavon [32] claim that dynamics of the holographic
dark energy in a spatially closed universe could solve the coincidence problem; however
the cosmological scale necessary for the definition of the holographic dark energy seems
to be incompatible with the particle horizon [7, 8, 33]. In a more recent paper F. Simpson
6Banks and Fischler propose a scenario where black holes of the maximum possible size –the size of
the particle horizon– coalesce saturating the holographic limit; this “fluid” evolves according to ω = 1.
7Work in progress.
[34] proposed an imaginative mechanism in which the non-monotonic evolution of the
particle horizon over a spatially closed universe controls the equation of state of the dark
energy. The abundant work in that line is still inconclusive but it seems to be a fairly
promising line of work.
4 Conclusions
It is usually believed that we live in a very complex and chaotic universe. The Holographic
Principle puts a bound for the complexity on our world arguing that a more complex
universe would undergo a gravitational collapse. So, one dare say that gravitational
interaction is responsible for the simplicity of our world. In this paper a measure of
the maximum complexity of the universe compatible with the FS prescription of the
Holographic Principle has been deduced. The maximum entropy density at the Planck
era under the assumption of a flat FRW universe (10) and a quintessence dominated
spatially closed FRW universe (28) has been computed as well.
One of the main points of this paper is to get over an extended prejudice which states that
the FS holographic prescription is, in general, incompatible with spatially closed cosmo-
logical models. Only two very particular solutions –[25] and [26]– solved the problem but
no physical arguments were given. It has been shown along this paper that cosmological
acceleration actually allows the verification of the FS prescription for a wide range of
spatially closed cosmological models.
Finally, let us take a further step, a step to a more clear suggestion. First let us assume
that the FS prescription is a correct method for the application of the Holographic Prin-
ciple in Cosmology, then if our universe is spatially closed (although almost flat) it should
be accelerated by virtue of the FS prescription. In this sense, the observed acceleration
[30] enforces the previous assumption. In fact, the experimental results are compatible
with k = 0 [31], but a very small positive curvature cannot be discarded [20, 30, 35, 36].
This reductionist use of the Holographic Principle is not usual in the literature. The most
common way is to search a valid prescription for every cosmological model and every sce-
nario (like the Bousso solution [4, 5]). However, the only possible world we have evidence
of is the one which is observed, and maybe it is so because the Holographic Principle does
not permit a different one.
Acknowledgements
We acknowledge R. Bousso criticism and suggestions. This work has been supported by
MCYT (Spain) under grant FPA 2003-02948.
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	Introduction
	Fischler-Susskind holography in flat universes
	Fischler-Susskind holography in closed universes
	Conventional matter dominated cosmological models
	Quintessence dominated cosmological models
	A more realistic cosmological model
	Discussion and related works
	Conclusions
