Entropy bounds render quantum corrections to the cosmological constant
$\Lambda$ finite. Under certain assumptions, the natural value of $\Lambda$ is
of order the observed dark energy density $\sim 10^{-10} {\rm eV}^4$, thereby
resolving the cosmological constant problem. We note that the dark energy
equation of state in these scenarios is $w \equiv p / \rho = 0$ over
cosmological distances, and is strongly disfavored by observational data.
Alternatively, $\Lambda$ in these scenarios might account for the diffuse dark
matter component of the cosmological energy density.Comment: 6 pages, Latex. Added discussion of non-cosmological limits on
  holographic dark energy. Version to appear in Physics Letters 

't Hooft

Banks

Bekenstein

Bousso

Cohen

Fischler

Horava

Spergel

Stephen D.H. Hsu

Susskind

Thomas

Turner

Weinberg

Physics Letters B

English

arXiv

Crossref

Entropy bounds and dark energy

Entropy bounds render quantum corrections to the cosmological constant Λ finite. Under certain assumptions, the natural value of Λ is of order the observed dark energy density ∼10^(−10) eV^4, thereby resolving the cosmological constant problem. We note that the dark energy equation of state in these scenarios is w≡p/ρ=0 over cosmological distances, and is strongly disfavored by observational data. Alternatively, Λ in these scenarios might account for the diffuse dark matter component of the cosmological energy density

Hsu, Stephen D.H.

Caltech Authors - Main

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Entropy Bounds and Dark Energy
Stephen D.H. Hsu∗
Department of Physics
California Institute of Technology, Pasadena, CA 91125
February 1, 2008
Abstract
Entropy bounds render quantum corrections to the cosmological constant Λ finite.
Under certain assumptions, the natural value of Λ is of order the observed dark energy
density ∼ 10−10 eV4, thereby resolving the cosmological constant problem. We note
that the dark energy equation of state in these scenarios is w ≡ p/ρ = 0 over cosmo-
logical distances, and is strongly disfavored by observational data. Alternatively, Λ in
these scenarios might account for the diffuse dark matter component of the cosmolog-
ical energy density.
∗Permanent address: Institute of Theoretical Science and Department of Physics, University of Oregon,
Eugene OR 97403. Email: hsu@duende.uoregon.edu
1
There is evidence that gravity limits the number of quantum states accessible to a system,
yielding non-extensive or holographic entropy bounds [1, 2, 3, 4]. These bounds require that
the entropy of a region increase less rapidly than its volume, generally as its area in Planck
units. Hence, they imply that the dimension of the Hilbert space (number of degrees of
freedom) describing a region is finite and much smaller than previously expected.
Entropy bounds have immediate implications for the cosmological constant problem [5].
In conventional quantum field theory (in which the entropy is extensive), the quantum
corrections to the vacuum energy are typically divergent. An extreme fine-tuning of bare
parameters, for which no satisfactory mechanism is known, is required to keep the vacuum
energy within observed limits.
As noted by several authors [6, 7], there is a connection between the entropy of a system
and the quantum correction to the vacuum energy. In d = 4 field theory, the classical vacuum
energy density receives quantum corrections
Λ = Λ0 +O(s
4/3) , (1)
where s = S/V is the entropy density per unit volume. This is because both Λ and s
are dominated by ultraviolet modes; indeed, in the simplest calculation the vacuum energy
correction is simply the zero point energy summed over all modes:
Λqm ∼
∫ M
d3k
√
~k2 +m2 ∼ M4 , (2)
where M is the UV cutoff. The corresponding entropy density is s ∼ M3. Rendering the
entropy density finite also renders the correction to the cosmological constant finite.
The naive estimate in (2), often used to characterize the severity of the cosmological
constant problem, is likely to be modified by gravitational effects when we consider length
scales of relevance to cosmology. The highest energy states of a system allowed by the
cutoff M : E ∼M4L3 for a system of size L, are already within their Schwarzschild radius if
L < Rs ∼ E, or L > M
−2. (We use Planckian units, where Newton’s constant is unity.) One
can see the self-gravitational effects of the vacuum energy explicitly in perturbation theory
as follows. Diagrammatically, the usual contribution to Λqm in (2) is given by a vacuum
bubble. Treating the graviton in perturbation theory, there is a correction to the vacuum
energy from the connected (but not 1PI) graph with a graviton line connecting two bubbles.
This graph is most easily evaluated in coordinate space, and has the form M8L2. It is a
large correction to the single vacuum bubble when M4L2 ∼ 1. Additional graphs containing
g graviton lines and g+1 bubbles contribute M4(M4L2)g to the vacuum energy. They show
that when L > M−2 there is a large gravitational back reaction. To eliminate these graphs
one needs to shift the metric, presumably about a classical de Sitter background.
2
One of the main ideas leading to holography is that black hole states must be treated
more carefully in quantum gravity. A correct evaluation of Λqm could yield a result which is
much smaller than (2), and dependent on length scale L . This effect is clearly related to the
entropy bounds resulting from gravity. By making specific assumptions, one can estimate
the natural size of the correction Λqm.
In [6], a relationship between the size L of the region under consideration (which provides
an IR cutoff) and the UV cutoff M was assumed. The relationship is deduced by requiring
that no state in the Hilbert space have energy E so large that the Schwarzschild radius
Rs ∼ E exceeds L. Under this requirement, the entropy grows no faster than A
3/4 ∼ L3/2
[2], where A is the area of the region. In physical terms, this corresponds to the assumption
that the effective field theory L(L,M) describe all states of the system excluding those for
which it has already collapsed to a black hole. Further, it is assumed that the black hole
states do not contribute to Λqm. Under these assumptions we obtain
Λqm ∼ s
4/3
∼
(L3/2
L3
)4/3
∼ L−2 . (3)
Note, the value of M obtained below satisfies mi > M for all standard model particles i
except the photon and perhaps the neutrinos. For these particles the result of (2) is modified
to Λqm ∼
∑
i mi M
3, and the corresponding relationship between L andM more complicated
than described above. Nevertheless the relationship between Λqm and L, which is central to
what follows, remains the same.
In [7], it was assumed that the entropy bound has the usual area form: S < A, but that
the delocalized states of the system have typical Heisenberg energy ∼ 1/L. This yields
Λqm ∼
s
L
∼
L2
L3L
∼ L−2 , (4)
which is the same scaling as in [6], but based on different assumptions. Evaluating (3),(4)
using the size of the observed universe (the current horizon size Ltoday ∼ 10Gy) yields a
result Λqm ∼ 10
−10 eV4, which would explain the observed dark energy density [8], assuming
that Λ0 ∼ Λqm. (Note, using the area law S < A to determine the L,M relation in [6] yields
a much larger estimate of Λqm. However it seems quite plausible that the black hole states
excluded in the A3/4 entropy bound do not contribute to the vacuum energy in the usual
way.)
While the holographic ideas discussed above yield the correct value of the observed cos-
mological constant, they do not yield the correct equation of state. Consider the universe
at some earlier time when the horizon size was L (L < Ltoday). By causality, gravitational
influences have not had time to propagate between regions separated by more than L. There-
fore, the vacuum energy which appears in the Einstein equations, driving the instantaneous
3
expansion, is Λ(L). However, because the cosmological constant is L-dependent, the dark
energy equation of state w ≡ p/ρ is not equal to −1. During the matter dominated epoch
to which the WMAP and supernova measurements are sensitive, the horizon size grows as
the Robertson-Walker scale factor R(t)3/2 , so (3) and (4) imply
Λ(L) ∼ R(t)−3 , (5)
or w = 0 at the largest scales (recall, for equation of state w the energy redshift behavior
is ρ(t) ∼ R(t)−3(1+w)). The WMAP data, which are sensitive to Λ over a redshift range of
roughly 103 (since decoupling), imply w < −0.78 (95% CL) [8]. In other words, the data
require a cosmological constant that is much more constant than obtained in holographic
scenarios. In fact, in the scenarios [6, 7] Λ(L) is at all times comparable to the radiation
+ matter energy density, which would be problematic for structure formation [9]. More
generally, if we take the entropy bound S < cLn and assume that the dark energy Λ(L) ∼
s4/3, the data requires that n > 2.7 (95% CL). This does not rule out holography per se, nor
a holographic improvement to the fine tuning problem, but does rule out a simple connection
between dark energy and holography.
It is difficult to see how holographic ideas can avoid this problem with the equation of
state. By linking IR and UV scales L and M through entropy bounds, holography does
provide an essential ingredient long believed necessary to solve the cosmological constant
problem1. However, observations indicate that the dark energy density is varying quite
slowly (if at all) with the size of our universe.
An alternative possibility is that Λ(L) is actually the diffuse dark matter, with w = 0,
while dark energy has some other origin. Dark matter is roughly 30 percent of the total
energy density of the universe - within a factor of two of the dark energy density itself. It is
an open question whether Λ(L) would behave on sub-horizon length scales as ordinary dark
matter (for example, clumping during structure formation) or rather as a smooth component
of energy density.
Finally, we describe another bound related to the possibility that Λ(L) might be a func-
tion of lengthscale2. Let
Λ(L) ∼ 10−10 eV4
(
Ltoday
L
)k
, (6)
1The solution proposed in [10], which promotes the cosmological constant to a dynamical field, implies
that our universe’s groundstate has zero cosmological constant, but does not explain the observed dark
energy density. Banks [11] and Fischler [12] have proposed that the cosmological constant is not a dynamical
consequence of quantum fluctuations, but rather an input parameter related to the number of degrees of
freedom in the universe.
2The author thanks M. Kamionkowski and B. Murray for discussions on this subject.
4
where Ltoday is the current horizon size. It is possible that entropy bounds only restrict
Λ on cosmological length scales, but there is no obvious reason that the holographic ideas
cannot be applied to smaller subsystems in our universe. For smaller L the corresponding
cosmological constant is larger, and leads to a repulsive force F (L) ∼ Λ(L)L. Perhaps
the best limit on such a force comes from planetary motion in our solar system, for which
Lsolar system ∼ 50 AU. A conservative bound on k can be deduced by requiring that F (L)
be much less than the the sun’s gravitational pull: F (L) << M⊙/L
2. We find that k = 2 is
ruled out by more than 10 orders of magnitude, and that k = 1 is just allowed. Note that
if holographic effects are not responsible for the dark energy, the overall coefficient in (6)
might be much smaller and the corresponding bound on k much weaker. It is also possible
that the manifestation of holographic ideas is more subtle than described here [11, 12].
The author would like to thank M. Graesser, M. Kamionkowski, B. Murray and M. Wise
for comments, and the Caltech theory group for its hospitality. This work was supported in
part under DOE contract DE-FG06-85ER40224.
References
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[2] G. ’t Hooft, “Dimensional Reduction In Quantum Gravity,” edited by A. Ali, J. El-
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Singapore,1993), [arXiv:gr-qc/9310026].
[3] L. Susskind, J. Math. Phys. 36 (1995) 6377, [arXiv:hep-th/9409089].
[4] W. Fischler and L. Susskind, [arXiv:hep-th/9806039]; R. Bousso, JHEP 9907 (1999)
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5
[11] T. Banks, [arXiv:hep-th/0007146], [arXiv:hep-th/0306074].
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and Biology (Geoff West festschrift), Santa Fe, Dec. 2000.
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