**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
*M*_{p} ~ 10^{18} 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}
*x*^{2} is *E*_{0} = 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
10^{120}, 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 *M*_{susy}. In a theory with broken
supersymmetry,
the vacuum energy is not expected to vanish, but to be of order

(73) |

with
_{vac} =
*M*_{vac}^{4}. What should *M*_{susy}
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
*M*_{vac} ~ *M*_{susy} 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 *M*_{P} is fifteen orders of magnitude larger
than *M*_{susy}, and *M*_{susy} is fifteen
orders of magnitude larger than
*M*_{vac}, 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 10^{100}) 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.