what was their (3 scientists) contribution to science related to astronomy?

-This year, the Nobel prize in Physics is awarded to APS Fellow James Peebles (Princeton University), Michel Mayor (University of Geneva), and Didier Queloz (University of Geneva; University of Cambridge).

-Discus the contribution of each winner to science and in particular to Astronomy.

–

https://www.aps.org/publications/apsnews/updates/n…

REVIEWS OF MODERN PHYSICS, VOLUME 75, APRIL 2003
The cosmological constant and dark energy
P. J. E. Peebles
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544
Bharat Ratra
Department of Physics, Kansas State University, Manhattan, Kansas 66506
(Published 22 April 2003)
Physics welcomes the idea that space contains energy whose gravitational effect approximates that of
Einstein’s cosmological constant, ⌳; today the concept is termed dark energy or quintessence. Physics
also suggests that dark energy could be dynamical, allowing for the arguably appealing picture of an
evolving dark-energy density approaching its natural value, zero, and small now because the
expanding universe is old. This would alleviate the classical problem of the curious energy scale of a
millielectron volt associated with a constant ⌳. Dark energy may have been detected by recent
cosmological tests. These tests make a good scientific case for the context, in the relativistic
Friedmann-Lemaı̂tre model, in which the gravitational inverse-square law is applied to the scales of
cosmology. We have well-checked evidence that the mean mass density is not much more than
one-quarter of the critical Einstein–de Sitter value. The case for detection of dark energy is not yet as
convincing but still serious; we await more data, which may be derived from work in progress. Planned
observations may detect the evolution of the dark-energy density; a positive result would be a
considerable stimulus for attempts at understanding the microphysics of dark energy. This review
presents the basic physics and astronomy of the subject, reviews the history of ideas, assesses the state
of the observational evidence, and comments on recent developments in the search for a fundamental
theory.
4.
The redshift-angular-size and redshiftmagnitude relations
5. Galaxy counts
6. The gravitational lensing rate
7. Dynamics and the mean mass density
8. The baryon mass fraction in clusters of
galaxies
9. The cluster mass function
10. Biasing and the development of nonlinear
mass density fluctuations
11. The anisotropy of the cosmic microwave
background radiation
12. The mass autocorrelation function and
nonbaryonic matter
13. The gravitational inverse-square law
C. The state of the cosmological tests
V. Concluding Remarks
Note added in proof
Acknowledgments
Appendix: Recent Dark-Energy Scalar Field Research
References
CONTENTS
I. Introduction
A. The issues for observational cosmology
B. The opportunity for physics
C. Some explanations
II. Basic Concepts
A. The Friedmann-Lemaı̂tre model
B. The cosmological constant
C. Inflation and dark energy
III. Historical Remarks
A. Einstein’s thoughts
B. The development of ideas
1. Early indications of ⌳
2. The coincidences argument against ⌳
3. Vacuum energy and ⌳
C. Inflation
1. The scenario
2. Inflation in a low-density universe
D. The cold-dark-matter model
E. Dark energy
1. The XCDM parametrization
2. Decay by emission of matter or radiation
3. Cosmic field defects
4. Dark-energy scalar field
IV. The Cosmological Tests
A. The theories
1. General relativity
2. The cold-dark-matter model for structure
formation
B. The tests
1. Thermal cosmic microwave background
radiation
2. Light-element abundances
3. Expansion times
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I. INTRODUCTION
There is significant observational evidence for the detection of Einstein’s cosmological constant, ⌳, or a component of the material content of the universe that varies only slowly with time and space and so acts like ⌳.
We shall use the term dark energy for ⌳ or a component
that acts like it. Detection of dark energy would be a
new clue to an old puzzle: the gravitational effect of the
zero-point energies of particles and fields. The total with
other energies, that are close to homogeneous and
nearly independent of time, acts as dark energy. What is
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P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
puzzling is that the value of the dark-energy density has
to be tiny compared to what is suggested by dimensional
analysis; the startling new evidence is that it may be different from the only other natural value, zero.
The main question to consider now is whether to accept the evidence for detection of dark energy. We outline the nature of the case in this section. After reviewing the basic concepts of the relativistic world model in
Sec. II, and in Sec. III reviewing the history of ideas, we
present in Sec. IV a more detailed assessment of the
cosmological tests and the evidence for detection of ⌳ or
its analog in dark energy.
There is little new to report on the big issue for
physics—why the dark-energy density is so small—since
Weinberg’s (1989) review in this journal.1 But there have
been analyses of a simpler idea: can we imagine that the
present dark-energy density is evolving, perhaps approaching zero? Models are introduced in Secs. II.C and
III.E, and recent work is summarized in more detail in
the Appendix. Feasible advances in cosmological tests
could detect evolution of the dark-energy density, and
perhaps its gravitational response to large-scale fluctuations in the mass distribution. This would substantially
motivate the search for a more fundamental physics
model for dark energy.
The reader is referred to Leibundgut’s (2001, Sec. 4) discussion of astrophysical hazards. Astronomers have
checks for this and other issues of interpretation when
considering the observations used in cosmological tests.
But it takes nothing away from this careful and elegant
work to note that the checks are seldom convincing, because the astronomy is complicated and what can be
observed is sparse. What is more, we do not know ahead
of time that the physics well tested on scales ranging
from the laboratory to the Solar System survives the
enormous extrapolation to cosmology.
The situation is by no means hopeless. We now have
significant cross-checks from the consistency of results
based on independent applications of the astronomy and
of the physics of the cosmological model. If the physics
or astronomy was faulty we would not expect consistency from independent lines of evidence—apart from
the occasional accident and the occasional tendency to
stop the analysis when it approaches the ‘‘right answer.’’
We have to demand abundant evidence of consistency,
and that is starting to appear.
The case for detection of ⌳ or dark energy commences with the Friedmann-Lemaı̂tre cosmological
model. In this model the expansion history of the universe is determined by a set of dimensionless parameters
whose sum is normalized to unity,
A. The issues for observational cosmology
We will make two points. First, cosmology has a substantial observational and experimental basis, which
supports many aspects of the standard model as almost
certainly being good approximations to reality. Second,
the empirical basis is not nearly as strong for cosmology
as it is for the standard model of particle physics: in
cosmology it is not yet a matter of measuring the parameters in a well-established theory.
To explain the second point we direct our attention to
those more accustomed to experiments in the laboratory
than to astronomy-related observations of astronomers’
Tantalus principle: one can look at distant objects but
never touch them. For example, the observations of supernovae in distant galaxies offer evidence of dark energy, under the assumption that distant and nearby supernovae are drawn from the same statistical sample
(that is, that they are statistically similar enough for the
purpose of this test). There is no direct way to check
this, and it is easy to imagine differences between distant
and nearby supernovae of the same nominal type. More
distant supernovae are seen in younger galaxies, because
of the travel time of light, and these younger galaxies
tend to have more massive rapidly evolving stars with
lower heavy-element abundances. How do we know that
the properties of the supernovae are not also different?
⍀ M0 ⫹⍀ R0 ⫹⍀ ⌳0 ⫹⍀ K0 ⫽1.
The first, ⍀ M0 , is a measure of the present mean mass
density in nonrelativistic matter, mainly baryons and
nonbaryonic dark matter. The second, ⍀ R0 ⬃1⫻10⫺4 , is
a measure of the present mass in the relativistic 3-K
thermal cosmic microwave background radiation, which
almost homogeneously fills space, and the accompanying
low-mass neutrinos. The third is a measure of ⌳ or the
present value of the dark-energy equivalent. The fourth,
⍀ K0 , is an effect of the curvature of space. We review
some details of these parameters in the next section, and
of their measurements in Sec. IV.
The most direct evidence for detection of dark energy
comes from observations of supernovae of a type whose
intrinsic luminosities are close to uniform (after subtle
astronomical corrections, a few details of which are discussed in Sec. IV.B.4). The observed brightness as a
function of the wavelength shift of the radiation probes
the geometry of spacetime, in what has come to be
called the redshift-magnitude relation.2 The measurements agree with the relativistic cosmological model
with ⍀ K0 ⫽0, meaning no space curvature, and ⍀ ⌳0
⬃0.7, meaning nonzero ⌳. A model with ⍀ ⌳0 ⫽0 is two
2
1
Sahni and Starobinsky (2000); Carroll (2001); Weinberg
(2001); Witten (2001); and Ellwanger (2002) present more recent reviews.
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
(1)
The apparent magnitude is m⫽⫺2.5 log10 f plus a constant,
where f is the detected energy flux density in a chosen wavelength band. The standard measure of the wavelength shift,
due to the expansion of the universe, is the redshift z defined
in Eq. (7) below.
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
or three standard deviations off the best fit, depending
on the data set and analysis technique. This is an important indication, but 2 to 3 ␴ is not convincing, even when
we can be sure that systematic errors are under reasonable control. And we have to consider that there may be
a significant systematic error from differences between
distant, high-redshift, and nearby, low-redshift, supernovae.
There is a check, based on the cold-dark-matter
(CDM) model3 for structure formation. The fit of the
model to the observations reviewed in Sec. IV.B yields
two key constraints. First, the angular power spectrum
of fluctuations in the temperature of the 3-K thermal
cosmic microwave background radiation across the sky
indicates that ⍀ K0 is small. Second, the power spectrum
of the spatial distribution of the galaxies requires ⍀ M0
⬃0.25. Similar estimates of ⍀ M0 follow from independent lines of observational evidence. The rate of gravitational lensing prefers a somewhat larger value (if ⍀ K0
is small), and some dynamical analyses of systems of
galaxies prefer lower ⍀ M0 . But the differences could all
result from measurement uncertainties. Since ⍀ R0 in Eq.
(1) is small, the conclusion is that ⍀ ⌳0 is large, in excellent agreement with the supernovae result.
Caution is in order, however, because this check
depends on the CDM model for structure formation.
We cannot see the dark matter, so we naturally assign
it the simplest properties possible. Maybe it is significant
that the model has observational problems with galaxy
formation, as discussed in Sec. IV.A.2, or maybe these
problems are only apparent, due to the complications of
the astronomy. We are going to have to determine which
is correct before we can have confidence in the role of
the CDM model in cosmological tests. We will get a
strong hint from current precision angular distribution
measurements of the 3-K thermal cosmic microwave
background radiation.4 If the results match precisely the
prediction of the relativistic model for cosmology and
the CDM model for structure formation, with parameter
choices that agree with the constraints from all the other
cosmological tests, there will be strong evidence that we
are approaching a good approximation to reality, and
the completion of the great program of cosmological
tests that commenced in the 1930s. But all that is in the
future.
We wish to emphasize that the advances in the empirical basis for cosmology already are very real and substantial. How firm the conclusion is depends on the issue, of course. Every competent cosmologist we know
accepts as established beyond reasonable doubt that
the universe is expanding and cooling in a near homo-
3
The model is named after the nonbaryonic cold dark matter
that is assumed to dominate the masses of galaxies in the
present universe. There are more assumptions in the CDM
model, of course; they are discussed in Secs. III.D and IV.A.2.
4
At the time of writing the Microwave Anisotropy Probe
(MAP) satellite is collecting data; the project is described in
http://map.gsfc.nasa.gov/
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
561
geneous and isotropic way from a hotter denser state:
how else could space, which is transparent now, have
been filled with radiation that has relaxed to a thermal
spectrum? The debate is when the expansion commenced or became a meaningful concept. Some whose
opinions and research we respect question the extrapolation of the gravitational inverse-square law, in
its use in estimates of masses in galaxies and systems of
galaxies, and of ⍀ M0 . We agree that this law is one of
the hypotheses to be tested. Our conclusion from the
cosmological tests of Sec. IV is that the law passes
significant, though not yet complete, tests, and that
we already have a strong scientific case, resting on the
abundance of cross-checks, that the matter density
parameter ⍀ M0 is about one-quarter. The case for
detection of ⍀ ⌳0 is significant too, but not yet as compelling.
For the most part the results of the cosmological tests
agree wonderfully well with accepted theory. But the observational challenges to the tests are substantial: we are
drawing profound conclusions from very limited information. We have to be liberal when considering ideas
about what the universe is like, and conservative when
accepting ideas into the established canon.
B. The opportunity for physics
Unless there is some serious and quite unexpected
flaw in our understanding of the principles of physics we
can be sure the zero-point energy of the electromagnetic
field at laboratory wavelengths is real and measurable,
as in the Casimir (1948) effect.5 Like all energy, this
zero-point energy has to contribute to the source term in
Einstein’s gravitational field equation. If, as seems likely,
the zero-point energy of the electromagnetic field is
close to homogeneous and independent of the velocity
of the observer, it manifests itself as a positive contribution to Einstein’s ⌳, or dark energy. The zero-point energies of the fermions make a negative contribution.
Other contributions, perhaps including the energy densities of fields that interact only with themselves and
5
See Bordag, Mohideen, and Mostepanenko (2001) for a recent review. The attractive Casimir force between two parallel
conducting plates results from the boundary condition that
suppresses the number of modes of oscillation of the electromagnetic field between the plates, thus suppressing the energy
of the system. One can understand the effect at small separation without reference to the quantum behavior of the electromagnetic field, such as in the analysis of the van der Waals
interaction in quantum mechanics, by taking account of the
term in the particle Hamiltonian for the Coulomb potential
energy between the charged particles in the two separate neutral objects. But a more complete treatment, as discussed by
Cohen-Tannoudji, Dupont-Roc, and Grynberg (1992), replaces
the Coulomb interaction with the coupling of the charged particles to the electromagnetic-field operator. In this picture the
van der Waals interaction is mediated by the exchange of virtual photons. With either way of looking at the Casimir
effect—the perturbation of the normal modes or the exchange
of virtual quanta of the unperturbed modes—the effect is the
same, the suppression of the energy of the system.
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P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
gravity, might have either sign. The value of the sum
suggested by dimensional analysis is much larger than
what is allowed by the relativistic cosmological model.
The only other natural value is ⌳⫽0. If ⌳ really is tiny
but not zero, this introduces a most stimulating though
enigmatic clue to the physics yet to be discovered.
To illustrate the problem we outline an example of a
contribution to ⌳. The energy density in the 3-K thermal
cosmic microwave background radiation, which amounts
to ⍀ R0 ⬃5⫻10⫺5 in Eq. (1) (ignoring the neutrinos),
peaks at wavelength ␭⬃2 mm. At this Wien peak the
photon occupation number is about one-fifteenth. The
zero-point energy amounts to half the energy of a photon at the given frequency. This means the zero-point
energy in the electromagnetic field at wavelengths ␭
⬃2 mm amounts to a contribution ␦ ⍀ ⌳0 ⬃4⫻10⫺4 to
the density parameter in ⌳ or the dark energy. The sum
over the modes scales as ␭ ⫺4 [as illustrated in Eq. (37)].
Thus a naive extrapolation to visible wavelengths determines that the contribution amounts to ␦ ⍀ ⌳0 ⬃5⫻1010,
already a ridiculous number.
The situation can be compared to the development of
the theory of weak interactions. The Fermi pointlike interaction model is strikingly successful for a considerable range of energies, but it was clear from the start
that the model fails at high energy. A fix was discussed—
mediate the interaction by an intermediate boson—and
eventually incorporated into the even more successful
electroweak theory. General relativity and quantum mechanics are extremely successful over a considerable
range of length scales, provided we agree not to use the
rules of quantum mechanics to count the zero-point energy density in the vacuum, even though we know we
have to count the zero-point energies in all other situations. There are thoughts on improving the situation,
though they seem to be less focused than was the case
for the Fermi model. Perhaps a new energy component
spontaneously cancels the vacuum energy density or the
new component varies slowly with position and here and
there happens to cancel the vacuum energy density well
enough to allow observers like us to exist. Whatever the
nature of the more perfect theory, it must reproduce the
successes of general relativity and quantum mechanics.
That includes the method of representing the material
content of the observable universe—all forms of mass
and energy—by the stress-energy tensor, and the relation between the stress-energy tensor and the curvature
of macroscopic spacetime. One part has to be adjusted.
The numerical values of the parameters in Eq. (1) also
are enigmatic, and possibly trying to tell us something.
The evidence is that the parameters have the approximate values
⍀ ⌳0 ⬃0.7,
⍀ DM0 ⬃0.2,
⍀ B0 ⬃0.05.
(2)
We have written ⍀ M0 in two parts: ⍀ B0 measures the
density of the baryons we know exist and ⍀ DM0 measures the hypothetical nonbaryonic cold dark matter we
need to fit the cosmological tests. The parameters ⍀ B0
and ⍀ DM0 have similar values but represent different
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
things—baryonic and nonbaryonic matter—and ⍀ ⌳0 ,
which is thought to represent something completely different, is not much larger. Also, if the parameters were
measured when the universe was one-tenth its present
size the time-independent ⌳ parameter would contribute ⍀ ⌳ ⬃0.003. That is, we seem to have come on the
scene just as ⌳ has become an important factor in the
expansion rate. These curiosities surely are in part accidental, but maybe in part physically significant. In particular, one might imagine that the dark-energy density
represented by ⌳ is rolling to its natural value, zero, but
is very small now because we measure it when the universe is very old. We shall discuss efforts along this line
to at least partially rationalize the situation.
C. Some explanations
We have to explain our choice of nomenclature. Basic
concepts of physics say that space contains homogeneous zero-point energy, and perhaps also energy that is
homogeneous or nearly so in other forms, real or effective (such as from counter terms in gravity physics,
which make the net energy density cosmologically acceptable). In the literature this near homogeneous energy has been termed vacuum energy, the sum of
vacuum energy and quintessence (Caldwell, Davé, and
Steinhardt, 1998), and dark energy (Turner, 1999). We
have adopted the last term, and we shall refer to the
dark-energy density ␳ ⌳ that manifests itself as an effective version of Einstein’s cosmological constant, but one
that may vary slowly with time and position.6
Our subject involves two quite different traditions, in
physics and astronomy. Each has familiar notation, and
familiar ideas that may be ‘‘in the air’’ but not in recent
literature. Our attempt to take account of these traditions commences with the summary in Sec. II of the basic notation with brief explanations. We expect that
readers will find some of these concepts trivial and others of some use, and that the useful parts will be different for different readers.
We offer in Sec. III our reading of the history of ideas
on ⌳ and its generalization to dark energy. This is a
fascinating and we think edifying illustration of how science may advance in unexpected directions. It is relevant to an understanding of the present state of research in cosmology, because traditions inform opinions,
and people have had mixed feelings about ⌳ ever since
Einstein (1917) introduced it 85 years ago. The concept
never entirely disappeared in cosmology because a series of observations hinted at its presence, and because
to some cosmologists ⌳ fits the formalism too well to be
ignored. The search for the physics of the vacuum, and
its possible relation to ⌳, has a long history too. Despite
6
The dark energy should of course be distinguished from a
hypothetical gas of particles with velocity dispersion large
enough that the distribution is close to homogeneous.
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
the common and strong suspicion that ⌳ must be negligibly small, because any other acceptable value is absurd, all this history has made the community well prepared for the recent observational developments that
argue for the detection of ⌳.
Our approach in Sec. IV to the discussion of the evidence for detection of ⌳, from the cosmological tests,
also requires explanation. One occasionally reads that
the tests will show us how the world will end. That certainly seems interesting, but it is not the main point: why
should we trust an extrapolation into the indefinite future of a theory that we can at best show is a good
approximation to reality?7 As we remarked in Sec. I.A,
the purpose of the tests is to check the approximation to
reality, by checking the physics and astronomy of the
standard relativistic cosmological model, along with any
viable alternatives that may be discovered. We take our
task to be the identification of the aspects of the standard theory that enter the interpretation of the measurements and thus are or may be empirically checked or
measured.
II. BASIC CONCEPTS
563
those who have not already thought to do so, to check
that Eq. (4) is required to preserve homogeneity and
isotropy.8
The rate of change of the distance in Eq. (4) is the
speed
v ⫽dl/dt⫽Hl,
H⫽ȧ/a,
(5)
where the overdot means the derivative with respect to
world time t and H is the time-dependent Hubble parameter. When v is small compared to the speed of light
this is Hubble’s law. The present value of H is Hubble’s
constant, H 0 . When needed we will use9
H 0 ⫽100h km s⫺1 Mpc⫺1 ⫽67⫾7 km s⫺1 Mpc⫺1
⫽ 共 15⫾2 Gyr兲 ⫺1 ,
(6)
at two standard deviations. The first equation defines the
dimensionless parameter h.
Another measure of the expansion follows by considering the stretching of the wavelength of light received
from a distant galaxy. The observed wavelength ␭ obs of a
feature in the spectrum that had wavelength ␭ em at emission satisfies
A. The Friedmann-Lemaı̂tre model
The standard world model is close to homogeneous
and isotropic on large scales, and lumpy on small
scales—the effect of mass concentrations in galaxies,
stars, people, etc. The length scale at the transition from
nearly smooth to strongly clumpy is about 10 Mpc. We
use here and throughout the standard astronomers’
length unit,
1 Mpc⫽3.1⫻1024 cm⫽3.3⫻106 light years.
(3)
To be more definite, imagine that many spheres of radius 10 Mpc are placed at random, and the mass within
each is measured. At this radius the rms fluctuation in
the set of values of masses is about equal to the mean
value. On smaller scales the departures from homogeneity are progressively more nonlinear; on larger scales the
density fluctuations are perturbations to the homogeneous model. From now on we mention these perturbations only when relevant for the cosmological tests.
The expansion of the universe means the distance l(t)
between two well-separated galaxies varies with world
time t as
l 共 t 兲 ⬀a 共 t 兲 ,
(4)
where the expansion or scale factor a(t) is independent
of the choice of galaxies. It is an interesting exercise, for
7
Observations may now have detected ⌳, at a characteristic
energy scale of a millielectron volt [Eq. (47)]. We have no
guarantee that an even lower-energy scale does not exist; such
a scale could first become apparent through cosmological tests.
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
1⫹z⫽␭ obs /␭ em⫽a 共 t obs兲 /a 共 t em兲 ,
8
(7)
We feel we have to comment on a few details about Eq. (4)
to avoid contributing to debates that are more intense than
seem warranted. Think of the world time t as the proper time
kept by each of a dense set of observers, each moving so that
all the others are isotropically moving away, and with the times
synchronized to a common energy density, ␳ (t), in the near
homogeneous expanding universe. The distance l(t) is the sum
of the proper distances between neighboring observers, all
measured at time t, and along the shortest distance between
the two observers. The rate of increase of the distance, dl/dt,
may exceed the velocity of light. This is no more problematic
in relativity theory than is the large speed at which the beam of
a flashlight on Earth may swing across the face of the Moon
(assuming an adequately tight beam). Space sections at fixed t
may be noncompact, and the total mass of a homogeneous
universe formally infinite. As far as is known this is not meaningful: we can only assert that the universe is close to homogeneous and isotropic over observable scales, and that what
can be observed is a finite number of baryons and photons.
9
The numerical values in Eq. (6) are determined from an
analysis of all available measurements of H 0 prior to mid-1999
(Gott et al., 2001). They are a very reasonable summary of
the current situation. For instance, the Hubble Space Telescope Key Project summary measurement value H 0 ⫽72
⫾8 km s⫺1 Mpc⫺1 (1␴ uncertainty; Freedman et al., 2001) is in
very good agreement with Eq. (6), as is the recent Tammann
et al. (2001) summary value H 0 ⫽60⫾6 km s⫺1 Mpc⫺1 (approximate 1␴ systematic uncertainty). This is an example of
the striking change in the observational situation over the previous five years: the uncertainty in H 0 has decreased by more
than a factor of 3, making it one of the better-measured cosmological parameters.
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P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
where the expansion factor a is defined in Eq. (4) and z
is the redshift. That is, the wavelength of freely traveling
radiation stretches in proportion to the factor by which
the universe expands. To understand this, imagine that a
large part of the universe is enclosed in a cavity with
perfectly reflecting walls. The cavity expands with the
general expansion, the widths proportional to a(t).
Electromagnetic radiation is a sum of the normal modes
that fit the cavity. At interesting wavelengths the mode
frequencies are much larger than the rate of expansion
of the universe, so adiabaticity says a photon in a mode
stays there, and its wavelength thus must vary as ␭
⬀a(t), as stated in Eq. (7). The cavity disturbs the longwavelength part of the radiation, but the disturbance can
be made exceedingly small by choosing a large cavity.
Equation (7) defines the redshift z. The redshift is a
convenient label for epochs in the early universe, where
z exceeds unity. A good exercise for the student is to
check that when z is small Eq. (7) reduces to Hubble’s
law, where ␭z is the first-order Doppler shift in the
wavelength ␭, and Hubble’s parameter H is given by Eq.
(5). Thus Hubble’s law may be written as cz⫽Hl (where
we have put in the speed of light).
These results follow from the symmetry of the cosmological model and conventional local physics; we do not
need general relativity theory. When zⲏ1 we need relativistic theory to compute the relations among the redshift and other observables. An example is the relation
between redshift and apparent magnitude used in the
supernova test. Other cosmological tests check consistency among these relations, and this checks the world
model.
In general relativity the second time derivative of the
expansion factor satisfies
ä
4
⫽⫺ ␲ G 共 ␳ ⫹3p 兲 .
a
3
(8)
The gravitational constant is G. Here and throughout
we choose units to set the velocity of light to unity. The
mean mass density, ␳ (t), and the pressure, p(t), counting all contributions including dark energy, satisfy the
local energy-conservation law
ȧ
␳˙ ⫽⫺3 共 ␳ ⫹p 兲 .
a
(9)
The first term on the right-hand side represents the decrease of mass density due to the expansion that more
broadly disperses the matter. The pdV work in the second term is a familiar local concept, and is meaningful in
general relativity. But one should note that energy does
not have a general global meaning in this theory.
The first integral of Eqs. (8) and (9) is the Friedmann
equation
8
ȧ 2 ⫽ ␲ G ␳ a 2 ⫹const.
3
It is conventional to rewrite this as
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
(10)
冉冊
ȧ
a
2
⫽H 20 E 共 z 兲 2
⫽H 20 关 ⍀ M0 共 1⫹z 兲 3 ⫹⍀ R0 共 1⫹z 兲 4
⫹⍀ ⌳0 ⫹⍀ K0 共 1⫹z 兲 2 兴 .
(11)
The first equation defines the function E(z) that is introduced for later use. The second equation assumes
constant ⌳; the time-dependent dark-energy case is reviewed in Secs. II.C and III.E. The first term in the last
part of Eq. (11) represents nonrelativistic matter with
negligibly small pressure; one sees from Eqs. (7) and (9)
that the mass density in this form varies with the expansion of the universe as ␳ M⬀a ⫺3 ⬀(1⫹z) 3 . The second
term represents radiation and relativistic matter, with
pressure p R⫽ ␳ R/3, whence ␳ R⬀(1⫹z) 4 . The third term
is the effect of Einstein’s cosmological constant, or a
constant dark-energy density. The last term, discussed in
more detail below, is the constant of integration in Eq.
(10). The four density parameters ⍀ i0 are the fractional
contributions to the square of Hubble’s constant, H 20 ,
that is, ⍀ i0 (t)⫽8 ␲ G ␳ i0 /(3H 20 ). At the present epoch,
z⫽0, the present value of ȧ/a is H 0 , and the ⍀ i0 sum to
unity [Eq. (1)].
In this notation, Eq. (8) is
ä
⫽⫺H 20 关 ⍀ M0 共 1⫹z 兲 3 /2⫹⍀ R0 共 1⫹z 兲 4 ⫺⍀ ⌳0 兴 .
a
(12)
The constant of integration in Eqs. (10) and (11) is
related to the geometry of spatial sections at constant
world time. Recall that in general relativity events in
spacetime are labeled by the four coordinates x ␮ of time
and space. Neighboring events at separation dx ␮ have
invariant separation ds defined by the line element
ds 2 ⫽g ␮ ␯ dx ␮ dx ␯ .
(13)
The repeated indices are summed, and the metric tensor
g ␮ ␯ is a function of position in spacetime. If ds 2 is positive then ds is the proper (physical) time measured by
an observer who moves from one event to the other; if
negative, 兩 ds 兩 is the proper distance between the events
measured by an observer who is moving so the events
are seen to be simultaneous.
In the flat spacetime of special relativity one can
choose coordinates so the metric tensor has the
Minkowskian form
␩ ␮␯⫽
冉
1
0
0
0
0
⫺1
0
0
0
0
⫺1
0
0
0
0
⫺1
冊
.
(14)
A freely falling, inertial observer can choose locally
Minkowskian coordinates, such that along the path of
the observer g ␮ ␯ ⫽ ␩ ␮ ␯ , and the first derivatives of g ␮ ␯
vanish.
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
In the homogeneous world model we can choose coordinates so the metric tensor is of the form that results
in the line element
ds 2 ⫽dt 2 ⫺a 共 t 兲 2
冋
dr 2
⫹r 2 共 d ␪ 2 ⫹sin2 ␪ d ␾ 2 兲
1⫹Kr 2
册
T ␮⌳␯ ⫽ ␳ ⌳ g ␮ ␯ ,
In the second expression, which assumes K⬎0, the radial coordinate is r⫽K ⫺1/2sinh ␹. The expansion factor
a(t) appears in Eq. (4). If a were constant and the constant K vanished this would represent the flat spacetime
of special relativity in polar coordinates. The key point
for now is that ⍀ K0 in Eq. (11), which represents the
constant of integration in Eq. (10), is related to the constant K:
⍀ K0 ⫽K/ 共 H 0 a 0 兲 2 ,
(16)
where a 0 is the present value of the expansion factor
a(t). Cosmological tests that are sensitive to the geometry of space constrain the value of the parameter ⍀ K0 ,
and ⍀ K0 and the other density parameters ⍀ i0 in Eq.
(11) determine the expansion history of the universe.
It is useful for what follows to recall that the metric
tensor in Eq. (15) satisfies Einstein’s field equation, a
differential equation we can write as
G ␮ ␯ ⫽8 ␲ GT ␮ ␯ .
(17)
The left side is a function of g ␮ ␯ and its first two derivatives; it represents the geometry of spacetime. The
stress-energy tensor T ␮ ␯ represents the material contents of the universe, including particles, radiation,
fields, and zero-point energies. An observer in a homogeneous and isotropic universe, moving so the universe
is observed to be isotropic, would measure the stressenergy tensor to be
T ␮␯⫽
冉 冊
␳
0
0
0
0
p
0
0
0
0
p
0
0
0
0
p
(19)
where ␳ ⌳ is a constant, in a general coordinate labeling.
When writing this contribution to the stress-energy tensor separately from the rest, we bring the field equation
(17) to
G ␮ ␯ ⫽8 ␲ G 共 T ␮ ␯ ⫹ ␳ ⌳ g ␮ ␯ 兲 .
⫽dt 2 ⫺K ⫺1 a 共 t 兲 2 关 d ␹ 2 ⫹sinh2 ␹ 共 d ␪ 2 ⫹sin2 ␪ d ␾ 2 兲兴 .
(15)
565
(20)
This is Einstein’s (1917) revision of the field equation of
general relativity, where ␳ ⌳ is proportional to his cosmological constant ⌳; his reason for writing down this equation is discussed in Sec. III.A. In many dark-energy scenarios ␳ ⌳ is a slowly varying function of time and its
stress-energy tensor differs slightly from Eq. (19), so the
observed properties of the vacuum depend on the observer’s velocity.
One sees from Eqs. (14), (18), and (19) that the new
component in the stress-energy tensor resembles an
ideal fluid with negative pressure,
p ⌳ ⫽⫺ ␳ ⌳ .
(21)
This fluid picture is of limited use, but the following
properties are worth noting.10
The stress-energy tensor of an ideal fluid with fourvelocity u ␮ generalizes from Eq. (18) to T ␮ ␯ ⫽( ␳
⫹p)u ␮ u ␯ ⫺pg ␮ ␯ . The equations of fluid dynamics follow from the vanishing of the divergence of T ␮ ␯ . Let us
consider the simple case of locally Minkowskian coordinates, meaning free fall, and a fluid that is close to homogeneous. By the latter we mean the fluid velocity
vជ —the space part of the four-velocity u ␮ —and the density fluctuation ␦␳ from homogeneity may be treated in
linear perturbation theory. Then the equations of energy
and momentum conservation are
␦ ␳˙ ⫹ 共 具 ␳ 典 ⫹ 具 p 典 兲 ⵜ• vជ ⫽0, 共 具 ␳ 典 ⫹ 具 p 典 兲vជ̇ ⫹c s2 ⵜ ␦ ␳ ⫽0,
(22)
c s2 ⫽dp/d ␳
.
(18)
This diagonal form is a consequence of the symmetry;
the diagonal components define the pressure and energy
density. With Eq. (18), the differential equation (17)
yields the expansion-rate equations (11) and (12).
B. The cosmological constant
Special relativity is very successful in laboratory physics. Thus one might guess that any inertial observer
would see the same vacuum. A freely moving inertial
observer represents spacetime in the neighborhood by
locally Minkowskian coordinates, with the metric tensor
␩ ␮ ␯ given in Eq. (14). A Lorentz transformation to an
inertial observer with another velocity does not change
this Minkowski form. The same must be true of the
stress-energy tensor of the vacuum, if all observers see
the same vacuum, so it has to be of the form
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
and the mean density and pressure are
where
具␳典 and 具 p 典 . These combine to
␦ ␳¨ ⫽c s2 ⵜ 2 ␦ ␳ .
(23)
If c s2 is positive this is a wave equation, and c s is the
speed of sound.
The first of Eqs. (22) is the local energy-conservation
law, as in Eq. (9). If p⫽⫺ ␳ , the pdV work cancels the
␳ dV part: the work done to increase the volume cancels
the effect of the increased volume. This has to be so for
a Lorentz-invariant stress-energy tensor, of course,
where all inertial observers see the same vacuum. Another way to see this is to note that the energy flux density in Eqs. (22) is ( 具 ␳ 典 ⫹ 具 p 典 ) vជ . This vanishes when
10
These arguments have been familiar, in some circles, for a
long time, though in our experience, discussed more often in
private than in the literature. Early statements of elements are
in Lemaı̂tre (1934) and McCrea (1951); see Kragh (1999, pp.
397 and 398) for a brief historical account.
566
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
p⫽⫺ ␳ : the streaming velocity loses meaning. When c s2
is negative Eq. (23) shows that the fluid is unstable, in
general. But when p⫽⫺ ␳ the vanishing divergence of
T ␮ ␯ becomes the condition shown in Eq. (22), that ␳
⫽ 具 ␳ 典 ⫹ ␦ ␳ is constant.
There are two measures of gravitational interactions
with a fluid: the passive gravitational mass density determines how the fluid streaming velocity is affected by an
applied gravitational field, and the active gravitational
mass density determines the gravitational field produced
by the fluid. When the fluid velocity is nonrelativistic the
expression for the former in general relativity is ␳ ⫹p, as
one can determine by writing out the covariant divergence of T ␮ ␯ . This vanishes when p⫽⫺ ␳ , consistent
with the loss of meaning of the streaming velocity. The
latter is ␳ ⫹3p, as one can see from Eq. (8). Thus a fluid
with p⫽⫺ ␳ /3, if somehow kept homogeneous and
static, would produce no gravitational field.11 In the
model in Eqs. (19) and (21) the active gravitational mass
density is negative when ␳ ⌳ is positive. When this positive ␳ ⌳ dominates the stress-energy tensor, ä is positive:
the rate of expansion of the universe increases. In the
language of Eq. (20), this cosmic repulsion is a gravitational effect of the negative active gravitational mass
density, not a new force law.
The homogeneous active mass represented by ⌳
changes the equation of relative motion of freely moving
test particles in the nonrelativistic limit to
d 2 rជ
dt 2
⫽gជ ⫹⍀ ⌳0 H 20 rជ ,
11
The negative active gravitational mass density associated with a positive cosmological constant is an early
precursor of the inflation picture of the early universe;
inflation in turn is one precursor of the idea that ⌳ might
generalize into evolving dark energy.
To begin, we review some aspects of causal relations
between events in spacetime. Neglecting space curvature, a light ray moves a proper distance dl⫽a(t)dx
⫽dt in time interval dt, so the integrated coordinate
displacement is
x⫽
(25)
Lest we contribute to a wrong problem for the student we
note that a fluid with p⫽⫺ ␳ /3 held in a container would have
net positive gravitational mass, from the pressure in the container walls required for support against the negative pressure
of the contents. We have finessed the walls by considering a
homogeneous situation. We believe Whittaker (1935) gives the
first derivation of the relativistic active gravitational mass density. Whittaker also presents an example of the general proposition that the active gravitational mass of an isolated stable
object is the integral of the time-time part of the stress-energy
tensor in the locally Minkowskian rest frame. Misner and Putman (1959) give the general demonstration.
12
This assumes that the particles are close enough for application of the ordinary operational definition of proper relative
position. The parameters in the last term follow from Eqs. (8)
and (21).
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
C. Inflation and dark energy
(24)
where gជ is the relative gravitational acceleration produced by the distribution of ordinary matter.12 For an
illustration of the size of the last term consider its effect
on our motion in a nearly circular orbit around the center of the Milky Way galaxy. The Solar System is moving
at speed v c ⫽220 km s⫺1 at radius r⫽8 kpc. The ratio of
the acceleration g ⌳ produced by ⌳ to the total gravitational acceleration g⫽ v 2c /r is
g ⌳ /g⫽⍀ ⌳0 H 20 r 2 / v 2c ⬃10⫺5 ,
a small number. Since we are near the edge of the luminous part of our galaxy, a search for the effect of ⌳ on
the internal dynamics of galaxies such as the Milky Way
does not look promising. The precision of celestial dynamics in the Solar System is much greater, but the effect of ⌳ is very much smaller; g ⌳ /g⬃10⫺22 for the orbit
of the Earth.
One can generalize Eq. (19) to a variable ␳ ⌳ , by taking p ⌳ to be negative but different from ⫺ ␳ ⌳ . But if the
dynamics were that of a fluid, with pressure a function of
␳ ⌳ , stability would require c s2 ⫽dp ⌳ /d ␳ ⌳ ⬎0, from Eq.
(23), which seems quite contrived. A viable working
model for a dynamical ␳ ⌳ is the dark energy of a scalar
field with self-interaction potential chosen to make the
variation of the field energy acceptably slow, as discussed next.
冕
dt/a 共 t 兲 .
(26)
If ⍀ ⌳0 ⫽0 this integral converges in the past—we see
distant galaxies that at the time of observation cannot
have seen us since the singular start of expansion at a
⫽0. This ‘‘particle horizon problem’’ is curious: how
could distant galaxies in different directions in the sky
know to look so similar? The inflation idea is that in the
early universe the expansion history approximates that
of de Sitter’s (1917) solution to Einstein’s field equation
for ⌳⬎0 and T ␮ ␯ ⫽0 in Eq. (20). We can choose the
coordinate labels in this de Sitter spacetime so space
curvature vanishes. Then Eqs. (11) and (12) show that
the expansion parameter is
a⬀e H ⌳ t ,
(27)
where H ⌳ is a constant. As one sees by working the
integral in Eq. (26), here everyone can have seen everyone else in the past. The details need not concern us; for
the following discussion two concepts are important.
First, the early universe acts like an approximation to de
Sitter’s solution because it is dominated by a large effective cosmological ‘‘constant,’’ or dark-energy density.
Second, the dark energy is modeled as that of a near
homogeneous field, ⌽.
In this scalar field model, motivated by grand unified
models of very-high-energy particle physics, the action
of the real scalar field ⌽ (in units chosen so that Planck’s
constant ប is unity) is
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
S⫽
冕
d 4 x 冑⫺g
冋
册
1 ␮␯
g ⳵ ␮ ⌽ ⳵ ␯ ⌽⫺V 共 ⌽ 兲 .
2
(28)
The potential-energy density V is a function of the field
⌽, and g is the determinant of the metric tensor. When
the field is spatially homogeneous [in the line element of
Eq. (15)], and space curvature may be neglected, the
field equation is
⌽̈⫹3
dV
ȧ
⌽̇⫹
⫽0.
a
d⌽
(29)
The stress-energy tensor of this homogeneous field is
diagonal (in the rest frame of an observer moving so
that the universe is seen to be isotropic), with time and
space parts along the diagonal
1
␳ ⌽ ⫽ ⌽̇ 2 ⫹V 共 ⌽ 兲 ,
2
1
p ⌽ ⫽ ⌽̇ 2 ⫺V 共 ⌽ 兲 .
2
(30)
If the scalar field varies slowly in time, so that ⌽̇ 2 ⰆV,
the field energy approximates the effect of Einstein’s
cosmological constant, with p ⌽ ⯝⫺ ␳ ⌽ .
The inflation picture assumes that the near exponential expansion of Eq. (27) in the early universe lasts long
enough so that every bit of the present observable universe has seen every other bit, and presumably has discovered how to relax to almost exact homogeneity. The
field ⌽ may then start varying rapidly enough to produce
the entropy of our universe, and the field or the entropy
may produce the baryons, leaving ␳ ⌽ small or zero. But
one can imagine that the late time evolution of ␳ ⌽ is
slow. If it is slower than the evolution in the mass density
in matter, there comes a time when ␳ ⌽ again dominates,
and the universe appears to have a cosmological constant.
A model for this late time evolution assumes a potential of the form
V⫽ ␬ /⌽ ␣ ,
(31)
where the constant ␬ has dimensions of mass raised to
the power ␣ ⫹4. For simplicity let us suppose the universe after inflation, but at high redshift, is dominated by
matter or radiation, with mass density ␳, that drives the
power-law expansion, a⬀t n . Then the power-law solution to the field Eq. (29) with the potential in Eq. (31) is
⌽⬀t 2/(2⫹ ␣ ) ,
(32)
and the ratio of the mass densities in the scalar field and
in matter or radiation is
␳ ⌽ / ␳ ⬀t 4/(2⫹ ␣ ) .
(33)
In the limit at which the parameter ␣ approaches zero,
␳ ⌽ is constant, and this model is equivalent to Einstein’s
⌳.
When ␣ ⬎0 the field ⌽ in this model grows arbitrarily
large at large time, so ␳ ⌽ →0, and the universe approaches the Minkowskian spacetime of special relativity. This is within a simple model, of course. It is easy to
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
567
imagine that in other models ␳ ⌽ approaches a constant
positive value at large time, and spacetime approaches
the de Sitter solution, or ␳ ⌽ passes through zero and
becomes negative, causing spacetime to collapse to a Big
Crunch.
The power-law model with ␣ ⬎0 has two properties
that seem desirable. First, the solution in Eq. (32) is said
to be an attractor (Ratra and Peebles, 1988) or a tracker
(Steinhardt, Wang, and Zlatev, 1999), meaning it is the
asymptotic solution for a broad range of initial conditions at high redshift. That includes relaxation to a near
homogeneous energy distribution even when gravity has
collected the other matter into nonrelativistic clumps.
Second, the energy density in the attractor solution decreases less rapidly than that of matter and radiation.
This allows us to realize the scenario: after inflation but
at high redshift the field energy density ␳ ⌽ is small so it
does not disturb the standard model for the origin of the
light elements, but eventually ␳ ⌽ dominates and the universe acts as if it had a cosmological constant, but one
that varies slowly with position and time. We comment
on details of this model in Sec III.E.
III. HISTORICAL REMARKS
These comments on what people were thinking are
gleaned from the literature and supplemented by private
discussions and our own recollections. More is required
for a complete history of the subject, of course, but we
hope we have captured the main themes and the way in
which these themes have evolved into the present appreciation of the situation.
A. Einstein’s thoughts
Einstein disliked the idea of an island universe in asymptotically flat spacetime, because a particle could
leave the island and move arbitrarily far from all the
other matter in the universe, yet preserve all its inertial
properties, which he considered a violation of Mach’s
idea of the relativity of inertia. Einstein’s (1917) cosmological model accordingly assumes that the universe is
homogeneous and isotropic, on average, thus removing
the possibility of arbitrarily isolated particles. Einstein
had no empirical support for this assumption, yet it
agrees with modern precision tests. There is no agreement as to whether this is more than a lucky guess.
Motivated by the observed low velocities of the then
known stars, Einstein assumed that the large-scale structure of the universe is static. He introduced the cosmological constant to reconcile this picture with his general
relativity theory. In the notation of Eq. (12), one sees
that a positive value of ⍀ ⌳0 can balance the positive
values of ⍀ M0 and ⍀ R0 for consistency with ä⫽0. The
balance is unstable: a small perturbation to the mean
mass density or the mass distribution causes expansion
or contraction of the whole or parts of the universe. One
sees this in Eq. (24): the mass distribution can be chosen
568
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
so the two terms on the right-hand side cancel, but the
balance can be upset by redistributing the mass.13
Einstein did not consider the cosmological constant to
be part of the stress-energy term: his form for the field
equation [in the streamlined notation of Eq. (17)] is
G ␮ ␯ ⫺8 ␲ G ␳ ⌳ g ␮ ␯ ⫽8 ␲ GT ␮ ␯ .
(34)
The left-hand side contains the metric tensor and its derivatives; a new constant of nature, ⌳, appears in the
addition to Einstein’s original field equation. One can
equally place Einstein’s new term on the right-hand side
of the equation, as in Eq. (20), and count ␳ ⌳ g ␮ ␯ as part
of the source term in the stress-energy tensor. The distinction becomes interesting when ␳ ⌳ takes part in the
dynamics, and the field equation is properly written with
␳ ⌳ , or its generalization, as part of the stress-energy tensor. One would then be able to say that the differential
equation of gravity physics has not changed from Einstein’s original form; instead there is a new component
in the content of the universe.
Having assumed that the universe is static, Einstein
did not write down the differential equation for a(t),
and so did not see the instability. Friedmann (1922,
1924) found the evolving homogeneous solution, but
had the misfortune to do so before the astronomy
became suggestive. Slipher’s measurements of the spectra of the spiral nebulae—galaxies of stars—showed
most are shifted toward the red, and Eddington (1924,
pp. 161 and 162) remarked that that might be a manifestation of the second, repulsive term in Eq. (24). Lemaı̂tre (1927) introduced the relation between Slipher’s redshifts and a homogeneous matter-filled expanding
relativistic world model. He may have been influenced
by Hubble’s work, which led to the publication (Hubble,
1929) of the linear redshift-distance relation [Eq. (5)]: as
a graduate student at MIT Lemaı̂tre attended a lecture
by Hubble.
In Lemaı̂tre’s (1927) solution, the expanding universe
traces asymptotically back to Einstein’s static case. Lemaı̂tre then turned to what he called the primeval atom,
which is now termed the Big Bang model. This solution
expands from densities so large that they require some
13
To help motivate the introduction of ⌳, Einstein (1917)
mentioned a modification of Newtonian gravity physics that
could render the theory well defined when the mass distribution is homogeneous. In Einstein’s example, similar to what
was considered by Seeliger and Neumann in the mid-1890s, the
modified field equation for the gravitational potential ␸ is
ⵜ 2 ␸ ⫺␭ ␸ ⫽4 ␲ G ␳ M . This allows the nonsingular homogeneous static solution ␸ ⫽⫺4 ␲ G ␳ M /␭. In this example the potential for an isolated point mass is the Yukawa form, ␸
⬀e ⫺ 冑␭r /r. Trautman (1965) pointed out that this is not the nonrelativistic limit of general relativity with the cosmological
term. Rather, Eq. (24) follows from ⵜ 2 ␸ ⫽4 ␲ G( ␳ M⫺2 ␳ ⌳ ),
where the active gravitational mass density of the ⌳ term is
␳ ⌳ ⫹3p ⌳ ⫽⫺2 ␳ ⌳ . Norton (1999) reviewed the history of ideas
of this Seeliger-Neumann Yukawa-type potential in gravity
physics.
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
sort of quantum treatment, passes through a quasistatic
approximation to Einstein’s solution, and then continues
expanding to de Sitter’s (1917) empty space solution. To
modern tastes, this ‘‘loitering’’ model requires incredibly
special initial conditions, as will be discussed. Lemaı̂tre
liked it because the loitering epoch allows the expansion
time to be acceptably long for Hubble’s (1929) estimate
of H 0 , which is an order-of-magnitude high.
The record shows Einstein never liked the ⌳ term. His
view of how general relativity might fit Mach’s principle
was disturbed by de Sitter’s (1917) solution to Eq. (34)
for empty space (T ␮ ␯ ⫽0) with ⌳⬎0.14,15 Pais (1982, p.
288) pointed out that Einstein, in a letter to Weyl in
1923, commented on the effect of ⌳ in Eq. (24): ‘‘According to De Sitter, two material points that are sufficiently far apart, continue to be accelerated and move
apart. If there is no quasistatic world, then away with the
cosmological term.’’ We do not know whether at this
time Einstein was influenced by Slipher’s redshifts or
Friedmann’s expanding world model.
14
North (1965) reviews the confused early history of ideas on
the possible astronomical significance of de Sitter’s solution for
an empty universe with ⌳⬎0; we add a few comments regarding the physics that contributed to the discovery of the expanding world model. Suppose an observer in de Sitter’s spacetime
holds a string tied to a source of light, so the source stays at
⫺1
fixed physical distance rⰆH ⌳
. The source is much less massive than the observer, the gravitational frequency shift due to
the observer’s mass may be neglected, and the observer is moving freely. Then the observer receives light from the source
shifted to the red by ␦ ␭/␭⫽⫺(H ⌳ r) 2 /2. The observed redshifts of particles moving on geodesics depend on the initial
conditions. Stars in the outskirts of our galaxy are held at fixed
mean distances from Earth by their motions. The mean shifts
of the spectra of light from these stars include this quadratic de
Sitter term as well as the much larger Doppler and ordinary
gravitational shifts. The prescription for initial conditions that
reproduces the linear redshift-distance relation for distant galaxies follows Weyl’s (1923) principle: the world particle geodesics trace back to a near common position in the remote past,
in the limiting case of the Friedmann-Lemaı̂tre model at ⍀ M0
→0. This spatially homogeneous coordinate labeling of de Sitter’s spacetime, with space sections with negative curvature,
already appears in de Sitter [1917, Eq. (15)], and is repeated in
Lanczos (1922). This line element is the second expression in
our Eq. (15) with a⬀cosh H⌳t. Lemaı̂tre (1925) and Robertson
(1928) present the coordinate labeling for the spatially flat
case, where the line element is ds 2 ⫽dt 2 ⫺e 2H ⌳ t (dx 2 ⫹dy 2
⫹dz 2 ) [in the choice of symbols and signature in Eqs. (15) and
(27)]. Lemaı̂tre (1925) and Robertson (1928) note that particles at rest in this coordinate system present a linear redshiftdistance relation, v ⫽H ⌳ r, at small v . Robertson (1928) estimated H ⌳ , and Lemaı̂tre (1927) its analog for the FriedmannLemaı̂tre model, from published redshifts and Hubble’s galaxy
distances. Their estimates are not far off Hubble’s (1929) published value.
15
To the present way of thinking the lengthy debate about the
singularity in de Sitter’s static solution, chronicled by North
(1965), seems surprising, because de Sitter (1917) and Klein
(1918) had presented de Sitter’s solution as a sphere embedded
in 4-plus-1-dimensional flat space, with no physical singularity.
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
The earliest published comments we have found on
Einstein’s opinion of ⌳ within the evolving world model
(Einstein, 1931; Einstein and de Sitter, 1932) make the
point that, since not all the terms in the expansion-rate
equation (11) are logically required, and the matter term
surely is present and likely dominates over radiation at
low redshift, a reasonable working model drops ⍀ K0 and
⍀ ⌳0 and ignores ⍀ R0 . This simplifies the expansion-rate
equation to what has come to be called the Einstein–de
Sitter model,
ȧ 2 8
⫽ ␲G␳M ,
a2 3
(35)
where ␳ M is the mass density in nonrelativistic matter;
here ⍀ M⫽8 ␲ G ␳ M /(3H 2 ) is unity. The left side is a
measure of the kinetic energy of expansion per unit
mass, and the right-hand side a measure of the negative
of the gravitational potential energy. In effect, this
model universe expands with escape velocity.
Einstein and de Sitter point out that Hubble’s estimate of H 0 and de Sitter’s estimate of the mean mass
density in galaxies are not inconsistent with Eq. (35)
(and since both quantities scale with distance in the
same way, this result is not affected by the error in the
distance scale that affected Hubble’s initial measurement of H 0 ). But the evidence shows now that the mass
density is about one-quarter of what is predicted by this
equation, as we will discuss.
Einstein and de Sitter (1932) remarked that the curvature term in Eq. (11) is ‘‘essentially determinable, and
an increase in the precision of the data derived from
observations will enable us in the future to fix its sign
and determine its value.’’ This is happening, 70 years
later. The cosmological constant term is measurable, in
principle, too, and may now have been detected. But
Einstein and de Sitter said only that the theory of an
expanding universe with finite mean mass density ‘‘can
be reached without the introduction of’’ ⌳.
Further to this point, in the appendix of the second
edition of his book, The Meaning of Relativity, Einstein
(1945, p. 127) states that the ‘‘introduction of the ‘cosmologic
member’ ’’—Einstein’s
terminology
for
⌳—‘‘into the equations of gravity, though possible from
the point of view of relativity, is to be rejected from the
point of view of logical economy,’’ and that if ‘‘Hubble’s
expansion had been discovered at the time of the creation of the general theory of relativity, the cosmologic
member would never have been introduced. It seems
now so much less justified to introduce such a member
into the field equations, since its introduction loses its
sole original justification,—that of leading to a natural
solution of the cosmologic problem.’’ Einstein knew that
without the cosmological constant the expansion time
derived from Hubble’s estimate of H 0 is uncomfortably
short compared to estimates of the ages of the stars, and
opined that that might be a problem with the star ages.
The big error, the value of H 0 , was corrected by 1960
(Sandage, 1958, 1962).
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
569
Gamow (1970, p. 44) recalls that ‘‘when I was discussing cosmological problems with Einstein, he remarked
that the introduction of the cosmological term was the
biggest blunder he ever made in his life.’’ This certainly
is consistent with all of Einstein’s written comments that
we have seen on the cosmological constant per se; we do
not know whether Einstein was also referring to the
missed chance to predict the evolution of the universe.
B. The development of ideas
1. Early indications of ⌳
In the classic book, The Classical Theory of Fields,
Landau and Lifshitz (1951, p. 338) second Einstein’s
opinion of the cosmological constant ⌳, stating there is
‘‘no basis whatsoever’’ for adjustment of the theory to
include this term. The empirical side of cosmology is not
much mentioned in this book, however (though there is
a perceptive comment on the limited empirical support
for the homogeneity assumption; p. 332). In the Supplementary Notes to the English translation of his book,
Theory of Relativity, Pauli (1958, p. 220) also endorses
Einstein’s position.
Discussions elsewhere in the literature on how one
might find empirical constraints on the values of the cosmological parameters usually take account of ⌳. The
continued interest was at least in part driven by indications that ⌳ might be needed to reconcile theory and
observations. Here are three examples.
First, the expansion time is uncomfortably short if ⌳
⫽0. Sandage’s recalibration of the distance scale in the
1960s indicates H 0 ⯝75 km s⫺1 Mpc⫺1 . If ⌳⫽0 this
shows that the time of expansion from densities too high
for stars to have existed is ⬍H ⫺1
0 ⯝13 Gyr, maybe less
than the ages of the oldest stars, then estimated to be
greater than about 15 Gyr. Sandage (1961a) points out
that the problem is removed by adding a positive ⌳. The
present estimates reviewed below (Sec. IV.B.3) are not
far from these numbers, but still too uncertain for a significant case for ⌳.
Second, counts of quasars as a function of redshift
show a peak at z⬃2, as would be produced by the loitering epoch in Lemaı̂tre’s ⌳ model (Petrosian, Salpeter,
and Szekeres, 1967; Shklovsky, 1967; Kardashev, 1967).
The peak is now well established, centered at z⬃2.5
(Croom et al., 2001; Fan et al., 2001). It is usually interpreted as the evolution in the rate of violent activity in
the nuclei of galaxies, though in the absence of a loitering epoch the indicated sharp variation in quasar activity
with time is curious (but certainly could be a consequence of astrophysics that is not well understood).
The third example is the redshift-magnitude relation.
Sandage’s (1961a) analysis indicates this is a promising
method of distinguishing world models. The Gunn and
Oke (1975) measurement of this relation for giant elliptical galaxies, with Tinsley’s (1972) correction for evolution of the star population from assumed formation at
high redshift, indicates curvature away from the linear
relation in the direction that, as Gunn and Tinsley
570
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
(1975) discuss, could only be produced by ⌳ (within general relativity theory). The new application of the
redshift-magnitude test, to type-Ia supernovae (Sec.
IV.B.4), is not inconsistent with the Gunn-Oke measurement; we do not know whether this agreement of the
measurements is significant, because Gunn and Oke
were worried about galaxy evolution.16
2. The coincidences argument against ⌳
An argument against an observationally interesting
value of ⌳, from our distrust of accidental coincidences,
has been in the air for decades, and became very influential in the early 1980s with the introduction of the
inflation scenario for the very early universe.
If the Einstein–de Sitter model in Eq. (35) were a
good approximation at the present epoch, an observer
measuring the mean mass density and Hubble’s constant
when the age of the universe was one-tenth the present
value, or ten times the present age, would reach the
same conclusion, that the Einstein–de Sitter model is a
good approximation. That is, we would flourish at a time
that is not special in the course of evolution of the universe. If, on the other hand, two or more of the terms in
the expansion-rate equation (11) made substantial contributions to the present value of the expansion rate, it
would mean that we are present at a special epoch, because each term in Eq. (11) varies with the expansion
factor in a different way. To put this in more detail, we
imagine that the physics of the very early universe, when
the relativistic cosmological model became a good approximation, set the values of the cosmological parameters. The initial values of the contributions to the
expansion-rate equation had to have been very different
from each other, and exceedingly specially fixed, to yield
two ⍀ i0 ’s with comparable values. This would be a most
remarkable and unlikely coincidence. The multiple coincidences required for the near vanishing of ȧ and ä at a
redshift not much larger than unity makes an even stronger case against Lemaı̂tre’s loitering model, with this line
of argument.
The earliest published comment we have found on
this point is by Bondi (1960, p. 166), in the second edition of his book Cosmology. Bondi notes the ‘‘remarkable property’’ of the Einstein–de Sitter model: the dimensionless parameter we now call ⍀ M is independent
of the time at which it is computed (since it is unity).
The coincidences argument follows and extends Bondi’s
comment. It is presented in McCrea (1971, p. 151).
When Peebles was a postdoctoral research associate, in
16
Early measurements of the redshift-magnitude relation
were meant in part to test the Steady State cosmology of
Bondi and Gold (1948) and Hoyle (1948). Since Steady State
cosmology assumes spacetime is independent of time its line
element has to have the form of the de Sitter solution with
⍀ K0 ⫽0 and the expansion parameter in Eq. (27). The measured curvature of the redshift-magnitude relation is in the
direction predicted by Steady State cosmology. But this cosmology fails other tests discussed in Sec. IV.B.
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
the early 1960s, in R. H. Dicke’s gravity research group,
the coincidences argument was discussed, but published
much later (Dicke, 1970, p. 62; Dicke and Peebles,
1979). We do not know its provenance in Dicke’s group,
whether from Bondi, McCrea, Dicke, or someone else.
We would not be surprised to learn others had similar
thoughts.
The coincidences argument is sensible but not a proof,
of course. The discovery of the 3-K thermal cosmic microwave background radiation gave us a term in the
expansion-rate equation that is down from the dominant
one by four orders of magnitude, not such a large factor
by astronomical standards. This might be counted as a
first step away from the argument. From the dynamics of
galaxies the evidence that ⍀ M0 is less than unity is another step (Peebles, 1984, p. 442; 1986). And yet another
is the development of the evidence that the ⌳ and darkmatter terms differ by only a factor of 3 [Eq. (2)]. This
last piece is the most curious, but the community has
come to accept it, for the most part. The precedent
makes Lemaı̂tre’s loitering model more socially acceptable.
A socially acceptable value of ⌳ cannot be such as to
make life impossible, of course.17 But perhaps the most
productive interpretation of the coincidences argument
is that it demands a search for a more fundamental underlying model. This is discussed further in Sec. III.E
and the Appendix.
3. Vacuum energy and ⌳
Another tradition to consider is the relation between
⌳ and the vacuum or dark-energy density. In one approach to the motivation for the Einstein field equation,
taken by McVittie (1956) and others, ⌳ appears as a
constant of integration (of the expression for local conservation of energy and momentum). McVittie (1956, p.
35) emphasizes that, as a constant of integration, ⌳ ‘‘cannot be assigned any particular value on a priori
grounds.’’ Interesting variants of this line of thought are
still under discussion (Weinberg, 1989; Unruh, 1989; and
references therein).
The notion of ⌳ as a constant of integration may be
related to the issue of the zero point of energy. In laboratory physics one measures and computes energy differences. But the net energy matters for gravity physics,
and one can imagine that ⌳ represents the difference
between the true energy density and the sum at which
one arrives by laboratory physics. Eddington (1939) and
Lemaı̂tre (1934, 1949) make this point.
If ⌳ were negative and the magnitude too large there would
not be enough time for the emergence of life such as ours. If ⌳
were positive and too large the universe would expand too
rapidly to allow galaxy formation. Our existence, which requires something resembling the Milky Way galaxy to contain
and recycle heavy elements, thus provides an upper bound on
the value of ⌳. Such anthropic considerations are discussed by
Weinberg (1987, 2001) and Vilenkin (2001), and references
therein.
17
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
Bronstein18 (1933) carries the idea further, allowing
for transfer of energy between ordinary matter and that
represented by ⌳. In our notation, Bronstein expresses
this picture by generalizing Eq. (9) to
ȧ
␳˙ ⌳ ⫽⫺ ␳˙ ⫺3 共 ␳ ⫹p 兲 ,
a
(36)
where ␳ and p are the energy density and pressure of
ordinary matter and radiation. Bronstein goes on to propose a violation of local energy conservation, a thought
that no longer seems interesting. North (1965, p. 81)
finds Eddington’s (1939) interpretation of the zero point
of energy also somewhat hard to defend. But for our
purpose the important point is that the idea of ⌳ as a
form of energy has been present, in at least some circles,
for many years.
The zero-point energy of fields contributes to the
dark-energy density. To make physical sense the sum
over the zero-point mode energies must be cut off at a
short distance or a high frequency up to which the
model under consideration is valid. The integral of the
zero-point energy (k/2) of normal modes (of wave number k) of a massless real bosonic scalar field (⌽), up to
the wave-number cutoff k c , gives the vacuum energy
density the quantum-mechanical expectation value19
18
Kragh (1996, p. 36) describes Bronstein’s motivation and
history. We discuss this model in more detail in Sec. III.E, and
comment on why decay of dark energy into ordinary matter or
radiation would be hard to reconcile with the thermal spectrum of the 3-K cosmic microwave background radiation. Decay into the dark sector may be interesting.
19
Equation (37), which usually figures in discussions of the
vacuum energy puzzle, gives a helpful indication of the situation: the zero-point energy of each mode is real and the sum is
large. The physics is seriously incomplete, however. The elimination of spatial momenta with magnitudes k⬎k c only makes
sense if there is a preferred reference frame in which k c is
defined. Magueijo and Smolin (2002) mention a related issue:
In which reference frame is the Planck momentum of a virtual
particle at the threshold for new phenomena? In both cases
one may implicitly choose the rest frame for the large-scale
distribution of matter and radiation. It seems strange to think
that the microphysics is concerned about large-scale structure,
but perhaps this happens in a sea of interacting fields. The
cutoff in Eq. (37) might be applied at a fixed comoving wave
number k c ⬀a(t) ⫺1 , or at a fixed physical value of k c . The first
prescription can be described by an action written as a sum of
2
2
terms ⌽̇ kជ /2⫹k 2 ⌽ kជ / 关 2a(t) 2 兴 for the allowed modes. The zeropoint energy of each mode scales with the expansion of the
universe as a(t) ⫺1 , and the sum over modes scales as ␳ ⌽
⬀a(t) ⫺4 , consistent with k c ⬀a(t) ⫺1 . In the limit of exact spatial homogeneity, an equivalent approach uses the spatial average of the standard expression for the field stress-energy tensor. Indeed, DeWitt (1975) and Akhmedov (2002) show that
the vacuum expectation value of the stress-energy tensor, expressed as an integral cutoff at k⫽k c , and computed in the
preferred coordinate frame, is diagonal with a space part p ⌽
⫽ ␳ ⌽ /3, for the massless field we are considering. That is, in
this prescription the vacuum zero-point energy acts like a homogeneous sea of radiation. This defines a preferred frame of
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
␳ ⌽⫽
冕
kc
0
k 4c
4 ␲ k 2 dk k
.
⫽
共 2 ␲ 兲 3 2 16␲ 2
571
(37)
Nernst (1916) seems to have been the first to write
down this equation, in connection with the idea that
the zero-point energy of the electromagnetic field
fills the vacuum, as a light aether, that could have physically significant properties.20 This was before Heisenberg and Schrödinger: Nernst’s hypothesis is that each
degree of freedom, to which classical statistical mechanics assigns energy kT/2, has Nullpunktsenergie h ␯ /2.
This would mean that the ground-state energy of a onedimensional harmonic oscillator is h ␯ , twice the correct
value. Nernst’s expression for the energy density in the
electromagnetic field thus differs from Eq. (37) by a factor of 2 (after taking account of the two polarizations),
which is wonderfully close. For a numerical example,
Nernst noted that if the cutoff frequency were ␯
⫽1020 Hz, or ⬃0.4 MeV, the energy density of the
Lichtäther (light aether) would be 1023 erg cm⫺3 , or
about 100 g cm⫺3 .
By the end of the 1920s Nernst’s hypothesis was replaced with the demonstration that in quantum mechanics the zero-point energy of the vacuum is as real as any
other. W. Pauli, in unpublished work in the 1920s,21 repeated Nernst’s calculation, with the correct factor of 2,
taking k c to correspond to the classical electron radius.
Pauli knew the value of ␳ ⌳ was quite unacceptable: the
radius of the static Einstein universe with this value of
␳ ⌳ ‘‘would not even reach to the moon’’ (Rugh and
Zinkernagel, 2002, p. 5).22 The modern version of this
‘‘physicists’ cosmological constant problem’’ is even
motion, where the stress-energy tensor is diagonal, which is
not unexpected because we need a preferred frame to define
k c . It is unacceptable as a model for the properties of dark
energy, of course. For example, if the dark-energy density were
normalized to the value now under discussion, and varied as
␳ ⌳ ⬀a(t) ⫺4 , it would quite mess up the standard model for the
origin of the light elements. We get a more acceptable model
for the behavior of ␳ ⌳ from the second prescription, with the
cutoff at a fixed physical momentum. If we also want to satisfy
local energy conservation we must take the pressure to be
p ⌽ ⫽⫺ ␳ ⌽ . This does not contradict the derivation of p ⌽ in the
first prescription, because the second situation cannot be described by an action: the pressure must be stipulated, not derived. What is worse is that the known fields at laboratory
momenta certainly do not allow this stipulation; they are well
described by analogs of the action in the first prescription. This
quite unsatisfactory situation illustrates how far we are from a
theory of vacuum energy.
20
A helpful discussion of Nernst’s ideas on cosmology is that
of Kragh (1996, pp. 151–157).
21
This is discussed by Enz and Thellung (1960); Enz (1974);
Rugh and Zinkernagel (2002, pp. 4 and 5), and Straumann
(2002).
22
In an unpublished letter in 1930, G. Gamow considered the
gravitational consequences of the Dirac sea (Dolgov, 1989, p.
230). We thank A. Dolgov for helpful correspondence on this
point.
572
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
more acute, because a natural value for k c is thought to
be much larger than what Nernst or Pauli used.23
While there was occasional discussion of this issue in
the middle of the 20th century (as in the quote from N.
Bohr in Rugh and Zinkernagel, 2002, p. 5), the modern
era begins with the paper by Zel’dovich (1967) that convinced the community to consider the possible connection between the vacuum energy density of quantum
physics and Einstein’s cosmological constant.24
If the physics of the vacuum looks the same to any
inertial observer its contribution to the stress-energy
tensor is the same as Einstein’s cosmological constant
[Eq. (19)]. Lemaı̂tre (1934) notes this: ‘‘in order that absolute motion, i.e., motion relative to the vacuum, may
not be detected, we must associate a pressure p⫽⫺ ␳ c 2
to the energy density ␳ c 2 of vacuum.’’ Gliner (1965)
goes further, presenting the relation between the metric
tensor and the stress-energy tensor of a vacuum that is
the same to any inertial observer. But it was Zel’dovich
(1968) who presented the argument clearly enough and
at the right time to catch the attention of the community.
With the development of the concept of broken symmetry in the now standard model for particle physics
came the idea that the expansion and cooling of the universe is accompanied by a sequence of first-order phase
4
In terms of an energy scale ⑀ ⌳ defined by ␳ ⌳ ⫽ ⑀ ⌳
, the
⫺1/2
Planck energy G
is about 30 orders of magnitude larger
than the ‘‘observed’’ value of ⑀ ⌳ . This is, of course, an extreme
case, since many of the theories of interest break down well
below the Planck scale. Furthermore, in addition to other contributions, one may add a counterterm to Eq. (37) to predict
any value of ␳ ⌳ . With reference to this point, it is interesting
to note that while Pauli did not publish his computation of ␳ ⌳ ,
he remarks in his famous 1933 Handbuch der Physik review on
quantum mechanics that it is more consistent to ‘‘exclude a
zero-point energy for each degree of freedom as this energy,
evidently from experience, does not interact with the gravitational field’’ (Rugh and Zinkernagel, 2002, p. 5). Pauli was fully
aware that one must take account of zero-point energies in the
binding energies of molecular structure, for example (and we
expect he was aware that what contributes to the energy contributes to the gravitational mass). He chose to drop the section with the above comment from the second (1958) edition
of the review (Pauli, 1980, pp. iv and v). In a globally supersymmetric field theory there are equal numbers of bosonic and
fermionic degrees of freedom, and the net zero-point vacuum
energy density ␳ ⌳ vanishes (Iliopoulos and Zumino, 1974;
Zumino, 1975). However, supersymmetry is not a symmetry of
low-energy physics, or even at the electroweak unification
scale. It must be broken at low energies, and the proper setting
for a discussion of the zero-point ␳ ⌳ in this case is locally supersymmetric supergravity. Weinberg (1989, p. 6) notes ‘‘it is
very hard to see how any property of supergravity or superstring theory could make the effective cosmological constant
sufficiently small.’’ Witten (2001) and Ellwanger (2002) review
more recent developments on this issue in the superstring/M
theory/branes scenario.
24
For subsequent more detailed discussions of this issue, see
Zel’dovich (1981), Weinberg (1989), Carroll, Press, and Turner
(1992), Sahni and Starobinsky (2000), Carroll (2001), and
Rugh and Zinkernagel (2000).
23
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
transitions accompanying the symmetry breaking. Each
first-order transition has a latent heat that appears as a
contribution to an effective time-dependent ⌳(t) or
dark-energy density.25 The decrease in value of the darkenergy density at each phase transition is much larger
than the acceptable present value (within relativistic cosmology); the natural presumption is that the dark energy
is negligible now. This final condition seems bizarre, but
the picture led to the very influential concept of inflation. We discussed the basic elements in connection with
Eq. (27); we now turn to some implications.
C. Inflation
1. The scenario
The deep issue addressed by inflation is the origin of
the large-scale homogeneity of the observable universe.
In a relativistic model with positive pressure we can see
distant galaxies that have not been in causal contact with
each other since the singular start of expansion [Sec.
II.C, Eq. (26)]; they are said to be outside each other’s
particle horizon. Why do apparently causally unconnected parts of space look so similar?26 Kazanas (1980),
Guth (1981), and Sato (1981a, 1981b) make the key
point: if the early universe were dominated by the energy density of a relatively flat real scalar field (inflaton)
potential V(⌽) that acts like ⌳, the particle horizon
could spread beyond the universe we can see. This
would allow for the possibility that microphysics during
inflation could smooth inhomogeneities sufficiently to
provide an explanation of the observed large-scale homogeneity. (We are unaware of a definitive demonstration of this idea, however.)
In the inflation scenario the field ⌽ rolls down its potential until eventually V(⌽) steepens enough to terminate inflation. Energy in the scalar field is supposed to
decay to matter and radiation, heralding the usual Big
Bang expansion of the universe. With the modifications
of Guth’s (1981) scenario by Linde (1982) and Albrecht
and Steinhardt (1982), the community quickly accepted
this promising and elegant way to understand the origin
of our homogeneous expanding universe.27
In Guth’s (1981) picture the inflation kinetic-energy
density is subdominant during inflaton, ⌽̇ 2 ⰆV(⌽), so
from Eqs. (30) the pressure p ⌽ is very close to the negative of the mass density ␳ ⌽ , and the expansion of the
universe approximates the de Sitter solution, a
⬀exp(H⌳t) [Eq. (27)].
25
Early references to this point were made by Linde (1974),
Dreitlein (1974), Kirzhnitz and Linde (1974), Veltman (1975),
Bludman and Ruderman (1977), Canuto and Lee (1977), and
Kolb and Wolfram (1980).
26
Early discussions of this question are reviewed by Rindler
(1956), more recent examples are Misner (1969); Dicke and
Peebles (1979); and Zee (1980).
27
Aspects of the present state of the subject are reviewed by
Guth (1997); Brandenberger (2001); and Lazarides (2002).
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
For our comments on the spectrum of mass density
fluctuations produced by inflation and the properties of
solutions of the dark-energy models in Sec. III.E we
shall find it useful to have another scalar field model.
Lucchin and Matarrese (1985a, 1985b) considered the
potential
V共 ⌽ 兲⫽
A
exp关 ⫺⌽ 冑8 ␲ qG 兴 ,
G2
(38)
where q and A are parameters.28 They showed that the
scale factor and the homogeneous part of the scalar field
evolve in time as
a 共 t 兲 ⫽a 0 关 1⫹Nt 兴 2/q ,
⌽共 t 兲⫽
1
冑2 ␲ qG
ln关 1⫹Nt 兴 ,
P 共 k 兲 ⫽ 具 兩 ␦ 共 k,t 兲 兩 2 典 ⫽AkT 2 共 k 兲 .
28
(40)
Similar exponential potentials appear in some higherdimensional Kaluza-Klein models. For an early discussion see
Shafi and Wetterich (1985).
29
Ratra (1989, 1992a) shows that spatial inhomogeneities do
not destroy this property, that is, for q⬍2 the spatially inhomogeneous scalar field perturbation has no growing mode.
30
The strong curvature of spacetime during inflation makes
the vacuum state quite different from that of Minkowski
spacetime (Ratra, 1985). This is somewhat analogous to how
the Casimir metal plates modify the usual Minkowski spacetime vacuum state.
31
For the development of these ideas see Hawking (1982);
Starobinsky (1982); Guth and Pi (1982); Bardeen, Steinhardt,
and Turner (1983); and Fischler, Ratra, and Susskind (1985).
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
Here ␦ (k,t) is the Fourier transform at wave number k
of the mass density contrast ␦ (xជ ,t)⫽ ␳ (xជ ,t)/ 具 ␳ (t) 典 ⫺1,
where ␳ is the mass density and 具␳典 the mean value. After inflation, but at very large redshifts, the spectrum in
this model is P(k)⬀k on all interesting length scales.
This means the curvature fluctuations produced by the
mass fluctuations diverge only as log k. The form P(k)
⬀k thus need not be cut off anywhere near observationally interesting lengths, and in this sense it is scale
invariant.32 The transfer function T(k) accounts for the
effects of radiation pressure and the dynamics of nonrelativistic matter on the evolution of ␦ (k,t), computed in
linear perturbation theory, at redshifts zⱗ104 . The constant A is determined by details of the chosen inflation
model we need not describe.
The exponential potential model in Eq. (38) produces
the power spectrum33
(39)
where N⫽2q 冑␲ A/ 冑G(6⫺q). If q⬍2 this model inflates. Halliwell (1987) and Ratra and Peebles (1988)
showed that the solution (39) of the homogeneous equation of motion has the attractor property29 mentioned in
connection with Eq. (31). This exponential potential is
of historical interest: it provided the first clear illustration of an attractor solution. We will return to this point
in Sec. III.E.
A signal achievement of inflation is that it offers a
theory for the origin of the departures from homogeneity. Inflation tremendously stretches length scales, so
that cosmologically significant lengths now correspond
to extremely short lengths during inflation. On these tiny
length scales quantum mechanics governs: the wavelengths of zero-point field fluctuations generated during
inflation are stretched by the inflationary expansion,30
and these fluctuations are converted to classical density
fluctuations in the late time universe.31
The power spectrum of the fluctuations depends on
the model for inflation. If the expansion rate during inflation is close to exponential [Eq. (27)], the zero-point
fluctuations are frozen into primeval mass density fluctuations with the power spectrum
573
P 共 k 兲 ⫽Ak n T 2 共 k 兲 ,
n⫽ 共 2⫺3q 兲 / 共 2⫺q 兲 .
(41)
When n⫽1(q⫽0) the power spectrum is said to be
tilted. This offers a parameter n to be adjusted to fit the
observations of the large-scale structure, though as we
will discuss, the simple scale-invariant case n⫽1 is close
to the best fit to the observations.
The mass fluctuations in these inflation models are
said to be adiabatic, because they are what results from
adiabatically compression or decompression of parts of
an exactly homogeneous universe. This means the initial
conditions for the mass distribution are described by one
function of position, ␦ (xជ ,t). This function is a realization of a spatially stationary random Gaussian process,
because it is frozen out of almost free quantum field
fluctuations. Thus the single function of position is statistically prescribed by its power spectrum, as in Eqs.
(40) and (41). More complicated models for inflation
produce density fluctuations that are not Gaussian, or do
not have simple power-law spectra, or have parts that
break adiabaticity, such as gravitational waves (Rubakov, Sazhin, and Veryaskin, 1982), magnetic fields
(Turner and Widrow, 1988; Ratra, 1992b), or new hypothetical fields. All these extra features may be invoked
to fit the observations, if needed. It may be significant
that none seem to be needed to fit the main cosmological constraints we have now.
2. Inflation in a low-density universe
We do need an adjustment from the simplest case—an
Einstein–de Sitter cosmology—to account for the measurements of the mean mass density. In the two models
that lead to Eqs. (40) and (41) the enormous expansion
factor during inflation suppresses the curvature of space
sections, making ⍀ K0 negligibly small. If ⌳⫽0, this fits
the Einstein–de Sitter model [Eq. (35)], which in the
32
The virtues of a spectrum that is scale invariant in this sense
were noted before inflation by Harrison (1970), Peebles and
Yu (1970), and Zel’dovich (1972).
33
This is discussed by Abbott and Wise (1984), Lucchin and
Matarrese (1985a, 1985b), and Ratra (1989, 1992a).
574
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
absence of data clearly is the elegant choice. But the
high mass density in this model was already seriously
challenged by the data available in 1983, on the low
streaming flow of the nearby galaxies toward the nearest
known large mass concentration, in the Virgo cluster of
galaxies, and the small relative velocities of galaxies outside the rich clusters of galaxies.34 A striking and long
familiar example of the latter is that the galaxies immediately outside the Local Group of galaxies, at distances
of a few megaparsecs, are moving away from us in a
good approximation to Hubble’s homogeneous flow, despite the very clumpy distribution of galaxies on this
scale.35 The options (within general relativity) are that
the mass density is low, so its clumpy distribution has
little gravitational effect, or the mass density is high and
the mass is more smoothly distributed than the galaxies.
We comment on the first option here, and the second in
connection with the cold-dark-matter model for structure formation in Sec. III.D.
Under the first option we have these choices: introduce a cosmological constant, or space curvature, or perhaps even both. In the conventional inflation picture
space curvature is unacceptable, but there is another line
of thought that leads to a universe with open space sections. Gott’s (1982) scenario commences with a large energy density in an inflaton at the top of its potential. This
behaves like Einstein’s cosmological constant and produces a near de Sitter universe expanding as a
⬀exp(H⌳t), with sufficient inflation to allow for a microphysical explanation of the large-scale homogeneity of
the observed universe. As the inflaton gradually rolls
down the potential it reaches a point where there is a
small bump in the potential. The inflaton tunnels
through this bump by nucleating a bubble. Symmetry
forces the interior of the bubble to have open spatial
sections (Coleman and De Luccia, 1980), and the con-
34
This is discussed by Davis and Peebles (1983a, 1983b) and
Peebles (1986). Relative velocities of galaxies in rich clusters
are large, but the masses in clusters are known to add up to a
modest mean mass density. Thus most of the Einstein–de Sitter mass would have to be outside the dense parts of the clusters, where the relative velocities are small.
35
The situation a half century ago is illustrated by the compilation of galaxy redshifts by Humason, Mayall, and Sandage
(1956). In this sample of 806 galaxies, 14 have negative redshifts (after correction for the rotation of the Milky Way galaxy and for the motion of the Milky Way toward the other
large galaxy in the Local Group, the Andromeda Nebula), indicating motion toward us. Nine are members of the Local
Group, at distances ⱗ1 Mpc. Four are in the direction of the
Virgo cluster, at redshift ⬃1200 km s⫺1 and distance
⬃20 Mpc. Subsequent measurements indicate two of these
four really have negative redshifts, and plausibly are members
of the Virgo cluster on the tail of the distribution of peculiar
velocities of the cluster members. (Astronomers use the term
peculiar velocity to denote the deviation from the uniform
Hubble expansion velocity.) The last of the 14, NGC 3077, is in
the M 81 group of galaxies at 3-Mpc distance. It is now known
to have a small positive redshift.
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
tinuing presence of a nonzero V(⌽) inside the bubble
acts like ⌳, resulting in an open inflating universe. The
potential is supposed to steepen, bringing the second
limited epoch of inflation to an end before space curvature has been completely redshifted away. The region
inside the open bubble at the end of inflation is a
radiation-dominated Friedmann-Lemaı̂tre open model,
with 0⬍⍀ K0 ⬍1 [Eq. (16)]. This can fit the dynamical
evidence for low ⍀ M0 with ⌳⫽0.36
The decision on which scenario, spatially flat or open,
is elegant, if either, depends ultimately on which Nature
has chosen, if either.37 But it is natural to make judgments in advance of the evidence. Since the early 1980s
there have been occasional explorations of the open
case, but the community generally has favored the flat
case, ⍀ K0 ⫽0, without or, more recently, with a cosmological constant, and indeed the evidence now shows
that space sections are close to flat. The earlier preference for the Einstein–de Sitter case with ⍀ K0 ⫽0 and
⍀ ⌳0 ⫽0 led to considerable interest in the picture of biased galaxy formation in the cold-dark-matter model, as
we now describe.
D. The cold-dark-matter model
Some of the present cosmological tests have been understood since the 1930s; others are based on new ideas
about structure formation. A decade ago a half dozen
models for structure formation were under discussion,38
now the known viable models have been winnowed to
36
Gott’s scenario is resurrected by Ratra and Peebles (1994,
1995). See Bucher and Turok (1995), Yamamoto, Sasaki, and
Tanaka (1995), and Gott (1997) for further discussions of this
model. In this case spatial curvature provides a second cosmologically relevant length scale (in addition to that set by the
Hubble radius H ⫺1 ), so there is no natural preference for a
power-law power spectrum (Ratra, 1994; Ratra and Peebles,
1995).
37
At present, high-energy physics considerations do not provide a compelling specific inflation model, but there are strong
indications that inflation occurs in a broad range of models, so
it might not be unreasonable to think that future advances in
high-energy physics could give us a compelling and observationally successful model of inflation that will determine
whether the scenario is flat or open.
38
A scorecard is given in Peebles and Silk (1990). Structureformation models that assume all matter is baryonic, and those
that augment baryons with hot dark matter such as low-mass
neutrinos, were already seriously challenged a decade ago. Vittorio and Silk (1985) showed that the Uson and Wilkinson
(1984) bound on the small-scale anisotropy of the 3-K cosmic
microwave background temperature rules out a baryondominated universe with adiabatic initial conditions. This is
because the dissipation of the baryon density fluctuations by
radiation drag as the primeval plasma combines to neutral hydrogen (at redshift z⬃1000) unacceptably suppresses structure
formation on the scale of galaxies. Cold dark matter avoids
this problem by eliminating radiation drag. This is one of the
reasons attention turned to the hypothetical nonbaryonic cold
dark matter. There has not been a thorough search for more
baroque initial conditions that might save the baryonic darkmatter model, however.
P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy
one class: cold dark matter (CDM) and variants. We
comment on the present state of tests of the CDM
model in Sec. IV.A.2, and in connection with the cosmological tests in Sec. IV.B.
The CDM model assumes that the mass of the universe now is dominated by dark matter that is nonbaryonic and acts like a gas of massive, weakly interacting
particles with negligibly small primeval velocity dispersion. Structure is supposed to have formed as a result of
the gravitational growth of primeval departures from
homogeneity that are adiabatic, scale invariant, and
Gaussian. The early discussions also assume an
Einstein–de Sitter universe. These features all are naturally implemented in simple models for inflation, and the
CDM model may have been inspired in part by the developing ideas of inflation. But the motivation for writing down this model was to find a simple way to show
that the observed present-day mass fluctuations can
agree with growing evidence that the anisotropy of the
3-K thermal cosmic microwave background radiation is
very small (Peebles, 1982). The first steps toward turning
this picture into a model for structure formation were
taken by Blumenthal et al. (1984).
In the decade commencing about 1985 the standard
cosmology for many active in research in this subject
was the Einstein–de Sitter model, and for good reason:
it eliminates the coincidences problem, it avoids the curiosity of nonzero dark energy, and it fits the condition
from conventional inflation that space sections have zero
curvature. But unease about the astronomical problems
with the high-mass density of the Einstein–de Sitter
model led to occasional discussions of a low-density universe with or without a cosmological constant, and the
CDM model played an important role in these considerations, as we now discuss.
When the CDM model was introduced it was known
that the observations disfavor the high-mass density of
the Einstein–de Sitter model, unless the mass is more
smoothly distributed than the visible matter (Sec. III.C).
Kaiser (1984) and Davis et al. (1985) showed that this
wanted biased distribution of visible galaxies, relative to
the distribution of all of the mass, can follow in a natural
way in the CDM theory. In short, where the mass density is high enough to lead to the gravitational assembly
of a large galaxy the mass density tends to be high
nearby, favoring the formation of neighboring large galaxies.
The biasing concept is important and certainly had to
be explored. But in 1985 there was little empirical evidence for the effect and significant arguments against it,
mainly involving the empty state of the voids between
the concentrations of large galaxies.39 In the biasing picture the voids contain most of the mass of an
Einstein–de Sitter universe, but few of the galaxies,
since galaxy formation there is supposed to have been
suppressed. But it is hard to see how galaxy formation
39
The issue is presented by Peebles (1986); the data and history of ideas are reviewed in Peebles (2001).
Rev. Mod. Phys., Vol. 75, No. 2, April 2003
575
could be entirely extinguished: the CDM model would
be expected to predict a void population of irregular
galaxies that show signs of a difficult youth. Many irregular galaxies are observed, but they avoid the voids.
The straightforward reading of the observations thus is
that the voids are empty, and that the dynamics of the
motions of the visible galaxies therefore show that ⍀ M0
is well below unity, and that the mass is not more
smoothly distributed than that of the visible galaxies.
In a low-density open universe, with ⍀ ⌳0 ⫽0 and positive ⍀ K0 , the growth of mass clustering is suppressed at
⫺1
⫺1. Thus to agree with the observed lowzⱗ⍀ M0
redshift mass distribution, density fluctuations at high
redshift must be larger in the open model than in the
Einstein–de Sitter case. This makes it harder to understand the small 3-K cosmic microwave background anisotropy. In a low-density spatially flat universe with
⍀ K0 ⫽0 and a cosmol…