The Cosmological Constant
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Abstract
This is a review of the physics and cosmology of the cosmological constant. Focusing on recent developments, I present a pedagogical overview of cosmology in the presence of a cosmological constant, observational constraints on its magnitude, and the physics of a small (and potentially nonzero) vacuum energy.
Keywords
cosmology cosmological constant vacuum energy1 Introduction
1.1 Truth and beauty
Science is rarely tidy. We ultimately seek a unified explanatory framework characterized by elegance and simplicity; along the way, however, our aesthetic impulses must occasionally be sacrificed to the desire to encompass the largest possible range of phenomena (i.e., to fit the data). It is often the case that an otherwise compelling theory, in order to be brought into agreement with observation, requires some apparently unnatural modification. Some such modifications may eventually be discarded as unnecessary once the phenomena are better understood; at other times, advances in our theoretical understanding will reveal that a certain theoretical compromise is only superficially distasteful, when in fact it arises as the consequence of a beautiful underlying structure.
General relativity is a paradigmatic example of a scientific theory of impressive power and simplicity. The cosmological constant, meanwhile, is a paradigmatic example of a modification, originally introduced [80] to help fit the data, which appears at least on the surface to be superfluous and unattractive. Its original role, to allow static homogeneous solutions to Einstein’s equations in the presence of matter, turned out to be unnecessary when the expansion of the universe was discovered [131], and there have been a number of subsequent episodes in which a nonzero cosmological constant was put forward as an explanation for a set of observations and later withdrawn when the observational case evaporated. Meanwhile, particle theorists have realized that the cosmological constant can be interpreted as a measure of the energy density of the vacuum. This energy density is the sum of a number of apparently unrelated contributions, each of magnitude much larger than the upper limits on the cosmological constant today; the question of why the observed vacuum energy is so small in comparison to the scales of particle physics has become a celebrated puzzle, although it is usually thought to be easier to imagine an unknown mechanism which would set it precisely to zero than one which would suppress it by just the right amount to yield an observationally accessible cosmological constant.
This checkered history has led to a certain reluctance to consider further invocations of a nonzero cosmological constant; however, recent years have provided the best evidence yet that this elusive quantity does play an important dynamical role in the universe. This possibility, although still far from a certainty, makes it worthwhile to review the physics and astrophysics of the cosmological constant (and its modern equivalent, the energy of the vacuum).
There are a number of other reviews of various aspects of the cosmological constant; in the present article I will outline the most relevant issues, but not try to be completely comprehensive, focusing instead on providing a pedagogical introduction and explaining recent advances. For astrophysical aspects, I did not try to duplicate much of the material in Carroll, Press and Turner [48], which should be consulted for numerous useful formulae and a discussion of several kinds of observational tests not covered here. Some earlier discussions include [85, 50, 221], and subsequent reviews include [58, 218, 246]. The classic discussion of the physics of the cosmological constant is by Weinberg [264], with more recent work discussed by [58, 218]. For introductions to cosmology, see [149, 160, 189].
1.2 Introducing the cosmological constant
The discovery by Hubble that the universe is expanding eliminated the empirical need for a static world model (although the Einstein static universe continues to thrive in the toolboxes of theorists, as a crucial step in the construction of conformal diagrams). It has also been criticized on the grounds that any small deviation from a perfect balance between the terms in (9) will rapidly grow into a runaway departure from the static solution.
Pandora’s box, however, is not so easily closed. The disappearance of the original motivation for introducing the cosmological constant did not change its status as a legitimate addition to the gravitational field equations, or as a parameter to be constrained by observation. The only way to purge Λ from cosmological discourse would be to measure all of the other terms in (8) to sufficient precision to be able to conclude that the Λ/3 term is negligibly small in comparison, a feat which has to date been out of reach. As discussed below, there is better reason than ever before to believe that Λ is actually nonzero, and Einstein may not have blundered after all.
1.3 Vacuum energy
The cosmological constant Λ is a dimensionful parameter with units of (length)^{2}. From the point of view of classical general relativity, there is no preferred choice for what the length scale defined by Λ might be. Particle physics, however, brings a different perspective to the question. The cosmological constant turns out to be a measure of the energy density of the vacuum — the state of lowest energy — and although we cannot calculate the vacuum energy with any confidence, this identification allows us to consider the scales of various contributions to the cosmological constant [277, 33].
Classically, then, the effective cosmological constant is the sum of a bare term Λ_{0} and the potential energy V(ϕ), where the latter may change with time as the universe passes through different phases. Ωuantum mechanics adds another contribution, from the zeropoint energies associated with vacuum fluctuations. Consider a simple harmonic oscillator, i.e. a particle moving in a onedimensional potential of the form \(V(x) = {\textstyle{1 \over 2}}{\omega ^2}{x^2}\). Classically, the “vacuum” for this system is the state in which the particle is motionless and at the minimum of the potential (x = 0), for which the energy in this case vanishes. Quantummechanically, however, the uncertainty principle forbids us from isolating the particle both in position and momentum, and we find that the lowest energy state has an energy \({E_0} = {\textstyle{1 \over 2}}\hbar \omega \) (where I have temporarily reintroduced explicit factors of ћ for clarity). Of course, in the absence of gravity either system actually has a vacuum energy which is completely arbitrary; we could add any constant to the potential (including, for example, \(  \frac{1}{2}\hbar \omega \)) without changing the theory. It is important, however, that the zeropoint energy depends on the system, in this case on the frequency ω.
2 Cosmology with a Cosmological Constant
2.1 Cosmological parameters
The ranges of values of the Ω_{ i }’s which are allowed in principle (as opposed to constrained by observation) will depend on a complete theory of the matter fields, but lacking that we may still invoke energy conditions to get a handle on what constitutes sensible values. The most appropriate condition is the dominant energy condition (DEC), which states that T_{ μν }l^{ μ }l^{ ν } ≥ 0, and T^{ μ }_{ ν }l^{ μ } is nonspacelike, for any null vector l^{ μ }; this implies that energy does not flow faster than the speed of light [117]. For a perfectfluid energymomentum tensor of the form (4), these two requirements imply that ρ + p ≥ 0 and ρ ≥ p, respectively. Thus, either the density is positive and greater in magnitude than the pressure, or the density is negative and equal in magnitude to a compensating positive pressure; in terms of the equationofstate parameter w, we have either positive ρ and w ≤ 1 or negative ρ and w = 1. That is, a negative energy density is allowed only if it is in the form of vacuum energy. (We have actually modified the conventional DEC somewhat, by using only null vectors l^{ μ } rather than null or timelike vectors; the traditional condition would rule out a negative cosmological constant, which there is no physical reason to do.)
There are good reasons to believe that the energy density in radiation today is much less than that in matter. Photons, which are readily detectable, contribute Ω_{ γ } ∼ 5 × 10^{5}, mostly in the 2.73 K cosmic microwave background [211, 87, 225]. If neutrinos are sufficiently low mass as to be relativistic today, conventional scenarios predict that they contribute approximately the same amount [149]. In the absence of sources which are even more exotic, it is therefore useful to parameterize the universe today by the values of Ω_{M} and Ω_{Λ}, with Ω_{ k } = 1  Ω_{M}  Ω_{Λ}, keeping the possibility of surprises always in mind.
Notice that positiveenergydensity sources with n > 2 cause the universe to decelerate while n < 2 leads to acceleration; the more rapidly energy density redshifts away, the greater the tendency towards universal deceleration. An empty universe (Ω = 0, Ω_{ k } = 1) expands linearly with time; sometimes called the “Milne universe”, such a spacetime is really flat Minkowski space in an unusual timeslicing.
2.2 Model universes and their fates
Figure 1 includes three fixed points, at (Ω_{M}, Ω_{Λ}) equal to (0,0), (0,1), and (1, 0). The attractor among these at (0,1) is known as de Sitter space — a universe with no matter density, dominated by a cosmological constant, and with scale factor growing exponentially with time. The fact that this point is an attractor on the diagram is another way of understanding the cosmological constant problem. A universe with initial conditions located at a generic point on the diagram will, after several expansion times, flow to de Sitter space if it began above the recollapse line, and flow to infinity and back to recollapse if it began below that line. Since our universe has expanded by many orders of magnitude since early times, it must have begun at a nongeneric point in order not to have evolved either to de Sitter space or to a Big Crunch. The only other two fixed points on the diagram are the saddle point at (Ω_{M}, Ω_{Λ}) = (0, 0), corresponding to an empty universe, and the repulsive fixed point at (Ω_{M}, Ω_{Λ}) = (1, 0), known as the Einsteinde Sitter solution. Since our universe is not empty, the favored solution from this combination of theoretical and empirical arguments is the Einsteinde Sitter universe. The inflationary scenario [113, 159, 6] provides a mechanism whereby the universe can be driven to the line Ω_{M} + Ω_{Λ} = 1 (spatial flatness), so Einsteinde Sitter is a natural expectation if we imagine that some unknown mechanism sets Λ = 0. As discussed below, the observationally favored universe is located on this line but away from the fixed points, near (Ω_{M}, Ω_{Λ}) = (0.3,0.7). It is fair to conclude that naturalness arguments have a somewhat spotty track record at predicting cosmological parameters.
2.3 Surveying the universe
2.4 Structure formation
The introduction of a cosmological constant changes the relationship between the matter density and expansion rate from what it would be in a matterdominated universe, which in turn influences the growth of largescale structure. The effect is similar to that of a nonzero spatial curvature, and complicated by hydrodynamic and nonlinear effects on small scales, but is potentially detectable through sufficiently careful observations.
The analysis of the evolution of structure is greatly abetted by the fact that perturbations start out very small (temperature anisotropies in the microwave background imply that the density perturbations were of order 10^{5} at recombination), and linearized theory is effective. In this regime, the fate of the fluctuations is in the hands of two competing effects: the tendency of selfgravity to make overdense regions collapse, and the tendency of test particles in the background expansion to move apart. Essentially, the effect of vacuum energy is to contribute to expansion but not to the selfgravity of overdensities, thereby acting to suppress the growth of perturbations [149, 189].
3 Observational Tests
It has been suspected for some time now that there are good reasons to think that a cosmology with an appreciable cosmological constant is the best fit to what we know about the universe [188, 248, 148, 79, 95, 147, 151, 181, 245]. However, it is only very recently that the observational case has tightened up considerably, to the extent that, as the year 2000 dawns, more experts than not believe that there really is a positive vacuum energy exerting a measurable effect on the evolution of the universe. In this section I review the major approaches which have led to this shift.
3.1 Type Ia supernovae
The most direct and theoryindependent way to measure the cosmological constant would be to actually determine the value of the scale factor as a function of time. Unfortunately, the appearance of Ω_{ k } in formulae such as (42) renders this difficult. Nevertheless, with sufficiently precise information about the dependence of a distance measure on redshift we can disentangle the effects of spatial curvature, matter, and vacuum energy, and methods along these lines have been popular ways to try to constrain the cosmological constant.
Recently, significant progress has been made by using Type Ia supernovae as “standardizable candles”. Supernovae are rare — perhaps a few per century in a MilkyWaysized galaxy — but modern telescopes allow observers to probe very deeply into small regions of the sky, covering a very large number of galaxies in a single observing run. Supernovae are also bright, and Type Ia’s in particular all seem to be of nearly uniform intrinsic luminosity (absolute magnitude M ∼ 19.5, typically comparable to the brightness of the entire host galaxy in which they appear) [36]. They can therefore be detected at high redshifts (z ∼ 1), allowing in principle a good handle on cosmological effects [236, 108].
The fact that all SNe Ia are of similar intrinsic luminosities fits well with our understanding of these events as explosions which occur when a white dwarf, onto which mass is gradually accreting from a companion star, crosses the Chandrasekhar limit and explodes. (It should be noted that our understanding of supernova explosions is in a state of development, and theoretical models are not yet able to accurately reproduce all of the important features of the observed events. See [274, 114, 121] for some recent work.) The Chandrasekhar limit is a nearlyuniversal quantity, so it is not a surprise that the resulting explosions are of nearlyconstant luminosity. However, there is still a scatter of approximately 40% in the peak brightness observed in nearby supernovae, which can presumably be traced to differences in the composition of the white dwarf atmospheres. Even if we could collect enough data that statistical errors could be reduced to a minimum, the existence of such an uncertainty would cast doubt on any attempts to study cosmology using SNe Ia as standard candles.
Fortunately, the observed differences in peak luminosities of SNe Ia are very closely correlated with observed differences in the shapes of their light curves: Dimmer SNe decline more rapidly after maximum brightness, while brighter SNe decline more slowly [200, 213, 115]. There is thus a oneparameter family of events, and measuring the behavior of the light curve along with the apparent luminosity allows us to largely correct for the intrinsic differences in brightness, reducing the scatter from 40% to less than 15% — sufficient precision to distinguish between cosmological models. (It seems likely that the single parameter can be traced to the amount of ^{56}Ni produced in the supernova explosion; more nickel implies both a higher peak luminosity and a higher temperature and thus opacity, leading to a slower decline. It would be an exaggeration, however, to claim that this behavior is wellunderstood theoretically.)
It is clear that the confidence intervals in the Ω_{M}Ω_{Λ} plane are consistent for the two groups, with somewhat tighter constraints obtained by the Supernova Cosmology Project, who have more data points. The surprising result is that both teams favor a positive cosmological constant, and strongly rule out the traditional (Ω_{M}, Ω_{Λ}) = (1,0) favorite universe. They are even inconsistent with an open universe with zero cosmological constant, given what we know about the matter density of the universe (see below).
Given the significance of these results, it is natural to ask what level of confidence we should have in them. There are a number of potential sources of systematic error which have been considered by the two teams; see the original papers [223, 214, 197] for a thorough discussion. The two most worrisome possibilities are intrinsic differences between Type Ia supernovae at high and low redshifts [75, 212], and possible extinction via intergalactic dust [2, 3, 4, 226, 241]. (There is also the fact that intervening weak lensing can change the distancemagnitude relation, but this seems to be a small effect in realistic universes [123, 143].) Both effects have been carefully considered, and are thought to be unimportant, although a better understanding will be necessary to draw firm conclusions. Here, I will briefly mention some of the relevant issues.
As thermonuclear explosions of white dwarfs, Type Ia supernovae can occur in a wide variety of environments. Consequently, a simple argument against evolution is that the highredshift environments, while chronologically younger, should be a subset of all possible lowredshift environments, which include regions that are “young” in terms of chemical and stellar evolution. Nevertheless, even a small amount of evolution could ruin our ability to reliably constrain cosmological parameters [75]. In their original papers [223, 214, 197], the supernova teams found impressive consistency in the spectral and photometric properties of Type Ia supernovae over a variety of redshifts and environments (e.g., in elliptical vs. spiral galaxies). More recently, however, Riess et al. [212] have presented tentative evidence for a systematic difference in the properties of high and lowredshift supernovae, claiming that the risetimes (from initial explosion to maximum brightness) were higher in the highredshift events. Apart from the issue of whether the existing data support this finding, it is not immediately clear whether such a difference is relevant to the distance determinations: first, because the risetime is not used in determining the absolute luminosity at peak brightness, and second, because a process which only affects the very early stages of the light curve is most plausibly traced to differences in the outer layers of the progenitor, which may have a negligible affect on the total energy output. Nevertheless, any indication of evolution could bring into question the fundamental assumptions behind the entire program. It is therefore essential to improve the quality of both the data and the theories so that these issues may be decisively settled.
Other than evolution, obscuration by dust is the leading concern about the reliability of the supernova results. Ordinary astrophysical dust does not obscure equally at all wavelengths, but scatters blue light preferentially, leading to the wellknown phenomenon of “reddening”. Spectral measurements by the two supernova teams reveal a negligible amount of reddening, implying that any hypothetical dust must be a novel “grey” variety. This possibility has been investigated by a number of authors [2, 3, 4, 226, 241]. These studies have found that even grey dust is highly constrained by observations: first, it is likely to be intergalactic rather than within galaxies, or it would lead to additional dispersion in the magnitudes of the supernovae; and second, intergalactic dust would absorb ultraviolet/optical radiation and reemit it at far infrared wavelengths, leading to stringent constraints from observations of the cosmological farinfrared background. Thus, while the possibility of obscuration has not been entirely eliminated, it requires a novel kind of dust which is already highly constrained (and may be convincingly ruled out by further observations).
According to the best of our current understanding, then, the supernova results indicating an accelerating universe seem likely to be trustworthy. Needless to say, however, the possibility of a heretofore neglected systematic effect looms menacingly over these studies. Future experiments, including a proposed satellite dedicated to supernova cosmology [154], will both help us improve our understanding of the physics of supernovae and allow a determination of the distance/redshift relation to sufficient precision to distinguish between the effects of a cosmological constant and those of more mundane astrophysical phenomena. In the meantime, it is important to obtain independent corroboration using other methods.
3.2 Cosmic microwave background
3.3 Matter density
Many cosmological tests, such as the two just discussed, will constrain some combination of Ω_{M} and Ω_{Λ}. It is therefore useful to consider tests of Ω_{M} alone, even if our primary goal is to determine Ω_{Λ}. (In truth, it is also hard to constrain Ω_{M} alone, as almost all methods actually constrain some combination of Ω_{M} and the Hubble constant h = H_{0}/(100 km/sec/Mpc); the HST Key Project on the extragalactic distance scale finds h = 0.71 ± 0.06 [175], which is consistent with other methods [88], and what I will assume below.)
For years, determinations of Ω_{M} based on dynamics of galaxies and clusters have yielded values between approximately 0.1 and 0.4 — noticeably larger than the density parameter in baryons as inferred from primordial nucleosynthesis, Ω_{B} = (0.019±0.001)h^{2} ≈ 0.04 [224, 41], but noticeably smaller than the critical density. The last several years have witnessed a number of new methods being brought to bear on the question; the quantitative results have remained unchanged, but our confidence in them has increased greatly.
A thorough discussion of determinations of Ω_{M} requires a review all its own, and good ones are available [66, 14, 247, 88, 206]. Here I will just sketch some of the important methods.
The traditional method to estimate the mass density of the universe is to “weigh” a cluster of galaxies, divide by its luminosity, and extrapolate the result to the universe as a whole. Although clusters are not representative samples of the universe, they are sufficiently large that such a procedure has a chance of working. Studies applying the virial theorem to cluster dynamics have typically obtained values Ω_{M} = 0.2 ± 0.1 [45, 66, 14]. Although it is possible that the global value of M/L differs appreciably from its value in clusters, extrapolations from small scales do not seem to reach the critical density [17]. New techniques to weigh the clusters, including gravitational lensing of background galaxies [227] and temperature profiles of the Xray gas [155], while not yet in perfect agreement with each other, reach essentially similar conclusions.
Another handle on the density parameter in matter comes from properties of clusters at high redshift. The very existence of massive clusters has been used to argue in favor of Ω_{M} ∼ 0.2 [15], and the lack of appreciable evolution of clusters from high redshifts to the present [16, 44] provides additional evidence that Ω_{M} < 1.0.
The story of largescale motions is more ambiguous. The peculiar velocities of galaxies are sensitive to the underlying mass density, and thus to Ω_{M}, but also to the “bias” describing the relative amplitude of fluctuations in galaxies and mass [66, 65]. Difficulties both in measuring the flows and in disentangling the mass density from other effects make it difficult to draw conclusions at this point, and at present it is hard to say much more than 0.2 ≤ Ω_{M} ≤ 1.0.
Finally, the matter density parameter can be extracted from measurements of the power spectrum of density fluctuations (see for example [187]). As with the CMB, predicting the power spectrum requires both an assumption of the correct theory and a specification of a number of cosmological parameters. In simple models (e.g., with only cold dark matter and baryons, no massive neutrinos), the spectrum can be fit (once the amplitude is normalized) by a single “shape parameter”, which is found to be equal to Γ = Ω_{M}h. (For more complicated models see [82].) Observations then yield Γ ∼ 0.25, or Ω_{M} ∼ 0.36. For a more careful comparison between models and observations, see [156, 157, 71, 205].
3.4 Gravitational lensing
The volume of space back to a specified redshift, given by (44), depends sensitively on Ω_{Λ}. Consequently, counting the apparent density of observed objects, whose actual density per cubic Mpc is assumed to be known, provides a potential test for the cosmological constant [109, 96, 244, 48]. Like tests of distance vs. redshift, a significant problem for such methods is the luminosity evolution of whatever objects one might attempt to count. A modern attempt to circumvent this difficulty is to use the statistics of gravitational lensing of distant galaxies; the hope is that the number of condensed objects which can act as lenses is less sensitive to evolution than the number of visible objects.
Analysis of lensing statistics is complicated by uncertainties in evolution, extinction, and biases in the lens discovery procedure. It has been argued [146, 83] that the existing data allow us to place an upper limit of Ω_{Λ} < 0.7 in a flat universe. However, other groups [52, 51] have claimed that the current data actually favor a nonzero cosmological constant. The near future will bring larger, more objective surveys, which should allow these ambiguities to be resolved. Other manifestations of lensing can also be used to constrain Ω_{Λ}, including statistics of giant arcs [275], deep weaklensing surveys [133], and lensing in the Hubble Deep Field [61].
3.5 Other tests

Observations of numbers of objects vs. redshift are a potentially sensitive test of cosmological parameters if evolutionary effects can be brought under control. Although it is hard to account for the luminosity evolution of galaxies, it may be possible to indirectly count dark halos by taking into account the rotation speeds of visible galaxies, and upcoming redshift surveys could be used to constrain the volume/redshift relation [176].

Alcock and Paczyński [7] showed that the relationship between the apparent transverse and radial sizes of an object of cosmological size depends on the expansion history of the universe. Clusters of galaxies would be possible candidates for such a measurement, but they are insufficiently isotropic; alternatives, however, have been proposed, using for example the quasar correlation function as determined from redshift surveys [201, 204], or the Lymana forest [134].

In a related effect, the dynamics of largescale structure can be affected by a nonzero cosmological constant; if a protocluster, for example, is anisotropic, it can begin to contract along a minor axis while the universe is matterdominated and along its major axis while the universe is vacuumdominated. Although small, such effects may be observable in individual clusters [153] or in redshift surveys [19].

A different version of the distanceredshift test uses extended lobes of radio galaxies as modified standard yardsticks. Current observations disfavor universes with Ω_{M} near unity ([112], and references therein).

Inspiralling compact binaries at cosmological distances are potential sources of gravitational waves. It turns out that the redshift distribution of events is sensitive to the cosmological constant; although speculative, it has been proposed that advanced LIGO (Laser Interferometric Gravitational Wave Observatory [215]) detectors could use this effect to provide measurements of Ω_{Λ} [262].

Finally, consistency of the age of the universe and the ages of its oldest constituents is a classic test of the expansion history. If stars were sufficiently old and H_{0} and Ω_{M} were sufficiently high, a positive Ω_{Λ} would be necessary to reconcile the two, and this situation has occasionally been thought to hold. Measurements of geometric parallax to nearby stars from the Hipparcos satellite have, at the least, called into question previous determinations of the ages of the oldest globular clusters, which are now thought to be perhaps 12 billion rather than 15 billion years old (see the discussion in [88]). It is therefore unclear whether the age issue forces a cosmological constant upon us, but by now it seems forced upon us for other reasons.
4 Physics Issues
Although the mechanism which suppresses the naive value of the vacuum energy is unknown, it seems easier to imagine a hypothetical scenario which makes it exactly zero than one which sets it to just the right value to be observable today. (Keeping in mind that it is the zerotemperature, latetime vacuum energy which we want to be small; it is expected to change at phase transitions, and a large value in the early universe is a necessary component of inflationary universe scenarios [113, 159, 6].) If the recent observations pointing toward a cosmological constant of astrophysically relevant magnitude are confirmed, we will be faced with the challenge of explaining not only why the vacuum energy is smaller than expected, but also why it has the specific nonzero value it does.
4.1 Supersymmetry
Although initially investigated for other reasons, supersymmetry (SUSY) turns out to have a significant impact on the cosmological constant problem, and may even be said to solve it halfway. SUSY is a spacetime symmetry relating fermions and bosons to each other. Just as ordinary symmetries are associated with conserved charges, supersymmetry is associated with “supercharges” Q_{ α }, where α is a spinor index (for introductions see [178, 166, 169]). As with ordinary symmetries, a theory may be supersymmetric even though a given state is not supersymmetric; a state which is annihilated by the supercharges, Q_{ α }ψ〉 = 0, preserves supersymmetry, while states with Q_{ α }ψ〉 ≠ 0 are said to spontaneously break SUSY.
 1.)
Supersymmetric states manifest a degeneracy in the mass spectrum of bosons and fermions, a feature not apparent in the observed world; and
 1.)
The above results imply that nonsupersymmetric states have a positivedefinite vacuum energy.
4.2 String theory
Unlike supergravity, string theory appears to be a consistent and welldefined theory of quantum gravity, and therefore calculating the value of the cosmological constant should, at least in principle, be possible. On the other hand, the number of vacuum states seems to be quite large, and none of them (to the best of our current knowledge) features three large spatial dimensions, broken supersymmetry, and a small cosmological constant. At the same time, there are reasons to believe that any realistic vacuum of string theory must be strongly coupled [70]; therefore, our inability to find an appropriate solution may simply be due to the technical difficulty of the problem. (For general introductions to string theory, see [110, 203]; for cosmological issues, see [167, 21]).
String theory is naturally formulated in more than four spacetime dimensions. Studies of duality symmetries have revealed that what used to be thought of as five distinct tendimensional superstring theories — Type I, Types IIA and IIB, and heterotic theories based on gauge groups E(8)×E(8) and SO(32) — are, along with elevendimensional supergravity, different lowenergy weakcoupling limits of a single underlying theory, sometimes known as Mtheory. In each of these six cases, the solution with the maximum number of uncompactified, flat spacetime dimensions is a stable vacuum preserving all of the supersymmetry. To bring the theory closer to the world we observe, the extra dimensions can be compactified on a manifold whose Ricci tensor vanishes. There are a large number of possible compactifications, many of which preserve some but not all of the original supersymmetry. If enough SUSY is preserved, the vacuum energy will remain zero; generically there will be a manifold of such states, known as the moduli space.
Of course, to describe our world we want to break all of the supersymmetry. Investigations in contexts where this can be done in a controlled way have found that the induced cosmological constant vanishes at the classical level, but a substantial vacuum energy is typically induced by quantum corrections [110]. Moore [174] has suggested that AtkinLehner symmetry, which relates strong and weak coupling on the string worldsheet, can enforce the vanishing of the oneloop quantum contribution in certain models (see also [67, 68]); generically, however, there would still be an appreciable contribution at two loops.

In three spacetime dimensions supersymmetry can remain unbroken, maintaining a zero cosmological constant, in such a way as to break the mass degeneracy between bosons and fermions [271]. This mechanism relies crucially on special properties of spacetime in (2+1) dimensions, but in string theory it sometimes happens that the strongcoupling limit of one theory is another theory in one higher dimension [272, 273].

More generally, it is now understood that (at least in some circumstances) string theory obeys the “holographic principle”, the idea that a theory with gravity in dimensions is equivalent to a theory without gravity in D1 dimensions [235, 234]. In a holographic theory, the number of degrees of freedom in a region grows as the area of its boundary, rather than as its volume. Therefore, the conventional computation of the cosmological constant due to vacuum fluctuations conceivably involves a vast overcounting of degrees of freedom. We might imagine that a more correct counting would yield a much smaller estimate of the vacuum energy [20, 57, 254, 222], although no reliable calculation has been done as yet.

The absence of manifest SUSY in our world leads us to ask whether the beneficial aspect of canceling contributions to the vacuum energy could be achieved even without a truly supersymmetric theory. Kachru, Kumar and Silverstein [139] have constructed such a string theory, and argue that the perturbative contributions to the cosmological constant should vanish (although the actual calculations are somewhat delicate, and not everyone agrees [136]). If such a model could be made to work, it is possible that small nonperturbative effects could generate a cosmological constant of an astrophysically plausible magnitude [116].

A novel approach to compactification starts by imagining that the fields of the Standard Model are confined to a (3+1)dimensional manifold (or “brane”, in string theory parlance) embedded in a larger space. While gravity is harder to confine to a brane, phenomenologically acceptable scenarios can be constructed if either the extra dimensions are any size less than a millimeter [216, 10, 124, 13, 140], or if there is significant spacetime curvature in a noncompact extra dimension [259, 207, 107]. Although these scenarios do not offer a simple solution to the cosmological constant problem, the relationship between the vacuum energy and the expansion rate can differ from our conventional expectation (see for example [32, 142]), and one is free to imagine that further study may lead to a solution in this context (see for example [231, 40]).
Of course, string theory might not be the correct description of nature, or its current formulation might not be directly relevant to the cosmological constant problem. For example, a solution may be provided by loop quantum gravity [98], or by a composite graviton [233]. It is probably safe to believe that a significant advance in our understanding of fundamental physics will be required before we can demonstrate the existence of a vacuum state with the desired properties. (Not to mention the equally important question of why our world is based on such a state, rather than one of the highly supersymmetric states that appear to be perfectly good vacua of string theory.)
4.3 The anthropic principle
The anthropic principle [25, 122] is essentially the idea that some of the parameters characterizing the universe we observe may not be determined directly by the fundamental laws of physics, but also by the truism that intelligent observers will only ever experience conditions which allow for the existence of intelligent observers. Many professional cosmologists view this principle in much the same way as many traditional literary critics view deconstruction — as somehow simultaneously empty of content and capable of working great evil. Anthropic arguments are easy to misuse, and can be invoked as a way out of doing the hard work of understanding the real reasons behind why we observe the universe we do. Furthermore, a sense of disappointment would inevitably accompany the realization that there were limits to our ability to unambiguously and directly explain the observed universe from first principles. It is nevertheless possible that some features of our world have at best an anthropic explanation, and the value of the cosmological constant is perhaps the most likely candidate.
In order for the tautology that “observers will only observe conditions which allow for observers” to have any force, it is necessary for there to be alternative conditions — parts of the universe, either in space, time, or branches of the wavefunction — where things are different. In such a case, our local conditions arise as some combination of the relative abundance of different environments and the likelihood that such environments would give rise to intelligence. Clearly, the current state of the art doesn’t allow us to characterize the full set of conditions in the entire universe with any confidence, but modern theories of inflation and quantum cosmology do at least allow for the possibility of widely disparate parts of the universe in which the “constants of nature” take on very different values (for recent examples see [100, 161, 256, 163, 118, 162, 251, 258]). We are therefore faced with the task of estimating quantitatively the likelihood of observing any specific value of A within such a scenario.
Thus, if one is willing to make the leap of faith required to believe that the value of the cosmological constant is chosen from an ensemble of possibilities, it is possible to find an “explanation” for its current value (which, given its unnaturalness from a variety of perspectives, seems otherwise hard to understand). Perhaps the most significant weakness of this point of view is the assumption that there are a continuum of possibilities for the vacuum energy density. Such possibilities correspond to choices of vacuum states with arbitrarily similar energies. If these states were connected to each other, there would be local fluctuations which would appear to us as massless fields, which are not observed (see Section 4.5). If on the other hand the vacua are disconnected, it is hard to understand why all possible values of the vacuum energy are represented, rather than the differences in energies between different vacua being given by some characteristic particlephysics scale such as M_{Pl} or M_{SUSY}. (For one scenario featuring discrete vacua with densely spaced energies, see [23].) It will therefore (again) require advances in our understanding of fundamental physics before an anthropic explanation for the current value of the cosmological constant can be accepted.
4.4 Miscellaneous adjustment mechanisms
The importance of the cosmological constant problem has engendered a wide variety of proposed solutions. This section will present only a brief outline of some of the possibilities, along with references to recent work; further discussion and references can be found in [264, 48, 218].
One approach which has received a great deal of attention is the famous suggestion by Coleman [59], that effects of virtual wormholes could set the cosmological constant to zero at low energies. The essential idea is that wormholes (thin tubes of spacetime connecting macroscopically large regions) can act to change the effective value of all the observed constants of nature. If we calculate the wave function of the universe by performing a Feynman path integral over all possible spacetime metrics with wormholes, the dominant contribution will be from those configurations whose effective values for the physical constants extremize the action. These turn out to be, under a certain set of assumed properties of Euclidean quantum gravity, configurations with zero cosmological constant at late times. Thus, quantum cosmology predicts that the constants we observe are overwhelmingly likely to take on values which imply a vanishing total vacuum energy. However, subsequent investigations have failed to inspire confidence that the desired properties of Euclidean quantum cosmology are likely to hold, although it is still something of an open question; see discussions in [264, 48].
Another route one can take is to consider alterations of the classical theory of gravity. The simplest possibility is to consider adding a scalar field to the theory, with dynamics which cause the scalar to evolve to a value for which the net cosmological constant vanishes (see for example [74, 230]). Weinberg, however, has pointed out on fairly general grounds that such attempts are unlikely to work [264, 265]; in models proposed to date, either there is no solution for which the effective vacuum energy vanishes, or there is a solution but with other undesirable properties (such as making Newton’s constant G also vanish). Rather than adding scalar fields, a related approach is to remove degrees of freedom by making the determinant of the metric, which multiplies Λ_{0} in the action (15), a nondynamical quantity, or at least changing its dynamics in some way (see [111, 270, 177] for recent examples). While this approach has not led to a believable solution to the cosmological constant problem, it does change the context in which it appears, and may induce different values for the effective vacuum energy in different branches of the wavefunction of the universe.
Along with global supersymmetry, there is one other symmetry which would work to prohibit a cosmological constant: conformal (or scale) invariance, under which the metric is multiplied by a spacetimedependent function, g_{ μν } → e^{ λ(x) }g_{ μν }. Like supersymmetry, conformal invariance is not manifest in the Standard Model of particle physics. However, it has been proposed that quantum effects could restore conformal invariance on length scales comparable to the cosmological horizon size, working to cancel the cosmological constant (for some examples see [240, 12, 11]). At this point it remains unclear whether this suggestion is compatible with a more complete understanding of quantum gravity, or with standard cosmological observations.
A final mechanism to suppress the cosmological constant, related to the previous one, relies on quantum particle production in de Sitter space (analogous to Hawking radiation around black holes). The idea is that the effective energymomentum tensor of such particles may act to cancel out the bare cosmological constant (for recent attempts see [242, 243, 1, 184]). There is currently no consensus on whether such an effect is physically observable (see for example [252]).
If inventing a theory in which the vacuum energy vanishes is difficult, finding a model that predicts a vacuum energy which is small but not quite zero is all that much harder. Along these lines, there are various numerological games one can play. For example, the fact that supersymmetry solves the problem halfway could be suggestive; a theory in which the effective vacuum energy scale was given not by M_{SUSY} ∼ 10^{3} GeV but by M _{SUSY} ^{2} /M_{Pl} ∼ 10^{3} eV would seem to fit the observations very well. The challenging part of this program, of course, is to devise such a theory. Alternatively, one could imagine that we live in a “false vacuum” — that the absolute minimum of the vacuum energy is truly zero, but we live in a state which is only a local minimum of the energy. Scenarios along these lines have been explored [250, 103, 152]; the major hurdle to be overcome is explaining why the energy difference between the true and false vacua is so much smaller than one would expect.
4.5 Other sources of dark energy
Although a cosmological constant is an excellent fit to the current data, the observations can also be accommodated by any form of “dark energy” which does not cluster on small scales (so as to avoid being detected by measurements of Ω_{M}) and redshifts away only very slowly as the universe expands [to account for the accelerated expansion, as per equation (33)]. This possibility has been extensively explored of late, and a number of candidates have been put forward.
There are many reasons to consider dynamical dark energy as an alternative to a cosmological constant. First and foremost, it is a logical possibility which might be correct, and can be constrained by observation. Secondly, it is consistent with the hope that the ultimate vacuum energy might actually be zero, and that we simply haven’t relaxed all the way to the vacuum as yet. But most interestingly, one might wonder whether replacing a constant parameter A with a dynamical field could allow us to relieve some of the burden of finetuning that inevitably accompanies the cosmological constant. To date, investigations have focused on scaling or tracker models of quintessence, in which the scalar field energy density can parallel that of matter or radiation, at least for part of its history [86, 62, 279, 158, 232, 278, 219]. (Of course, we do not want the dark energy density to redshift away as rapidly as that in matter during the current epoch, or the universe would not be accelerating.) Tracker models can be constructed in which the vacuum energy density at late times is robust, in the sense that it does not depend sensitively on the initial conditions for the field. However, the ultimate value ρ_{vac} ∼ (10^{3} eV)^{4} still depends sensitively on the parameters in the potential. Indeed, it is hard to imagine how this could help but be the case; unlike the case of the axion solution to the strongCP problem, we have no symmetry to appeal to that would enforce a small vacuum energy, much less a particular small nonzero number.
Nevertheless, these naturalness arguments are by no means airtight, and it is worth considering specific particlephysics models for the quintessence field. In addition to the pseudoGoldstone boson models just mentioned, these include models based on supersymmetric gauge theories [31, 170], supergravity [37, 5], small extra dimensions [29, 24], large extra dimensions [28, 22], quantum field theory effects in curved spacetime [185, 186], and nonminimal couplings to the curvature scalar [217, 253, 8, 198, 199, 64, 30]. Finally, the possibility has been raised that the scalar field responsible for driving inflation may also serve as quintessence [90, 191, 192, 106], although this proposal has been criticized for producing unwanted relics and isocurvature fluctuations [84].
There are other models of dark energy besides those based on nearlymassless scalar fields. One scenario is “solid” dark matter, typically based on networks of tangled cosmic strings or domain walls [255, 229, 39, 27]. Strings give an effective equationofstate parameter w_{string} = 1/3, and walls have w_{wall} = 2/3, so walls are a better fit to the data at present. There is also the idea of dark matter particles whose masses increase as the universe expands, their energy thus redshifting away more slowly than that of ordinary matter [99, 9] (see also [126]). The cosmological consequences of this kind of scenario turn out to be difficult to analyze analytically, and work is still ongoing.
5 Conclusions: The Preposterous Universe
Observational evidence from a variety of sources currently points to a universe which is (at least approximately) spatially flat, with (Ω_{M}, Ω_{Λ}) ≈ (0.3,0.7). The nucleosynthesis constraint implies that Ω_{B} ∼ 0.04, so the majority of the matter content must be in an unknown nonbaryonic form.
Apart from confirming (or disproving) this picture, a major challenge to cosmologists and physicists in the years to come will be to understand whether these apparently distasteful aspects of our universe are simply surprising coincidences, or actually reflect a beautiful underlying structure we do not as yet comprehend. If we are fortunate, what appears unnatural at present will serve as a clue to a deeper understanding of fundamental physics.
Notes
Acknowledgments
I wish to thank Greg Anderson, Tom Banks, Robert Caldwell, Joanne Cohn, Gordon Chalmers, Michael Dine, George Field, Peter Garnavich, Christophe Grojean, Jeff Harvey, Dragan Huterer, Steuard Jensen, Gordy Kane, Manoj Kaplinghat, Bob Kirshner, Lloyd Knox, Finn Larsen, Laura Mersini, UeLi Pen, Saul Perlmutter, Joe Polchinski, Ted Pyne, Brian Schmidt, and Michael Turner for numerous useful conversations, Patrick Brady, Deryn Fogg and Clifford Johnson for rhetorical encouragement, and Bill Press and Ed Turner for insinuating me into this formerlydisreputable subject. This work was supported in part by the U.S. Department of Energy.
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