We point out that for a large class of universes, holography implies that the
most probable value of the cosmological constant is zero. In four spacetime
dimensions, the probability distribution takes the Baum-Hawking form,
$dP\sim\exp(cM_p^2/\Lambda)d\Lambda$.Comment: 8 pages, harvmac (b

Horava, Petr

Minic, Djordje

English

arXiv

Petr Hořava

Djordje Minic

Crossref

Physical Review Letters

Probable Values of the Cosmological Constant in a Holographic Theory

We point out that for a large class of universes, holography implies that the most probable value for the cosmological constant is zero. In four space-time dimensions, the probability distribution takes the Baum-Hawking form, dP similar to exp(cM(p)^(2)/Lambda)d Lambda

Hořava, Petr

Caltech Authors - Main

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hep-th/0001145
CALT-68-2257
CITUSC/00-006
Probable Values of the Cosmological Constant
in a Holographic Theory
Petr Horˇava1,2 and Djordje Minic1,3
1CIT-USC Center for Theoretical Physics
2California Institute of Technology, Pasadena, CA 91125, USA
horava@theory.caltech.edu
3Department of Physics and Astronomy, University of Southern California
Los Angeles, CA 90089-0484, USA
minic@physics.usc.edu
We point out that for a large class of universes, holography implies that the most probable
value for the cosmological constant is zero. In four spacetime dimensions, the probability
distribution takes the Baum-Hawking form, dP ∼ exp(cM2p/Λ)dΛ.
January 2000
1. Introduction
One of the central problems of theoretical physics is why the cosmological constant is
small [1].
The cosmological constant problem has a twofold meaning: it is a problem of fun-
damental physics, because the value of the cosmological constant Λ is tied to vacuum
energy density. On the other hand, the cosmological constant tells us something about
the large scale behaviour of the universe, since a small cosmological constant implies that
the observable universe is big and (nearly) flat. The problem is that there is an enormous
discrepancy between the value of the vacuum energy density as predicted by quantum field
theory of the standard-model degrees of freedom, and the cosmologically observed value of
Λ [2]. This discrepancy occurs already at very low energy scales, of order eV, and clearly
represents the most flagrant naturalness problem in today’s physics.
Thus, the cosmological constant relates the properties of the microscopic physics of the
vacuum to the long-distance physics on cosmic scales.1 Therefore, the observable smallness
of the cosmological constant should tell us something fundamental about the underlying
microscopic theory of nature.
In this note we study implications of holography [6,7], taken as a fundamental prop-
erty of the microscopic theory of quantum gravity, for the cosmological constant problem.
Assuming that the cosmological constant is a dynamical variable, and that holographic
entropy can be given a Botzmannian interpretation, we point out that the most probable
value of the cosmological constant in a holographic theory is zero, in ensembles of universes
with finite-area holographic screens.
The argument is very simple, but apparently has not been presented in the literature
before.
2. Holography and the Cosmological Constant
It has been suggested on rather general grounds [8,9,10] that holography should be
relevant for solving the cosmological constant problem. In its simplest heuristic form, this
argument can be stated as follows. The cosmological constant problem in local quantum
field theory is a naturalness problem, following from a gross overcount of the degrees
1 This general philosophy was stressed in the wormhole approach to the cosmological constant
problem (see [3] for the original references and [4,5] for a critique of this approach).
1
of freedom of the vacuum. In a holographic theory, the Bekenstein-Hawking bound [7]
imposes a natural limit on the number of degrees of freedom, which subsequently reduces
the vacuum energy density in the microscopic theory. Notice that this intuitive argument
does not suggest that Λ should be zero; instead, it would make a small but non-zero
cosmological constant natural.2
In order to make this argument work on a practical level, we may need a precise
microscopic model that exhibits holography in a manifest way. A phenomenological model
of holography has been proposed in [9], in the context of M-theory. In that model one is
lead, schematically, to the following expansion of the low-energy effective action in four
space-time dimensions3
S ≈ NM2
∫
d4x
√
g
(
M2 +R +
1
M2
R2 + . . .
)
(2.1)
(in this expansion we ignore various multiplicative constants of order one). Here M is
an infrared scale, essentially the inverse size of the system; N is tied to the number of
degrees of freedom in the theory; and R2 symbolically denotes the terms quadratic in the
Riemann tensor. The low-energy (super)gravity regime is defined as the regime in which
the Einstein-Hilbert term is kept finite; this determines the Newton constant GN in terms
of the infrared mass M and the number of degrees of freedom N
GN
−1 ≈ NM2. (2.2)
This can be interpreted as the Bekenstein-Hawking formula for N in terms of the area
of the surface surrounding the system, measured in Planck units. Another indication for
holography comes from the following observation. The Einstein-Hilbert term is dominant
only if the higher-order curvature terms are surpressed, which happens for R ≪ M2.
This bound coincides with the Bekenstein bound [9]. For exactly the same reason the
cosmological constant term in (2.1) is naturally small, of order Λ ≈ M2 – the “Hubble
radius” of the system.
Thus, the phenomenological approach of [9] seems to suggest at a somewhat semi-
quantitative level that the cosmological constant problem – as a naturalness problem –
2 Various other aspects of the cosmological constant problem in the light of holography have
been recently studied in [11].
3 The phenomenological theory in [9] is formulated in eleven space-time dimensions; here we
have implicitly compactified the theory to four dimensions on T 7.
2
could indeed be solved in a holographic theory (for other arguments from a different point
of view, see [8]). However, the problem of finding a truly microscopic theory that manifestly
satisfies holography still remains a fascinating challenge that has not been satisfactorily
met so far.
Short of a microscopic formulation of a holographic theory, we choose a different
strategy to address the cosmological constant problem. We simply assume that holography
is valid, and show that in combination with certain robust thermodynamic arguments,
holography indeed implies that the most probable value of the cosmological constant is
zero.
For concreteness we work in four space-time dimensions, but the argument can be
easily generalized to other cases as well.
At large scales gravity is described by the most general local effective action which
incorporates the four-dimensional diffeomorphism invariance (the Einstein-Hilbert action,
the cosmological constant term, plus higher order terms in the curvature tensor and its
derivatives)
Seff = −
∫
d4x
√
g
(
1
8piG
Λ+
1
16piG
R + aRµνR
µν + bR2 + cRµνρσR
µνρσ + . . .
)
. (2.3)
Note that Λ denotes an effective cosmological constant, which also takes into account the
vacuum energy density of matter. We also assume that Λ is a dynamical variable, without
specifying a particular mechanism that leads to a dynamical Λ. Several such mechanisms
are available in the literature [1]: coupling gravity to a three-form gauge field [12], topology
change [3], and chaotic inflation [13], to name a few.
According to the holographic principle [6,14], the entire information about the space-
time can be stored on particular hypersurfaces (called holographic screens) [14], such that
the total number of degrees of freedom living on these holographic screens does not exceed
the Bekenstein-Hawking bound. At present this is the only available formulation of the
holographic principle, but it will be sufficient for our argument.
Let us concentrate on a class of universes with well-defined holographic screens of
finite area, examples of which were explicitly constructed for various space-time geome-
tries in [14]. In this class of universes, a closer look at Einstein’s equations leads to the
following scaling relation between the characteristic size r of the preferred screen and the
cosmological constant
Λr2 ≈ 1. (2.4)
3
As can be seen in [14], this relation between Λ and r is valid for a broad class of holographic
screens in spacetimes with non-negative Λ, including the de Sitter spacetime, the Einstein
static universe, and a class of the Robertson-Walker cosmologies. In the de Sitter case, for
example, the preferred screen is the cosmological horizon [14], whose area is given by [15]
A =
12pi
Λ
. (2.5)
In the anti-de Sitter spacetime, the size of the holographic screen is infinite, and we do
not know whether our argument can be extended to the cases with holographic screens of
infinite area.
The holographic principle asserts that the total number of degrees of freedom, or
entropy S, living on the holographic screen is bounded by one quarter of the area in
Planck units,
S ≈ r2M2p . (2.6)
Of course, this formula is just an upper bound, and should be really treated as an inequality.
We will take this into account below.
Using the scaling relation (2.4) valid for the holographic screens, together with the
Bekenstein-Hawking bound (2.6), we get an intriguing formula which relates the cosmo-
logical constant and the holographic gravitational entropy S
ΛS ≈M2p . (2.7)
More precisely, this formula should be viewed as an inequality
S ≤ M
2
p
Λ
. (2.8)
Thus, the holographic gravitational entropy (or the total number of degrees of freedom
living on the holographic screens) S increases with a decreasing cosmological constant.
We use this observation to argue that the most probable value for the cosmological
constant is zero. The argument proceeds as follows.
At large distances the cosmological constant could be treated as a classical variable (its
fluctuations can be neglected). Thus, following the Boltzmann principle from statistical
mechanics [16], the probabity distribution for the cosmological constant Λ to have a value
in the interval from Λ to Λ + dΛ is given by
w(Λ)dΛ = const exp {S(Λ)} (2.9)
4
(with the Boltzmann constant set to one). In (2.9), S(Λ) is the holographic entropy, which
can be formally regarded as a function of Λ using (2.8). We conclude that
w(Λ)dΛ = const exp
{
cM2p
Λ
}
. (2.10)
Here c denotes a constant of order one which takes into account the neglected numerical
factors. This formula tells us that the probability distribution is strongly peaked around
the value Λ = 0+. Hence, the most probable value for the cosmological constant, as
implied by holography and thermodynamics, is zero.
We have obtained this formula by saturating the Bekenstein-Hawking bound, but it
is easy to see that even if the holographic bound is taken as a true inequality the same
conclusion follows, due to the properties of the function exp{ cM
2
p
Λ
}, and the fact that the
probability distributions have to be normalized to one. Notice also that we have implicitly
worked with a microcanonical ensemble of universes, given our Boltzmannian interpretation
of the holographic entropy.4
3. Discussion
Here we compare our result with the Baum-Hawking mechanism for the vanishing of
the cosmological constant.
Our probability distribution (2.10) exhibits the same exponential dependence on 1/Λ
as the probability formula obtained in Euclidean quantum gravity by Baum and Hawking
[12] .5 In the case of de Sitter universes, the exact value of the numerical constant c in (2.10)
follows from (2.5), and is found to be c = 3pi. Thus, for de Sitter universes, our holographic
probability distribution exactly coincides with the Baum-Hawking distribution.
Recall that the Baum-Hawking mechanism asumes the validity of the Euclidean effec-
tive action formalism in quantum gravity; the Minkowski action would lead to an oscillating
factor exp[i
M2
p
Λ
] which completely changes the conclusion about the most probable value for
the cosmological constant. Also, the Euclidean action for Einstein’s gravity is well known
4 Some aspects of the microcanonical ensemble for gravity have been discussed in [17].
5 In D spacetime dimensions, our probability distribution generalizes to exp
{
c˜M
D−2
p Λ
(2−D)/2
}
,
and therefore gives a functional dependence on Λ which again agrees with the result of the Eu-
clidean path integral approach.
5
to be unbounded from below; this fact renders the Baum-Hawking mechanism rather prob-
lematic. Moreover, it is not clear – at least within the Hamiltonian approach as presented
in [5] – whether the Baum-Hawking factor exp[
cM2
p
Λ
] can be interpreted as a probability
distribution in the context of Euclidean quantum gravity with spacetime topology change.
By contrast, our argument is based on the assumptions of holography (without speci-
fying its microscopic origin) and thermodynamics. According to the holographic principle,
the holographic entropy counts the total number of degrees of freedom in quantum gravity.
The use of the Boltzmann formula is therefore justified by the microscopic definition of
entropy. Likewise, the use of the problematic Euclidean formalism of quantum gravity has
been completely avoided by the use of the holographic principle. Nevertheless, we find it
intriguing that the same probability distribution is found in both cases.
In this paper, we have presented a simple argument suggesting that in a holographic
theory, the cosmological constant can be naturally small. This conclusion follows from the
simple but somewhat subtle fact that by maximizing the entropy in a holographic theory,
one minimizes the vacuum energy density, which indeed seems contrary to the intuitive
expectation based on our experience with local field theory, where large entropy gives a
large contribution to the vacuum energy density.
We wish to thank T. Banks, J. Polchinski, A. Strominger, and E. Witten for useful discus-
sions. The work of P.H. has been supported in part by a Sherman Fairchild Prize Fellow-
ship, and by DOE grant DE-FG03-92-ER 40701. The work of D.M. has been supported
in part by the US Department of Energy under grant number DE-FG03-84ER 40168.
6
References
[1] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1;
S.M. Carroll, W.H. Press and E.L. Turner, Ann. Rev. Astron. and Astrophys. 30
(1992) 499.
[2] A nice review of recent data is N.A. Bachall, J.P. Ostriker, S. Pelmutter and P.J.
Steinhardt, Science 284 (1999) 1481.
[3] S. Coleman, Nucl. Phys. B307 (1988) 867; Nucl. Phys. B310 (1988) 643; S. Giddings
and A. Stominger, Nucl. Phys. B306 (1988) 890; Nucl. Phys. B307 (1988) 854; T.
Banks, Nucl. Phys. B309 (1988) 493; T. Banks, I. Klebanov and L. Susskind, Nucl.
Phys. B317 (1989) 665.
[4] J. Polchinski, Phys. Lett. B219 (1989) 251; V. Kaplunovsky, unpublished; W. Fischler
and L. Susskind, Phys. Lett. B217 (1989) 48.
[5] W. Fischler, I. Klebanov, J. Polchinski and L. Susskind, Nucl. Phys. B327 (1989) 157.
[6] G. ’t Hooft, gr-qc/9310026; L. Susskind, J. Math. Phys. 36 (1995) 6377, hep-
th/9409089.
[7] S.W. Hawking, Phys. Rev. D13 (1976) 191; J.D. Bekenstein, Phys. Rev. D23 (1981)
287.
[8] T. Banks, hep-th/9601151.
[9] P. Horˇava, Phys. Rev. D59 (1999) 046004, hep-th/9712130.
[10] A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Rev. Lett. 82 (1999) 4971, hep-
ph/9803132.
[11] C.P. Burgess, R.C. Myers and F. Quevedo, hep-th/9911164; H. Verlinde and E. Ver-
linde, hep-th/9912018; C. Schmidhuber, hep-th/9912156.
[12] E. Baum, Phys. Lett. B133 (1983) 185; S. W. Hawking, Phys. Lett. B134 (1984) 403.
[13] A.D. Linde, Phys. Lett. B175 (1986) 395, Phys. Lett. B202 (1988) 194.
[14] R. Bousso, hep-th/9905177; hep-th/9906022; hep-th/9911002.
[15] G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2738.
[16] L.D. Landau and E.M. Lifshitz, Statistical Physics, part 1 (Pergamon, 1988).
[17] J.D. Brown and J.W. York, Phys. Rev. D47 (1993) 1420, gr-qc/9209014.
7


Probable values of the cosmological constant in a holographic theory

VOLUME 85, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 AUGUST 2000
1610Probable Values of the Cosmological Constant in a Holographic Theory
Petr Hořava1,2,* and Djordje Minic1,3,†
1CIT-USC Center for Theoretical Physics
2California Institute of Technology, Pasadena, California 91125
3Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089-0484
(Received 25 January 2000)
We point out that for a large class of universes, holography implies that the most probable value for
the cosmological constant is zero. In four space-time dimensions, the probability distribution takes the
Baum-Hawking form, dP  expcM2pLdL.
PACS numbers: 98.80.Cq, 04.70.Dy, 11.10.Kk1. Introduction.—One of the central problems of
theoretical physics is why the cosmological constant is
small [1].
The cosmological constant problem has a twofold mean-
ing: It is a problem of fundamental physics, because the
value of the cosmological constant L is tied to vacuum en-
ergy density. On the other hand, the cosmological constant
tells us something about the large scale behavior of the
universe, since a small cosmological constant implies that
the observable universe is big and (nearly) flat. The prob-
lem is that there is an enormous discrepancy between the
value of the vacuum energy density as predicted by quan-
tum field theory of the standard-model degrees of freedom
and the cosmologically observed value of L [2]. This dis-
crepancy occurs already at very low-energy scales, of the
order of eV, and clearly represents the most flagrant natu-
ralness problem in today’s physics.
Thus, the cosmological constant relates the properties of
the microscopic physics of the vacuum to the long-distance
physics on cosmic scales. [This general philosophy
was stressed in the wormhole approach to the cosmologi-
cal constant problem (see [3] for the original references
and [4,5] for a critique of this approach).] Therefore,
the observable smallness of the cosmological constant
should tell us something fundamental about the underlying
microscopic theory of nature.
In this note we study implications of holography [6,7],
taken as a fundamental property of the microscopic theory
of quantum gravity, for the cosmological constant problem.
Assuming that the cosmological constant is a random vari-
able, and that holographic entropy can be given a Boltz-
mannian interpretation, we point out that the most probable
value of the cosmological constant in a holographic theory
is zero, in ensembles of universes with finite-area holo-
graphic screens.
It is not the aim of this paper to discuss the microscopic
origin of any of these assumptions. In particular, the mi-
croscopic origin of holography and the randomness of the
cosmological constant are clearly difficult problems whose
solutions we do not claim to have understood. It is our goal
to show that once these assumptions are satisfied, a simple
and robust phenomenological argument implies a naturally
small cosmological constant.0031-90070085(8)1610(4)$15.002. Holography and the cosmological constant.—It has
been suggested on rather general grounds [8–10] that
holography should be relevant for solving the cosmologi-
cal constant problem. In its simplest heuristic form, this
argument can be stated as follows. The cosmological con-
stant problem in local quantum field theory is a naturalness
problem, following from a gross overcount of the degrees
of freedom of the vacuum. In a holographic theory, the
Bekenstein-Hawking bound [7] imposes a natural limit on
the number of degrees of freedom, which subsequently
reduces the vacuum energy density in the microscopic
theory. Notice that this intuitive argument does not
suggest that L should be zero; instead, it would make a
small but nonzero cosmological constant natural. (Various
other aspects of the cosmological constant problem in the
light of holography have been recently studied in [11].)
In order to make this argument work on a practical level,
we may need a precise microscopic model that exhibits
holography in a manifest way. A phenomenological model
of holography has been proposed in [9], in the context of
M theory. In that model one is led, schematically, to the
following expansion of the low-energy effective action in
four space-time dimensions
S  NM2
Z
d4x
p
g
µ
M2 1 R 1
1
M2
R2 1 · · ·
∂
(1)
(in this expansion we ignore various multiplicative con-
stants of the order of 1). (The phenomenological theory in
[9] is formulated in eleven space-time dimensions; here we
have implicitly compactified the theory to four dimensions
on T7.) Here M is an infrared scale, essentially the inverse
size of the system; N is tied to the number of degrees of
freedom in the theory; and R2 symbolically denotes the
terms quadratic in the Riemann tensor. The low-energy
(super)gravity regime is defined as the regime in which
the Einstein-Hilbert term is kept finite; this determines the
Newton constant GN in terms of the infrared mass M and
the number of degrees of freedom N
GN
21  NM2. (2)
This can be interpreted as the Bekenstein-Hawking for-
mula for N in terms of the area of the surface surrounding© 2000 The American Physical Society
VOLUME 85, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 AUGUST 2000the system, measured in Planck units. Another indication
for holography comes from the following observation.
The Einstein-Hilbert term is dominant only if the higher-
order curvature terms are suppressed, which happens for
R ø M2. This bound coincides with the Bekenstein
bound [9]. For exactly the same reason the cosmo-
logical term in (1) is naturally small, of the order of
L  M2 — the “Hubble radius” of the system.
Thus, the phenomenological approach of [9] seems
to suggest at a somewhat semiquantitative level that
the cosmological constant problem—as a naturalness
problem—could indeed be solved in a holographic theory
(for other arguments from a different point of view, see
[8]). However, the problem of finding a truly microscopic
theory that manifestly satisfies holography still remains a
fascinating challenge that has not been satisfactorily met
so far.
Short of a microscopic formulation of a holographic
theory, we choose a different strategy to address the
cosmological constant problem. We simply assume that
holography is valid, and show that in combination with
certain robust thermodynamic arguments, holography
indeed implies that the most probable value of the cosmo-
logical constant is zero.
For concreteness we work in four space-time dimen-
sions, but the argument can be easily generalized to other
cases as well.
At large scales gravity is described by the most
general local effective action which incorporates
the four-dimensional diffeomorphism invariance (the
Einstein-Hilbert action, the cosmological constant term,
plus higher-order terms in the curvature tensor and its
derivatives)
Seff  2
Z
d4x
p
g
µ
1
8pG
L 1
1
16pG
R 1 aRmnR
mn
1 bR2 1 cRmnrsR
mnrs 1 · · ·
∂
.
(3)
Note that L denotes an effective cosmological constant,
which also takes into account the vacuum energy density
of matter. We also assume that L is a random variable,
without specifying a particular mechanism that leads to a
stochastic L. Several such mechanisms are available in
the literature [1]: coupling gravity to a three-form gauge
field [12], topology change [3], and chaotic inflation [13]
to name a few.
According to the holographic principle [6,14], the en-
tire information about the space-time can be stored on
particular hypersurfaces (called holographic screens) [14],
such that the total number of degrees of freedom living on
these holographic screens does not exceed the Bekenstein-
Hawking bound. At present this is the only available
formulation of the holographic principle, but it will be
sufficient for our argument.Let us concentrate on a class of universes with well-
defined holographic screens of finite area, examples of
which were explicitly constructed for various space-time
geometries in [14]. In this class of universes, a closer
look at Einstein’s equations leads to the following scaling
relation between the characteristic size r of the preferred
screen and the cosmological constant
Lr2  1 . (4)
As can be seen in [14], this relation between L and r is
valid for a broad class of holographic screens in space-
times with non-negative L, including the de Sitter space-
time, the Einstein static universe, and a class of the
Robertson-Walker cosmologies. In the de Sitter case, for
example, the preferred screen is the cosmological horizon
[14], whose area is given by [15]
A 
12p
L
. (5)
In the anti–de Sitter space-time, the size of the holographic
screen is infinite, and we do not know whether our argu-
ment can be extended to the cases with holographic screens
of infinite area.
The holographic principle asserts that the total number
of degrees of freedom, or entropy S, living on the holo-
graphic screen is bounded by one quarter of the area in
Planck units,
S  r2M2p . (6)
Of course, this formula is just an upper bound and should
be really treated as an inequality. We will take this into
account below.
Using the scaling relation (4) valid for the holographic
screens, together with the Bekenstein-Hawking bound (6),
we get an intriguing formula which relates the cosmologi-
cal constant and the holographic gravitational entropy S
LS  M2p . (7)
More precisely, this formula should be viewed as an
inequality
S #
M2p
L
. (8)
Thus, the holographic gravitational entropy (or the total
number of degrees of freedom living on the holographic
screens) S increases with a decreasing cosmological
constant.
We use this observation to argue that the most probable
value for the cosmological constant is zero. The argument
proceeds as follows.
At large distances the cosmological constant could be
treated as a classical variable (its fluctuations can be ne-
glected). Thus, following the Boltzmann principle from
statistical mechanics [16], the probability distribution for
the cosmological constant L to have a value in the interval1611
VOLUME 85, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 AUGUST 2000from L to L 1 dL is given by
wLdL  const 3 expSL (9)
(with the Boltzmann constant set to 1). In (9), SL is the
holographic entropy, which can be formally regarded as a
function of L using (8). We conclude that
wLdL  const 3 exp
Ω
cM2p
L
æ
. (10)
Here c denotes a constant of the order of 1 which takes
into account the neglected numerical factors. This formula
tells us that the probability distribution is strongly peaked
around the value L  01. Hence, the most probable value
for the cosmological constant, as implied by holography
and thermodynamics, is zero.
We have obtained this formula by saturating the
Bekenstein-Hawking bound, but it is easy to see that even
if the holographic bound is taken as a true inequality the
same conclusion follows, due to the properties of the
function expcM2pL, and the fact that the probability
distributions have to be normalized to 1. Notice also that
we have implicitly worked with a microcanonical ensemble
of universes, given our Boltzmannian interpretation of the
holographic entropy. (Some aspects of the microcanoni-
cal ensemble for gravity have been discussed in [17].)
3. Discussion.—Here we compare our result with the
Baum-Hawking mechanism for the vanishing of the cos-
mological constant.
Our probability distribution (10) exhibits the same ex-
ponential dependence on 1L as the probability formula
obtained in Euclidean quantum gravity by Baum and
Hawking [12]. (In D space-time dimensions, our proba-
bility distribution generalizes to expc̃MD22p L22D2,
and therefore gives a functional dependence on L which
again agrees with the result of the Euclidean path integral
approach.) In the case of de Sitter universes, the exact
value of the numerical constant c in (10) follows from (5),
and is found to be c  3p. Thus, for de Sitter universes,
our holographic probability distribution exactly coincides
with the Baum-Hawking distribution.
Recall that the Baum-Hawking mechanism assumes the
validity of the Euclidean effective action formalism in
quantum gravity; the Minkowski action would lead to an
oscillating factor expiM2pL	 which completely changes
the conclusion about the most probable value for the cos-
mological constant. Also, the Euclidean action for Ein-
stein’s gravity is well known to be unbounded from below;
this fact renders the Baum-Hawking mechanism rather
problematic. Moreover, it is not clear—at least within the
Hamiltonian approach as presented in [5]—whether the
Baum-Hawking factor expcM2pL	 can be interpreted
as a probability distribution in the context of Euclidean
quantum gravity with space-time topology change.
By contrast, our argument is based on the assumptions of
holography (without specifying its microscopic origin) and
thermodynamics. According to the holographic principle,1612the holographic entropy counts the total number of degrees
of freedom in quantum gravity. The use of the Boltzmann
formula is therefore justified by the microscopic definition
of entropy. Likewise, the use of the problematic Euclidean
formalism of quantum gravity has been completely avoided
by the use of the holographic principle. Nevertheless, we
find it intriguing that the same probability distribution is
found in both cases.
In this paper, we have presented a simple argument sug-
gesting that in a holographic theory, the cosmological con-
stant can be naturally small. This conclusion follows from
the simple but somewhat subtle fact that by maximizing
the entropy in a holographic theory, one minimizes the vac-
uum energy density. Indeed, small vacuum energy density
(hence small L) implies a large universe, and therefore
more degrees of freedom. As we have argued, this simple
correspondence is natural in a holographic theory, yet it
is contrary to our experience with conventional local field
theory, where large entropy gives a large contribution to
the vacuum energy density.
We wish to thank T. Banks, J. Polchinski, A. Strominger,
and E. Witten for useful discussions. The work of P. H. has
been supported in part by a Sherman Fairchild Prize Fel-
lowship and by DOE Grant No. DE-FG03-92-ER 40701.
The work of D. M. has been supported in part by the U.S.
Department of Energy under Grant No. DE-FG03-84ER
40168.
*Email address: horava@theory.caltech.edu
†Email address: minic@physics.usc.edu
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Probable Values of the Cosmological Constant in a Holographic Theory

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