3.4. The Cosmological Constant Problem(s)

In classical general relativity the cosmological constant is a completely free parameter. It has dimensions of [length]-2 (while the energy density has units [energy/volume]), and hence defines a scale, while general relativity is otherwise scale-free. Indeed, from purely classical considerations, we can't even say whether a specific value of is "large" or "small"; it is simply a constant of nature we should go out and determine through experiment.

The introduction of quantum mechanics changes this story somewhat. For one thing, Planck's constant allows us to define the reduced Planck mass Mp ~ 1018 GeV, as well as the reduced Planck length

 (65)

Hence, there is a natural expectation for the scale of the cosmological constant, namely

 (66)

or, phrased as an energy density,

 (67)

We can partially justify this guess by thinking about quantum fluctuations in the vacuum. At all energies probed by experiment to date, the world is accurately described as a set of quantum fields (at higher energies it may become strings or something else). If we take the Fourier transform of a free quantum field, each mode of fixed wavelength behaves like a simple harmonic oscillator. ("Free" means "noninteracting"; for our purposes this is a very good approximation.) As we know from elementary quantum mechanics, the ground-state or zero-point energy of an harmonic oscillator with potential V(x) = 1/2 2 x2 is E0 = 1/2 . Thus, each mode of a quantum field contributes to the vacuum energy, and the net result should be an integral over all of the modes. Unfortunately this integral diverges, so the vacuum energy appears to be infinite. However, the infinity arises from the contribution of modes with very small wavelengths; perhaps it was a mistake to include such modes, since we don't really know what might happen at such scales. To account for our ignorance, we could introduce a cutoff energy, above which ignore any potential contributions, and hope that a more complete theory will eventually provide a physical justification for doing so. If this cutoff is at the Planck scale, we recover the estimate (67).

The strategy of decomposing a free field into individual modes and assigning a zero-point energy to each one really only makes sense in a flat spacetime background. In curved spacetime we can still "renormalize" the vacuum energy, relating the classical parameter to the quantum value by an infinite constant. After renormalization, the vacuum energy is completely arbitrary, just as it was in the original classical theory. But when we use general relativity we are really using an effective field theory to describe a certain limit of quantum gravity. In the context of effective field theory, if a parameter has dimensions [mass]n, we expect the corresponding mass parameter to be driven up to the scale at which the effective description breaks down. Hence, if we believe classical general relativity up to the Planck scale, we would expect the vacuum energy to be given by our original guess (67).

However, we claim to have measured the vacuum energy (58). The observed value is somewhat discrepant with our theoretical estimate:

 (68)

This is the famous 120-orders-of-magnitude discrepancy that makes the cosmological constant problem such a glaring embarrassment. Of course, it is a little unfair to emphasize the factor of 10120, which depends on the fact that energy density has units of [energy]4. We can express the vacuum energy in terms of a mass scale,

 (69)

so our observational result is

 (70)

The discrepancy is thus

 (71)

We should think of the cosmological constant problem as a discrepancy of 30 orders of magnitude in energy scale.

In addition to the fact that it is very small compared to its natural value, the vacuum energy presents an additional puzzle: the coincidence between the observed vacuum energy and the current matter density. Our best-fit universe (64) features vacuum and matter densities of the same order of magnitude, but the ratio of these quantities changes rapidly as the universe expands:

 (72)

As a consequence, at early times the vacuum energy was negligible in comparison to matter and radiation, while at late times matter and radiation are negligible. There is only a brief epoch of the universe's history during which it would be possible to witness the transition from domination by one type of component to another.

To date, there are not any especially promising approaches to calculating the vacuum energy and getting the right answer; it is nevertheless instructive to consider the example of supersymmetry, which relates to the cosmological constant problem in an interesting way. Supersymmetry posits that for each fermionic degree of freedom there is a matching bosonic degree of freedom, and vice-versa. By "matching" we mean, for example, that the spin-1/2 electron must be accompanied by a spin-0 "selectron" with the same mass and charge. The good news is that, while bosonic fields contribute a positive vacuum energy, for fermions the contribution is negative. Hence, if degrees of freedom exactly match, the net vacuum energy sums to zero. Supersymmetry is thus an example of a theory, other than gravity, where the absolute zero-point of energy is a meaningful concept. (This can be traced to the fact that supersymmetry is a spacetime symmetry, relating particles of different spins.)

We do not, however, live in a supersymmetric state; there is no selectron with the same mass and charge as an electron, or we would have noticed it long ago. If supersymmetry exists in nature, it must be broken at some scale Msusy. In a theory with broken supersymmetry, the vacuum energy is not expected to vanish, but to be of order

 (73)

with vac = Mvac4. What should Msusy be? One nice feature of supersymmetry is that it helps us understand the hierarchy problem - why the scale of electroweak symmetry breaking is so much smaller than the scales of quantum gravity or grand unification. For supersymmetry to be relevant to the hierarchy problem, we need the supersymmetry-breaking scale to be just above the electroweak scale, or

 (74)

In fact, this is very close to the experimental bound, and there is good reason to believe that supersymmetry will be discovered soon at Fermilab or CERN, if it is connected to electroweak physics.

Unfortunately, we are left with a sizable discrepancy between theory and observation:

 (75)

Compared to (71), we find that supersymmetry has, in some sense, solved the problem halfway (on a logarithmic scale). This is encouraging, as it at least represents a step in the right direction. Unfortunately, it is ultimately discouraging, since (71) was simply a guess, while (75) is actually a reliable result in this context; supersymmetry renders the vacuum energy finite and calculable, but the answer is still far away from what we need. (Subtleties in supergravity and string theory allow us to add a negative contribution to the vacuum energy, with which we could conceivably tune the answer to zero or some other small number; but there is no reason for this tuning to actually happen.)

But perhaps there is something deep about supersymmetry which we don't understand, and our estimate Mvac ~ Msusy is simply incorrect. What if instead the correct formula were

 (76)

In other words, we are guessing that the supersymmetry-breaking scale is actually the geometric mean of the vacuum scale and the Planck scale. Because MP is fifteen orders of magnitude larger than Msusy, and Msusy is fifteen orders of magnitude larger than Mvac, this guess gives us the correct answer! Unfortunately this is simply optimistic numerology; there is no theory that actually yields this answer (although there are speculations in this direction [75]). Still, the simplicity with which we can write down the formula allows us to dream that an improved understanding of supersymmetry might eventually yield the correct result.

As an alternative to searching for some formula that gives the vacuum energy in terms of other measurable parameters, it may be that the vacuum energy is not a fundamental quantity, but simply our feature of our local environment. We don't turn to fundamental theory for an explanation of the average temperature of the Earth's atmosphere, nor are we surprised that this temperature is noticeably larger than in most places in the universe; perhaps the cosmological constant is on the same footing. This is the idea commonly known as the "anthropic principle."

To make this idea work, we need to imagine that there are many different regions of the universe in which the vacuum energy takes on different values; then we would expect to find ourselves in a region which was hospitable to our own existence. Although most humans don't think of the vacuum energy as playing any role in their lives, a substantially larger value than we presently observe would either have led to a rapid recollapse of the universe (if vac were negative) or an inability to form galaxies (if vac were positive). Depending on the distribution of possible values of vac, one can argue that the observed value is in excellent agreement with what we should expect [76, 77, 78, 79, 80, 81, 82].

The idea of environmental selection only works under certain special circumstances, and we are far from understanding whether those conditions hold in our universe. In particular, we need to show that there can be a huge number of different domains with slightly different values of the vacuum energy, and that the domains can be big enough that our entire observable universe is a single domain, and that the possible variation of other physical quantities from domain to domain is consistent with what we observe in ours.

Recent work in string theory has lent some support to the idea that there are a wide variety of possible vacuum states rather than a unique one [83, 84, 85, 86, 87, 88]. String theorists have been investigating novel ways to compactify extra dimensions, in which crucial roles are played by branes and gauge fields. By taking different combinations of extra-dimensional geometries, brane configurations, and gauge-field fluxes, it seems plausible that a wide variety of states may be constructed, with different local values of the vacuum energy and other physical parameters. An obstacle to understanding these purported solutions is the role of supersymmetry, which is an important part of string theory but needs to be broken to obtain a realistic universe. From the point of view of a four-dimensional observer, the compactifications that have small values of the cosmological constant would appear to be exactly the states alluded to earlier, where one begins with a supersymmetric state with a negative vacuum energy, to which supersymmetry breaking adds just the right amount of positive vacuum energy to give a small overall value. The necessary fine-tuning is accomplished simply by imagining that there are many (more than 10100) such states, so that even very unlikely things will sometimes occur. We still have a long way to go before we understand this possibility; in particular, it is not clear that the many states obtained have all the desired properties [89].

Even if such states are allowed, it is necessary to imagine a universe in which a large number of them actually exist in local regions widely separated from each other. As is well known, inflation works to take a small region of space and expand it to a size larger than the observable universe; it is not much of a stretch to imagine that a multitude of different domains may be separately inflated, each with different vacuum energies. Indeed, models of inflation generally tend to be eternal, in the sense that the universe continues to inflate in some regions even after inflation has ended in others [90, 91]. Thus, our observable universe may be separated by inflating regions from other "universes" which have landed in different vacuum states; this is precisely what is needed to empower the idea of environmental selection.

Nevertheless, it seems extravagant to imagine a fantastic number of separate regions of the universe, outside the boundary of what we can ever possibly observe, just so that we may understand the value of the vacuum energy in our region. But again, this doesn't mean it isn't true. To decide once and for all will be extremely difficult, and will at the least require a much better understanding of how both string theory (or some alternative) and inflation operate - an understanding that we will undoubtedly require a great deal of experimental input to achieve.