DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

tions, and in some modern theories of the early
universe it must be nonzero. (See inflation.) A
positive cosmological constant implies a universal
repulsive force acting on all objects in
the universe. The force is very weak at short
distances, but can outbalance the gravitational
force at large distances, hence the existence of
the static Einstein universe. The Einstein equations
written with the cosmological constant are
Rµν − 1
2 gµνR−gµν = 8πGTµν
whereRµν is the Ricci tensor,gµν is the metric
tensor, R is the curvature scalar, and Tµν is the
stress-energy tensor.
cosmological constant problem In quantum
field theory, all the fields contribute to the vacuum
energy density, that is, to the cosmological
constant and, therefore, the theoretical value
for it is quite big. From dimensional analysis
one expects the density of the cosmological constant
in the universe to be of the order of (mass
of heaviest particle) 4 , and very heavy particles
have been detected, for instance the top-quark
with mass about 175 GeV. At the same time,
the observable value of cosmological constant
is zero, or at least it is very small. The astronomical
observations do not give any definite
lower bounds for the cosmological constant, but
instead put the upper bound for this density corresponding
to a mass of about 10 −47 GeV. Thus,
there is an explicit discrepancy between the theoretically
based matter field’s contributions to
the induced cosmological constant and observations.
To eliminate this discrepancy one has
to introduce the vacuum cosmological constant,
which must cancel the induced one with tremendous
precision. The exactness of this cancellation
is the cosmological constant problem.
Suggestions for the solution of the cosmological
constant problem typically reduce the finetuning
of the vacuum cosmological constant to
the fine-tuning of some other quantities like the
potential for a cosmic scalar field. Other suggestions
are based on supersymmetry, which can
prevent the contributions of matter fields to the
induced cosmological constant via the cancellation
of the contributions between bosons and
fermions. However, supersymmetry, if it exists
in nature, is believed to be broken at the energies
© 2001 by CRC Press LLC
cosmological principle
of the orderMF , the Fermi scale, ≈ (250) 4 GeV.
Thus, there is no acceptable and reliable solution
of the cosmological constant problem at the moment.
A solution has been postulated in terms
of an anthropic many-world hypothesis. According
to this hypothesis our universe is just a
single one among many others, and we live in it
because the small cosmological constant lets us
do so. Most of the other universes are strongly
compactified because of the large cosmological
constant.
One can also mention that some cosmological
and astrophysical theories require small but
non-zero cosmological constant. The density
of the cosmological constant may, in such theories,
serve as a dark matter and, in particular,
provide a desirable age of the universe. Recent
observations of Ia supernovae in distant galaxies
suggest the existence of dark matter in the form
ofaverysmall, repulsivecosmologicalconstant.
Thissmallcosmologicalconstanthasapparently
little to do with the cosmological constant problembecauseitspossibleexistencecanreducethe
necessary exactness of the cancellation between
the induced and vacuum cosmological constants
by at most one order of magnitude. See cosmological
constant, induced gravity, spontaneous
symmetry breaking.
cosmological model A solution of Einstein’s
equations (see also metric) that can be used to
describe the geometry and large-scale evolution
of matter in the universe. The term is sometimes
misused (for various reasons) to denote
such models of spacetime that do not apply to
the real universe. See Bianchi classification, homogeneity,
inhomogeneous models, perturbative
solution.
cosmological principle An assumption that
says that every observer in the universe would
see the same large-scale distribution of matter
around him (see homogeneity), and that for a
fixed observer, the large-scale distribution of
matter would be the same along every direction
(see isotropy). Philosophically, the cosmological
principle is the extreme opposite of
the pre-Copernican view that the Earth was the
center of the universe; according to the cosmological
principle all positions in the universe are
equivalent. The cosmological principle clearly