**E. Dark energy**

The idea that the universe contains close to homogeneous dark energy
that approximates a time-variable cosmological "constant" arose
in particle physics, through the discussion of phase transitions in
the early universe and through the search for a dynamical
cancellation of the vacuum energy density;
in cosmology, through the discussions of how to reconcile a
cosmologically flat universe with the small mass density
indicated by galaxy peculiar velocities; and on both
sides by the thought that
might be very
small now because it has been rolling toward zero for a very long time.
^{44}

The idea that the dark energy is decaying by emission of matter or radiation is now strongly constrained by the condition that the decay energy must not significantly disturb the spectrum of the 3 K cosmic microwave background radiation. But the history of the idea is interesting, and decay to dark matter still a possibility, so we comment on both here. The picture of dark energy in the form of defects in cosmic fields has not received much attention in recent years, in part because the computations are difficult, but might yet prove to be productive. Much discussed nowadays is dark energy in a slowly varying scalar field. The idea is reviewed at some length here and in even more detail in the Appendix. We begin with another much discussed approach: prescribe the dark energy by parameters in numbers that seem fit for the quality of the measurements.

In the XCDM parametrization
the dark energy interacts only with itself and gravity,
the dark energy density _{X}(*t*) > 0 is approximated as a
function of world time alone, and the pressure is written as

(43) |

an expression that has come to be known as the cosmic
equation of state. ^{45}
Then the local energy conservation equation (9) is

(44) |

If *w*_{X} is constant the dark energy density scales with
the expansion factor as

(45) |

If *w*_{X} < - 1/3 the dark energy makes a positive
contribution
to /*a*
(Eq. [8]). If *w*_{X} = - 1/3 the dark energy has no effect on
, and
the energy density varies as
_{X}
1 /
*a*^{2}, the same as the space curvature term in
^{2} /
*a*^{2} (Eq. [11]).
That is, the expansion time histories are the same in an open
model with no dark energy and in a spatially-flat model with
*w*_{X} = - 1/3, although the spacetime geometries
differ. ^{46} If
*w*_{X} < - 1 the dark energy density is increasing.
^{47}

Equation (45) with constant *w*_{X} has the great
advantage of simplicity. An appropriate generalization for the
more precise measurements to come might be guided by the idea that
the dark energy density is close to homogeneous, spatial
variations rearranging themselves at close to the speed of light,
as in the scalar field models discussed below. Then for most of
the cosmological tests we have an adequate general description of
the dark energy if we let *w*_{X} be a free function of
time. ^{48} In scalar field
pictures *w*_{X} is derived from
the field model; it can be a complicated function of time
even when the potential is a simple function of the scalar field.

The analysis of the large-scale anisotropy of the 3 K cosmic microwave
background radiation requires a prescription for how the spatial
distribution of the dark energy is gravitationally related to
the inhomogeneous distribution of other matter and radiation
(Caldwell et al., 1998).
In XCDM this requires at least one
more parameter, an effective speed of sound, with
*c*^{2}_{sX} > 0
(for stability, as discussed in Sec. II.B),
in addition to *w*_{X}.

**2. Decay by emission of matter or radiation**

Bronstein (1933)
introduced the idea that the dark energy density
_{} is
decaying by the emission of matter or
radiation. The continuing discussions of this and the associated
idea of decaying dark matter
(Sciama, 2001,
and references therein) are testimony to the appeal. Considerations in
the decay of dark energy include the
effect on the formation of light elements at
*z* ~ 10^{10}, the contribution to the
-ray or
optical extragalactic
background radiation, and the perturbation to the spectrum of
the 3 K cosmic microwave background radiation.
^{49}

The effect on the 3 K cosmic microwave background was of
particular interest a
decade ago, as a possible explanation of indications of a
significant departure from a Planck spectrum. Precision
measurements now show the spectrum is very close to thermal.
The measurements and their interpretation are discussed by
Fixsen et al. (1996).
They show that the allowed addition to the
3 K cosmic microwave background energy density
_{R}
is limited to just
_{R} /
_{R}
10^{-4}
since redshift *z* ~ 10^{5}, when the interaction between
matter and radiation was last strong enough for thermal relaxation. The
bound on
_{R} /
_{R} is
not inconsistent with what
the galaxies are thought to produce, but it is well below an
observationally interesting dark energy density.

Dark energy could decay by emission of dark matter, cold or hot,
without disturbing the spectrum of the 3 K cosmic microwave background
radiation. For example, let us suppose the dark energy
equation of state is *w*_{X} = - 1, and hypothetical
microphysics causes the dark energy density to decay as
_{}
*a*^{-n}
by the production of nonrelativistic dark matter. Then Bronstein's
Eq. (36) says the dark matter density varies with time as

(46) |

where *A* is a constant and 0 < *n* < 3. In the late
time limit the
dark matter density is a fixed fraction of the dark energy. But
for the standard interpretation of the measured anisotropy of the
3 K background we would have to suppose the first term on the
right hand side of Eq. (46) is not much smaller than
the second, so the coincidences issue discussed in
Sec. III.B.2
is not much relieved. It does help relieve the problem with the small
present value of
_{} (to
be discussed in connection with Eq. [47]).

We are not aware of any work on this decaying dark energy picture. Attention instead has turned to the idea that the dark energy density evolves without emission, as illustrated in Eq. (45) and the two classes of physical models to be discussed next.

The physics and cosmology of topological defects produced at
phase transitions in the early universe are reviewed by
Vilenkin and Shellard
(1994).
An example of dark energy is a tangled web
of cosmic string, with fixed mass per unit length, which
self-intersects without reconnection. In
Vilenkin's (1984)
analysis ^{50}
the mean mass density in strings scales as
_{string}
(*ta*(*t*))^{-1}. When ordinary matter is
the dominant contribution to
^{2} /
*a*^{2}, the ratio of mass densities is
_{string}
/
*t*^{1/3}. Thus at late
times the string mass dominates. In this limit,
_{string}
*a*^{-2}, *w*_{X} = - 1/3 for the XCDM
parametrization of Eq. (45), and the universe expands as
*a* *t*.
Davis (1987) and
Kamionkowski and Toumbas
(1996)
propose the same behavior for a texture model.
One can also imagine domain walls fill space densely enough not
to be dangerous. If the domain walls are fixed in comoving
coordinates the domain wall energy density scales as
_{X}
*a*^{-1}
(Zel'dovich, Kobzarev, and
Okun, 1974;
Battye, Bucher, and
Spergel, 1999).
The corresponding
equation of state parameter is *w*_{X} = - 2/3, which is
thought to be easier to reconcile
with the supernova measurements than *w*_{X} = - 1/3
(Garnavich et al.,
1998;
Perlmutter et al.,
1999a).
The cosmological tests of defects models for the dark energy have
not been very thoroughly explored, at least in part because an
accurate treatment of the behavior of the dark energy is
difficult (as seen, for example, in
Spergel and Pen, 1997;
Friedland, Muruyama, and
Perelstein, 2002),
but this class of models is worth bearing in mind.

At the time of writing the popular picture for dark energy is a
classical scalar field with a self-interaction potential
*V*()
that is shallow enough that the field energy density decreases
with the expansion of the universe more slowly than the energy
density in matter. This idea grew in part out of the inflation
scenario, in part from ideas from particle physics. Early examples are
Weiss (1987) and
Wetterich (1988).
^{51}
The former considers a quadratic potential with an ultralight
effective mass, an idea that reappears in
Frieman et al. (1995).
The latter considers the time variation
of the dark energy density in the case of the
Lucchin and Matarrese
(1985a)
exponential self-interaction potential
(Eq. [38]). ^{52}

In the exponential potential model the scalar field
energy density varies with time in constant proportion to the
dominant energy density. The evidence is that radiation dominates at
redshifts in the range
10^{3}
*z*
10^{10}, from the success of the standard model for light element
formation, and matter dominates at
1 *z*
10^{3},
from the success of the standard model for the gravitational growth of
structure. This would leave the dark energy
subdominant today, contrary to what is wanted. This led to the
proposal of the inverse power-law potential in
Eq. (31) for a single real scalar field.
^{53}

We do not want the hypothetical field
to couple too
strongly to baryonic matter and fields, because that would
produce a "fifth force" that is not observed.
^{54}^{,} ^{55}
Within quantum field theory the inverse
power-law scalar field potential makes the model non-renormalizable
and thus pathological. But the model is meant to describe what might
emerge out of a more fundamental quantum theory, which maybe also
resolves the physicists' cosmological constant problem
(Sec. III.B), as the effective
classical description of the dark energy.
^{56} The potential of this
classical effective field is chosen *ad hoc*,
to fit the scenario. But one can adduce analogs within supergravity,
superstring/M, and brane theory, as reviewed in the
Appendix.

The solution for the mass fraction in dark energy in the inverse
power-law potential model (in Eq. [33] when
_{} <<
, and the
numerical solution at lower
redshifts) is not unique, but it behaves as what has come to be
termed an attractor or tracker: it is the asymptotic solution
for a broad range of initial conditions.
^{57}
The solution also has the property that
_{} is
decreasing, but less rapidly than the mass densities in matter
and radiation. This may help alleviate two troubling aspects of
the cosmological constant. The coincidences issue is discussed
in Sec. III.B. The other is the
characteristic energy scale set
by the value of ,

(47) |

when
_{R0} and
_{K0} may be
neglected. In the limit of constant
dark energy density, cosmology seems to indicate new
physics at an energy scale more typical of chemistry. If
_{} is
rolling toward zero the energy
scale might look more reasonable, as follows
(Peebles and Ratra,
1988;
Steinhardt et al., 1999;
Brax et al., 2000).

Suppose that as conventional inflation ends the scalar field
potential switches over to the inverse power-law form in
Eq. (31). Let the energy scale at the end of inflation be
(*t*_{I})
= (*t*_{I})^{1/4},
where (*t*_{I}) is the energy density in matter and
radiation at the end of inflation, and let
_{}(*t*_{I}) be the energy
scale of the dark energy at the end of inflation. Since the
present value _{}(*t*_{0}) of the dark
energy scale (Eq. [47]) is comparable to the
present energy scale belonging to the matter, we have from Eq. (33)

(48) |

For parameters of common inflation models,
(*t*_{I})
~ 10^{13} GeV, and
_{}(*t*_{0}) /
(*t*_{I})
~ 10^{-25}. If, say,
= 6, then

(49) |

As this example illustrates, one can arrange the scalar field
model so it has a characteristic energy scale that exceeds the
energy ~ 10^{3} GeV below which physics is thought to be
well understood: in this model cosmology does not
force upon us the idea that there is as yet undiscovered
physics at the very small energy in Eq. (47).
Of course, where the factor ~ 10^{-6} in Eq. (49)
comes from still is an open question, but, as discussed in the
Appendix, perhaps easier to resolve than the
origin of the factor ~ 10^{-25} in the constant
case.

When we can describe the dynamics of the departure from a
spatially homogeneous field in linear perturbation theory,
a scalar field model generally is characterized by the
time-dependent values of *w*_{X} (Eq. [43]) and
the speed of sound *c*_{sX} (e.g.,
Ratra, 1991;
Caldwell et al., 1998).
In the inverse power-law
potential model the relation between the power-law index
and the equation of state parameter in the matter-dominated epoch is
independent of time
(Ratra and Quillen,
1992),

(50) |

When the dark energy density starts to make an appreciable
contribution to the expansion rate the parameter *w*_{X}
starts to evolve. The use of a constant value of *w*_{X} to
characterize the inverse power-law potential model thus can be
misleading. For example,
Podariu and Ratra (2000,
Fig. 2) show that, when applied to the Type Ia supernova measurements, the
XCDM parametrization in Eq. (50) leads to a significantly tighter
apparent upper limit on *w*_{X}, at fixed
_{M0}, than
is warranted by the
results of a computation of the evolution of the dark energy
density in this scalar field model.
Caldwell et al. (1998)
deal with the relation between scalar field models and the XCDM
parametrization by fixing *w*_{X}, as a constant or some
function of redshift, deducing the scalar field potential
*V*() that produces
this *w*_{X}, and then computing the
gravitational response of
to the large-scale mass
distribution.

^{44} This last idea is similar in spirit to
Dirac's (1937,
1938)
attempt to explain the large dimensionless numbers of physics.
He noted that the gravitational force between two
protons is much smaller than the electromagnetic force,
and that that might be because the gravitational constant *G*
is decreasing in inverse proportion to the world time. This is the
earliest discussion we know of what has come to be called the
hierarchy problem, that is, the search for a mechanism that might
be responsible for the large ratio between a possibly more fundamental
high energy scale, for example, that of grand unification or the
Planck scale
(where quantum gravitational effects become significant) and a
lower possibly less fundamental energy scale, for example that of
electroweak unification (see, for example,
Georgi, Quinn, and
Weinberg, 1974).
The hierarchy problem in particle physics
may be rephrased as a search for a mechanism to prevent the
light electroweak symmetry breaking Higgs scalar field mass from being
large because of a quadratically divergent quantum mechanical
correction (see, for example,
Susskind, 1979).
In this sense it is similar in spirit to the physicists' cosmological
constant problem of Sec. III.B.
Back.

^{45} Other parametrizations of dark energy
are discussed by
Hu (1998)
and Bucher and Spergel
(1999).
The name, XCDM, for the case
*w*_{X} < 0 in Eq. (43), was introduced by
Turner and White (1997).
There is a long history in
cosmology of applications of such an equation of state, and the
related evolution of
_{};
examples are
Canuto et al. (1977),
Lau (1985),
Huang (1985),
Fry (1985),
Hiscock (1986),
Özer and Taha
(1986),
and Olson and Jordan
(1987).
See Ratra and Peebles
(1988)
for references to other early work
on a time-variable
and
Overduin and Cooperstock
(1998)
and Sahni and Starobinsky
(2000)
for reviews. More recent discussions of this and related models may
be found in
John and Joseph (2000),
Zimdahl et al. (2001),
Dalal et al. (2001),
Gudmundsson and
Björnsson (2002),
Bean and Melchiorri
(2002),
Mak, Belinchón, and
Harko (2002),
and Kujat et al. (2002),
through which other recent work may be traced.
Back.

^{46} As discussed in
Sec. IV, it
appears difficult to reconcile the case *w*_{X} = - 1/3
with the Type Ia supernova apparent magnitude data
(Garnavich et al., 1998;
Perlmutter et al.,
1999a).
Back.

^{47} This is quite a step from the thought
that the dark energy
density is small because it has been rolling to zero for a long
time, but the case has found a context
(Caldwell, 2002;
Maor et al., 2002).
Such models were first discussed in the context of inflation (e.g.,
Lucchin and Matarrese,
1985b),
where it was shown that
the *w*_{X} < - 1 component could be modeled as a scalar
field with a negative kinetic energy density
(Peebles, 1989a).
Back.

^{48} The availability of a free function
greatly complicates the search for tests as opposed to curve fitting!
This is clearly illustrated by
Maor et al. (2002).
For more examples see
Perlmutter, Turner, and
White (1999b)
and Efstathiou (1999).
Back.

^{49} These
considerations generally are phenomenological: the evolution of
the dark energy density, and its related coupling to matter or
radiation, is assigned rather than derived from an action
principle. Recent discussions include
Pollock (1980),
Kazanas (1980),
Freese et al. (1987),
Gasperini (1987),
Sato, Terasawa, and
Yokoyama (1989),
Bartlett and Silk (1990),
Overduin, Wesson, and
Bowyer (1993),
Matyjasek (1995),
and Birkel and Sarkar
(1997).
Back.

^{50} The string flops at speeds comparable
to light, making the coherence
length comparable to the expansion time *t*. Suppose a string
randomly walks across a region of physical size
*a*(*t*)*R* in *N* steps, where
*aR* ~ *N*^{1/2}*t*. The total length of this
string within the region *R* is *l* ~ *Nt*. Thus the mean
mass density of the string scales with time as
_{string}
*l* /
*a*^{3}
(*ta*(*t*))^{-1}. One
randomly walking string does not fill space, but we can imagine
many randomly placed strings produce a nearly smooth mass distribution.
Spergel and Pen (1997)
compute the 3 K cosmic microwave background radiation
anisotropy in a related model, where the string network is
fixed in comoving coordinates so the mean mass density scales as
_{string}
*a*^{-2}.
Back.

^{51} Other early examples include those
cited in
Ratra and Peebles (1988)
as well as
Endo and Fukui (1977),
Fujii (1982),
Dolgov (1983),
Nilles (1985),
Zee (1985),
Wilczek (1985),
Bertolami (1986),
Ford (1987),
Singh and Padmanabhan
(1988),
and Barr and Hochberg
(1988).
Back.

^{52} For recent discussions of this model see
Ferreira and Joyce
(1998),
Ott (2001),
Hwang and Noh (2001),
and references therein.
Back.

^{53} In what follows we focus on this
model, which was proposed by
Peebles and Ratra
(1988).
The model assumes a conventionally
normalized scalar field kinetic energy and spatial gradient term
in the action, and it assumes the scalar field is coupled only to
itself and gravity. The model is then completely characterized by
the form of the potential (in addition to all the other usual
cosmological parameters, including initial conditions). Models
based on other forms for
*V*(), with a more
general kinetic energy and spatial gradient term, or with more general
couplings to gravity and other fields, are discussed in the
Appendix.
Back.

^{54} The current value of the mass
associated with spatial inhomogeneities in the field is
*m*_{}(*t*_{0}) ~ *H*_{0} ~
10^{-33} eV, as one would expect from the dimensions. More
explicitly, one arrives at this mass by writing the field as
(*t*,
) =
<>(*t*) +
(*t*,
) and Taylor expanding
the scalar field potential energy density
*V*() about the
homogeneous mean background
<> to quadratic
order in the spatially inhomogeneous part
, to get
*m*_{}^{2} =
*V*"(<>).
Within the
context of the inverse power-law model, the tiny value of the mass
follows from the requirements that *V* varies slowly with the field
value and that the current value of *V* be observationally
acceptable. The difference between the roles of
*m*_{} and the constant *m*_{q} in the quadratic
potential model *V* = *m*_{q}^{2}
^{2} / 2 is
worth noting. The mass
*m*_{q} has an assigned and arguably fine-tuned value. The
effective mass *m*_{} ~ *H* belonging to
*V*
^{-} is a
derived quantity, that evolves as the universe expands. The
small value of
*m*_{}(*t*_{0}) explains why the scalar field
energy cannot be concentrated with the non-relativistic mass in galaxies
and clusters of galaxies. Because of the tiny mass a scalar field
would mediate a new long-range fifth force if it were
not weakly coupled to ordinary matter. Weak coupling
also ensures that the contributions to coupling constants (such
as the gravitational constant) from the exchange of dark energy
bosons are small, so such coupling constants are not significantly
time variable in this model. See, for example,
Carroll (1998),
Chiba (1999),
Horvat (1999),
Amendola (2000),
Bartolo and Pietroni
(2000),
and Fujii (2000)
for recent discussions of this and related issues.
Back.

^{55} Coupling between dark energy and
dark matter is not constrained by conventional fifth force
measurements. An example is discussed by
Amendola and
Tocchini-Valentini (2002).
Perhaps the first consideration is
that the fifth-force interaction between neighboring dark matter halos
must not be so strong as to shift regular galaxies of stars away
from the centers of their dark matter halos.
Back.

^{56} Of course, the zero-point energy of
the quantum-mechanical
fluctuations around the mean field value contributes to the physicists'
cosmological constant problem, and renormalization of the
potential could destroy the attractor solution (however, see
Doran and Jäckel,
2002)
and could generate couplings between
the scalar field and other fields leading to an observationally
inconsistent "fifth force". The problems within quantum field
theory with the idea that the energy of a classical scalar field
is the dark energy, or drives inflation, are further discussed
in the Appendix. The best we can hope is
that the effective classical model is a useful approximation to what
actually is happening, which might lead us to a more satisfactory
theory.
Back.

^{57} A recent discussion is in
Brax and Martin (2000).
Brax, Martin, and Riazuelo
(2000)
present a thorough analysis of the evolution of
spatial inhomogeneities in the inverse power-law scalar field potential
model and confirm that these inhomogeneities do not destroy the homogeneous
attractor solution. For other recent discussions of attractor solutions
in a variety of contexts see
Liddle and Scherrer
(1999),
Uzan (1999),
de Ritis et al. (2000),
Holden and Wands
(2000),
Baccigalupi, Matarrese,
and Perrotta (2000),
and Huey and Tavakol
(2002).
Back.