1 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episodes in which a nonzero cosmological constant was put forward as an explanation for a set of observations and later withdrawn when the observational case evaporated.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 [48Jump To The Next Citation Point], which should be consulted for numerous useful formulae and a discussion of several kinds of observational tests not covered here. Some earlier discussions include [8550221Jump To The Next Citation Point], and subsequent reviews include [58Jump To The Next Citation Point218Jump To The Next Citation Point246]. The classic discussion of the physics of the cosmological constant is by Weinberg [264Jump To The Next Citation Point], with more recent work discussed by [58Jump To The Next Citation Point218Jump To The Next Citation Point]. For introductions to cosmology, see [149Jump To The Next Citation Point160189Jump To The Next Citation Point].

1.2 Introducing the cosmological constant

Einstein’s original field equations are:

1- R μν − 2Rg μν = 8πGT μν. (1 )
(I use conventions in which c = 1, and will also set ¯h = 1 in most of the formulae to follow, but Newton’s constant will be kept explicit.) On very large scales the universe is spatially homogeneous and isotropic to an excellent approximation, which implies that its metric takes the Robertson–Walker form,
[ dr2 ] ds2 = − dt2 + a2(t)R20------2-+ r2d Ω2 , (2 ) 1 − kr
where dΩ2 = d 𝜃2 + sin2𝜃d ϕ2 is the metric on a two-sphere. The curvature parameter k takes on values +1, 0, or − 1 for positively curved, flat, and negatively curved spatial sections, respectively. The scale factor characterizes the relative size of the spatial sections as a function of time; we have written it in a normalized form a(t) = R(t)∕R 0, where the subscript 0 will always refer to a quantity evaluated at the present time. The redshift z undergone by radiation from a comoving object as it travels to us today is related to the scale factor at which it was emitted by
---1--- a = (1 + z). (3 )

The energy-momentum sources may be modeled as a perfect fluid, specified by an energy density ρ and isotropic pressure p in its rest frame. The energy-momentum tensor of such a fluid is

Tμν = (ρ + p)U μUν + pgμν, (4 )
where Uμ is the fluid four-velocity. To obtain a Robertson–Walker solution to Einstein’s equations, the rest frame of the fluid must be that of a comoving observer in the metric (2View Equation); in that case, Einstein’s equations reduce to the two Friedmann equations,
( ˙a)2 8πG k H2 ≡ -- = ----ρ − -2-2-, (5 ) a 3 a R0
where we have introduced the Hubble parameter H ≡ ˙a∕a, and
¨a 4πG --= − ----(ρ + 3p). (6 ) a 3

Einstein was interested in finding static (˙a = 0) solutions, both due to his hope that general relativity would embody Mach’s principle that matter determines inertia, and simply to account for the astronomical data as they were understood at the time. (This account gives short shrift to the details of what actually happened; for historical background see [264Jump To The Next Citation Point].) A static universe with a positive energy density is compatible with (5View Equation) if the spatial curvature is positive (k = +1) and the density is appropriately tuned; however, (6View Equation) implies that ¨a will never vanish in such a spacetime if the pressure p is also nonnegative (which is true for most forms of matter, and certainly for ordinary sources such as stars and gas). Einstein therefore proposed a modification of his equations, to

1- R μν − 2Rg μν + Λgμν = 8πGT μν, (7 )
where Λ is a new free parameter, the cosmological constant. Indeed, the left-hand side of (7View Equation) is the most general local, coordinate-invariant, divergenceless, symmetric, two-index tensor we can construct solely from the metric and its first and second derivatives. With this modification, the Friedmann equations become
8 πG Λ k H2 = -----ρ + --− -----, (8 ) 3 3 a2R20
and
¨a 4 πG Λ --= − -----(ρ + 3p) + --. (9 ) a 3 3
These equations admit a static solution with positive spatial curvature and all the parameters ρ, p, and Λ nonnegative. This solution is called the “Einstein static universe.”

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 (9View Equation) 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 (8View Equation) 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 [27733].

Consider a single scalar field ϕ, with potential energy V (ϕ). The action can be written

∫ √ ---[ 1 ] S = d4x − g -gμν∂μϕ∂νϕ − V (ϕ) (10 ) 2
(where g is the determinant of the metric tensor gμν), and the corresponding energy-momentum tensor is
1- 1- ρσ Tμν = 2∂μϕ∂νϕ + 2 (g ∂ρϕ∂σϕ)gμν − V (ϕ)gμν. (11 )
In this theory, the configuration with the lowest energy density (if it exists) will be one in which there is no contribution from kinetic or gradient energy, implying ∂μϕ = 0, for which T μν = − V (ϕ0)gμν, where ϕ0 is the value of ϕ which minimizes V(ϕ ). There is no reason in principle why V (ϕ ) 0 should vanish. The vacuum energy-momentum tensor can thus be written
vac Tμν = − ρvacgμν, (12 )
with ρ vac in this example given by V (ϕ ) 0. (This form for the vacuum energy-momentum tensor can also be argued for on the more general grounds that it is the only Lorentz-invariant form for vac Tμν.) The vacuum can therefore be thought of as a perfect fluid as in (4View Equation), with
pvac = − ρvac. (13 )
The effect of an energy-momentum tensor of the form (12View Equation) is equivalent to that of a cosmological constant, as can be seen by moving the Λgμν term in (7View Equation) to the right-hand side and setting
-Λ--- ρvac = ρΛ ≡ 8πG . (14 )
This equivalence is the origin of the identification of the cosmological constant with the energy of the vacuum. In what follows, I will use the terms “vacuum energy” and “cosmological constant” essentially interchangeably.

It is not necessary to introduce scalar fields to obtain a nonzero vacuum energy. The action for general relativity in the presence of a “bare” cosmological constant Λ0 is

∫ --1--- 4 √--- S = 16πG d x − g(R − 2Λ0), (15 )
where R is the Ricci scalar. Extremizing this action (augmented by suitable matter terms) leads to the equations (7View Equation). Thus, the cosmological constant can be thought of as simply a constant term in the Lagrange density of the theory. Indeed, (15View Equation) is the most general covariant action we can construct out of the metric and its first and second derivatives, and is therefore a natural starting point for a theory of gravity.

Classically, then, the effective cosmological constant is the sum of a bare term Λ0 and the potential energy V (ϕ), where the latter may change with time as the universe passes through different phases. Quantum mechanics adds another contribution, from the zero-point energies associated with vacuum fluctuations. Consider a simple harmonic oscillator, i.e. a particle moving in a one-dimensional potential of the form V(x ) = 1ω2x2 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. Quantum-mechanically, 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 E0 = 1¯h ω 2 (where I have temporarily re-introduced explicit factors of ¯h 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, 1 − 2¯hω) without changing the theory. It is important, however, that the zero-point energy depends on the system, in this case on the frequency ω.

A precisely analogous situation holds in field theory. A (free) quantum field can be thought of as a collection of an infinite number of harmonic oscillators in momentum space. Formally, the zero-point energy of such an infinite collection will be infinite. (See [264Jump To The Next Citation Point48Jump To The Next Citation Point] for further details.) If, however, we discard the very high-momentum modes on the grounds that we trust our theory only up to a certain ultraviolet momentum cutoff kmax, we find that the resulting energy density is of the form

4 ρΛ ∼ ¯hkmax. (16 )
This answer could have been guessed by dimensional analysis; the numerical constants which have been neglected will depend on the precise theory under consideration. Again, in the absence of gravity this energy has no effect, and is traditionally discarded (by a process known as “normal-ordering”). However, gravity does exist, and the actual value of the vacuum energy has important consequences. (And the vacuum fluctuations themselves are very real, as evidenced by the Casimir effect [49].)

The net cosmological constant, from this point of view, is the sum of a number of apparently disparate contributions, including potential energies from scalar fields and zero-point fluctuations of each field theory degree of freedom, as well as a bare cosmological constant Λ0. Unlike the last of these, in the first two cases we can at least make educated guesses at the magnitudes. In the Weinberg-Salam electroweak model, the phases of broken and unbroken symmetry are distinguished by a potential energy difference of approximately M ∼ 200 GeV EW (where 1 GeV = 1.6 × 10−3 erg); the universe is in the broken-symmetry phase during our current low-temperature epoch, and is believed to have been in the symmetric phase at sufficiently high temperatures at early times. The effective cosmological constant is therefore different in the two epochs; absent some form of prearrangement, we would naturally expect a contribution to the vacuum energy today of order

ρEW ∼ (200 GeV )4 ∼ 3 × 1047 erg∕cm3. (17 ) Λ
Similar contributions can arise even without invoking “fundamental” scalar fields. In the strong interactions, chiral symmetry is believed to be broken by a nonzero expectation value of the quark bilinear ¯qq (which is itself a scalar, although constructed from fermions). In this case the energy difference between the symmetric and broken phases is of order the QCD scale MQCD ∼ 0.3 GeV, and we would expect a corresponding contribution to the vacuum energy of order
QCD 4 36 3 ρΛ ∼ (0.3 GeV ) ∼ 1.6 × 10 erg∕cm . (18 )
These contributions are joined by those from any number of unknown phase transitions in the early universe, such as a possible contribution from grand unification of order 16 MGUT ∼ 10 GeV. In the case of vacuum fluctuations, we should choose our cutoff at the energy past which we no longer trust our field theory. If we are confident that we can use ordinary quantum field theory all the way up to the Planck scale MPl = (8πG )− 1∕2 ∼ 1018 GeV, we expect a contribution of order
Pl 18 4 110 3 ρΛ ∼ (10 GeV ) ∼ 2 × 10 erg∕cm . (19 )
Field theory may fail earlier, although quantum gravity is the only reason we have to believe it will fail at any specific scale.

As we will discuss later, cosmological observations imply

(obs) − 12 4 − 10 3 |ρΛ | ≤ (10 GeV ) ∼ 2 × 10 erg∕cm , (20 )
much smaller than any of the individual effects listed above. The ratio of (19View Equation) to (20View Equation) is the origin of the famous discrepancy of 120 orders of magnitude between the theoretical and observational values of the cosmological constant. There is no obstacle to imagining that all of the large and apparently unrelated contributions listed add together, with different signs, to produce a net cosmological constant consistent with the limit (20View Equation), other than the fact that it seems ridiculous. We know of no special symmetry which could enforce a vanishing vacuum energy while remaining consistent with the known laws of physics; this conundrum is the “cosmological constant problem”. In Section 4 we will discuss a number of issues related to this puzzle, which at this point remains one of the most significant unsolved problems in fundamental physics.


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