**5.1. The variable cosmological "constant"**

The cosmological-constant problem is essentially the problem of reconciling
the very high vacuum-energy densities expected on the basis of quantum
field theory with the small (but nonzero) dark-energy density now inferred
from cosmological observation (Sec. 4.5).
Many authors have sought to bridge the gap by looking for a mechanism
that would allow the energy density
_{v} of
the vacuum to *decay* with time. Since
*c*^{2} = 8 *G*
_{v}
from (23), this
means replacing Einstein's cosmological constant by a variable
"cosmological term." With such a mechanism in hand, the problem
would be reduced to explaining why the Universe is of intermediate age:
old enough that
has relaxed from primordial values like those
suggested by quantum field theory to the values which we measure now,
but young enough that
_{}
_{v} /
_{crit}
has not yet reached its asymptotic value of unity.

Energy conservation requires that any decrease in the energy density of the vacuum be made up by a corresponding increase somewhere else. In some scenarios, dark energy goes into the kinetic energy of new forms of matter such as scalar fields, which have yet to be observed in nature. In others it is channelled instead into baryons, photons or neutrinos. Baryonic decays would produce equal amounts of matter and antimatter, whose subsequent annihilation would flood the Universe with -rays. Radiative decays would similarly pump photons into intergalactic space, but are harder to constrain because they could in principle involve any part of the electromagnetic spectrum. As we will see, however, robust limits can be set on any such process under conservative assumptions.

But how can ,
originally introduced by Einstein in 1917 as a
constant of nature akin to *c* and *G*, be allowed to vary? To
answer this, we go back to the field equations of general relativity:

(106) |

The covariant derivative of these equations can be written in the following
form with the help of the Bianchi identities, which read
^{}(_{µ} - 1/2
*g*_{µ}) = 0:

(107) |

Within Einstein's theory, it follows that
= constant as long
as matter and energy (as contained in
_{µ}) are conserved.

In variable- theories, one must therefore do one of three things: abandon matter-energy conservation, modify general relativity, or stretch the definition of what is conserved. The first of these routes was explored as early as 1933 by Bronstein [145], who sought to connect energy non-conservation with the cosmological arrow of time. Bronstein was executed in Stalin's Soviet Union a few years later, and his work is not widely known [146].

Today, few physicists would be willing to sacrifice energy conservation outright. Some, however, would be willing to modify general relativity, or to consider new forms of matter and energy. Historically, these two approaches have sometimes been seen as distinct, with one being a change to the "geometry of nature" while the other is concerned with the material content of the Universe. The modern tendency, however, is to regard them as equivalent. This viewpoint is best personified by Einstein, who in 1936 compared the left-hand (geometrical) and right-hand (matter) sides of his field equations to "fine marble" and "low-grade wooden" wings of the same house [147]. In a more complete theory, he argued, matter fields of all kinds would be seen to be just as geometrical as the gravitational one.

**5.2. Models based on scalar fields**

Let us see how this works in one of the oldest and simplest
variable-
theories: a modification of general relativity in
which the metric tensor
*g*_{µ} is supplemented by a scalar field
whose
coupling to matter is determined by a parameter
.
Ideas of this kind go back to Jordan in 1949
[148],
Fierz in 1956
[149]
and Brans and Dicke in 1961
[150].
In those days, of course, new scalar fields were not bandied about
as freely as they are today, and all these authors sought to associate
with a
known quantity.
Various lines of argument (notably Mach's principle) pointed to an
identification with Newton's gravitational "constant" such that
*G* ~ 1 /
. By 1968
it was appreciated that
and
too would depend on
in general
[151].
The original Brans-Dicke theory (with
= 0) has subsequently
been extended to *generalized scalar-tensor theories* in which
=
()
[152],
=
(),
=
()
[153]
and =
(,
),
=
() where
^{µ}
^{µ}
[154].
In the last and most general of these cases, the field equations read

(108) |

where
^{µ}(^{µ}
) is the
D'Alembertian. These reduce to Einstein's equations (106) when
= const
= 1/*G*.

If we now repeat the exercise on the previous page and take the covariant derivative of the field equations (108) with the Bianchi identities, we obtain a generalized version of the equation (107) faced by Bronstein:

(109) |

Now energy conservation
(^{}
_{µ} = 0) no longer
requires
= const. In fact,
it is generally *incompatible* with
constant , unless
an extra condition is imposed on the terms inside
the curly brackets in (109). (This cannot come from the wave equation for
, which
merely confirms that the terms inside the
curly brackets sum to zero, in agreement with energy conservation.)
Similar conclusions hold for other scalar-tensor theories in
which is
no longer associated with *G*. Examples include
models with non-minimal couplings between
and the
curvature scalar
[155],
conformal rescalings of the metric tensor by functions of
[156]
and nonzero potentials
*V*()
[157,
158,
159].
(Theories of this last kind
are now known as *quintessence* models
[160]).
In each of these scenarios, the cosmological "constant"
becomes a dynamical variable.

In the modern approach to
variable-
cosmology, which goes back to Zeldovich in 1968
[161],
all extra terms of the kind just described -- including
-- are moved to the
right-hand side of the field equations (108), leaving only the
Einstein tensor
(_{µ} - 1/2
*g*_{µ})
to make up the "geometrical" left-hand side. The cosmological term,
along with scalar (or other) additional fields, are thus effectively
reinterpreted as *new kinds of matter*. Eqs. (108) then read

(110) |

Here ^{eff}_{µ} is an effective energy-momentum tensor
describing
the sum of ordinary matter plus whatever scalar (or other) fields have
been added to the theory. For generalized scalar-tensor theories as
described above, this could be written as
^{eff}_{µ}
_{µ} +
^{}_{µ} where
_{µ} refers to
ordinary matter and ^{}_{µ} to the scalar field. For the case with
=
() and
=
(),
for instance, the latter would be defined by (108) as

(111) |

The covariant derivative of the field equations (110) now reads

(112) |

Eq. (112) carries the same physical content as (109),
but is more general in form and can readily be extended to other theories.
Physically, it says that energy *is* conserved in
variable- cosmology --
where "energy" is now understood to refer to the energy of ordinary
matter along with that in any additional fields which may be present,
*and along with that in the vacuum*, as represented by
.
In general, the latter parameter can vary as it likes, so long as the
conservation equation (112) is satisfied.

It was noted at least as early as 1977 by Endo and Fukui
[152]
that the evolution of
in theories of
this kind can help with
the cosmological "constant" problem, in the sense of dropping from large
primordial values to ones like those seen today. These authors found
solutions for
(*t*)
such that
*t*^{-2}
when =
() and
= constant.
In precursors to the modern quintessence scenarios, Barr
[157]
found models in which
*t*^{-} at late times, while Peebles and Ratra
[158]
discussed a theory in which
*R*^{-m} at early ones (here
and *m* are powers).
There is now a rich literature on
-decay laws of
this kind
[113].
Their appeal is easy to understand, and can be
illustrated with a simple dimensional argument for the case with
*R*^{-2}
[162].
Since already has
dimensions of *L*^{-2}, the proportionality factor in this
case is a pure number (,
say) which is presumably of order unity. Taking
~ 1 and identifying
*R* with a suitable length scale
in cosmology (namely the Hubble distance *c* / *H*_{0}),
one finds that
_{0} ~
*H*_{0}^{2} / *c*^{2}. The present
vacuum density parameter is then
_{,0}
_{0}
*c*^{2} / 3*H*_{0}^{2} ~ 1/3, close to
the values implied by by the supernovae data
(Sec. 4.5). The most natural choice
*R* ~ _{Pl}
gives a primordial
-term of
_{Pl} ~
_{Pl}^{-2}.
It then follows that
_{Pl} /
_{0} ~
(*c* / *H*_{0}
_{Pl})^{2} ~
10^{122}, in good agreement
with the values suggested by Table 3.

**5.3. Theoretical and observational challenges**

While this would seem to be a promising approach, two cautions must be
kept in mind. The first is theoretical. Insofar as the mechanisms
discussed so far are entirely classical, they do not address the
underlying problem. For this, one would also need to explain why net
contributions to
from the *quantum vacuum* do not remain
at the primordial level, or how they are suppressed with time.
Polyakov
[163]
and S.L. Adler
[164]
in 1982 were the first
to speculate explicitly that such a suppression might come about if the
"bare" cosmological term implied by quantum field theory were
progressively screened by an "induced" counterterm of opposite sign,
driving the effective value of
(*t*) toward
zero at late times.
Many theoretical adjustment mechanisms have now been identified
as potential sources of such a screening effect, beginning with a 1983
suggestion by Dolgov
[165]
based on non-minimally coupled scalar
fields. Subsequent proposals have involved scalar fields
[166,
167,
168],
fields of higher spin
[169,
170,
171],
quantum effects during inflation
[172,
173,
174]
and other phenomena
[175,
176,
177].
In most of these cases, no analytic expression is found for
in
terms of time or other cosmological parameters; the intent is merely
to demonstrate that decay (and preferably near-cancellation) of the
cosmological term is possible in principle. None of these mechanisms
has been widely accepted as successful to date. In fact, there
is a general argument due to Weinberg to the effect that a successful
mechanism based on scalar fields would necessarily be so finely-tuned
as to be just as mysterious as the original problem
[178].
Similar concerns have been raised in the case of vector and tensor-based
proposals
[179].
Nevertheless, the idea of the adjustment
mechanism remains feasible in principle, and continues to attract
more attention than any other approach to the cosmological-constant
problem.

The second caution is empirical. Observational data place increasingly
strong restrictions on the way in which
can vary with time.
Among the most important are early-time bounds on the dark-energy
density
_{}
*c*^{2} =
*c*^{4} / 8
*G*. The success of standard primordial
nucleosynthesis theory implies that
_{} was
smaller than
_{r}
and _{m}
during the radiation-dominated era, and large-scale structure
formation could not have proceeded in the conventional way unless
_{}
<
_{m}
during the early matter-dominated era. Since
_{r}
*R*^{-4}
and _{m}
*R*^{-3}
from (32),
these requirements mean in practice that the dark-energy density must
climb *less steeply than* *R*^{-3} in the past
direction, if it is comparable to that of matter or radiation at present
[180,
181].
The variable- term
must also satisfy late-time bounds like those
which have been placed on the cosmological constant
(Sec. 4.5). Tests of this kind have
been carried out using data on the age of the Universe
[182,
183],
structure formation
[184,
185,
186],
galaxy number counts
[187],
the CMB power spectrum
[188,
189],
gravitational lensing statistics
[189,
190,
191]
and Type Ia supernovae
[189,
192].
Some of these tests are less restrictive in the case of a variable
-term than they
are for = const,
and this can open up
new regions of parameter space. Observation may even be compatible with
some nonsingular models whose expansion originates in a hot, dense
"big bounce" rather than a big bang
[193],
a possibility which can be ruled out on general grounds if
= constant.

A third group of limits comes from asking what the vacuum decays
*into*. In quintessence theories, dark energy is transferred
to the kinetic energy of a scalar field as it "rolls" down a gradient
toward the minimum of its potential. This may have observable consequences
if the scalar field is coupled strongly to ordinary matter, but is hard
to constrain in general. A simpler situation is that in which the
vacuum decays into known particles such as baryons, photons or neutrinos.
The baryonic decay channel would produce excessive levels of
-ray
background radiation due to matter-antimatter annihilation
unless the energy density of the vacuum component is less than
3 × 10^{-5} times that of matter
[180].
This limit can be weakened if the decay process violates baryon number,
or if it takes place in such a way that matter and antimatter are
segregated on large scales, but such conditions are hard to arrange
in a natural way. The radiative decay channel is more promising,
but also faces a number of tests. The decay process should meet
certain criteria of thermodynamic stability
[194]
and adiabaticity
[195].
The *shape* of the spectrum of decay photons must
not differ too much from that of pre-existing background radiation, or
distortions will arise. Freese have argued on this basis that
the energy density of a vacuum decaying primarily into low-energy photons
could not exceed 4 × 10^{-4} times that of radiation
[180].

It may be, however, that vacuum-decay photons blend into the spectrum
of background radiation without distorting it.
Fig. 1
shows that the best place to "hide" the evidence of such a process
would be the microwave region, where the energy density of background
radiation is highest. Could all or part of the CMB be due to dark-energy
decay? We know from the COBE satellite that its spectrum
is very nearly that of a perfect blackbody
[196].
Freese pointed out that vacuum-decay photons
would be thermalized efficiently by brehmsstrahlung and double-Compton
scattering in the early Universe, and might continue to assume a
blackbody spectrum at later times if pre-existing CMB photons played
a role in "inducing" the vacuum to decay
[180].
Subsequent work has shown that this would require a special combination
of thermodynamical parameters
[197].
This possibility is important
in practice, however, because it leads to the most conservative limits
on the theory. Even if the radiation produced by decaying dark energy
does not distort the background, it will contribute to the latter's
*absolute intensity*. We can calculate the size of these
contributions to the background radiation using the methods that have
been laid out in Secs. 2 and
3.

The first step in this problem is to solve the field equations and conservation equations for the energy density of the decaying vacuum. We will do this in the context of a general phenomenological model. This means that we retain the field equations (110) and the conservation law (112) without specifying the form of the effective energy-momentum tensor in terms of scalar (or other) fields. These equations may be written

(113) | |||

(114) |

Here _{}
*c*^{ 2}
*c*^{ 4} / 8
*G* from (23) and we have replaced 1 /
with
*G* (assumed to be constant in what follows).
Eqs. (113) and (114) have the same form as their
counterparts (106) and (29) in standard cosmology, the key
difference being that the cosmological term has migrated to the right-hand
side and is *no longer necessarily constant*. Its evolution
is now governed by the conservation equations (114), which require
only that any change in
_{}
*c*^{2}
*g*_{µ} be balanced by an equal and
opposite change in the energy-momentum tensor
^{eff}_{µ}.

While the latter is model-dependent in general, it is reasonable to assume in the context of isotropic and homogeneous cosmology that its form is that of a perfect fluid, as given by (26):

(115) |

Comparison of Eqs. (114) and (115) shows that
the conserved quantity in (114) must then *also* have the
form of a perfect-fluid energy-momentum tensor, with density and
pressure given by

(116) |

The conservation law (114) may then be simplified at once by analogy with Eq. (29):

(117) |

This reduces to the standard result (30) for the case of a constant
cosmological term,
_{} =
const. Throughout Sec. 5, we allow
the cosmological term to contain both a constant part *and*
a time-varying part so that

(118) |

Let us assume in addition that the perfect fluid described by
^{eff}_{µ}
consists of a mixture of dust-like matter
(*p*_{m} = 0) and radiation
(*p*_{r} = 1/3
_{r}
*c*^{ 2}):

(119) |

The conservation equation (117) then reduces to

(120) |

From this equation it is clear that one (or both) of the radiation and
matter densities can no longer obey the usual relations
_{r}
*R*^{-4}
and _{m}
*R*^{-3}
in a theory with
const. Any change in
(or
_{})
must be accompanied by a change in radiation and/or matter densities.

To go further, some simplifying assumptions must be made. Let us take to begin with:

(121) |

This is just conservation of particle number, as may be seen by
replacing "galaxies" with "particles" in Eq. (6). Such an
assumption is well justified during the matter-dominated era by the
stringent constraints on matter creation discussed in
Sec. 5.1. It is equally well justified during the
radiation-dominated era, when the matter density is small so that the
_{m}
term is of secondary importance compared to the other terms in (120)
in any case.

In light of Eqs. (120) and (121), the vacuum
can exchange energy only with radiation. As a model for this process,
let us follow Pollock in 1980
[198]
and assume that it takes
place in such a way that the energy density of the decaying vacuum
component remains proportional to that of radiation,
_{v}
_{r}.
We adopt the notation of Freese in 1987 and write the
proportionality factor as *x* / (1 - *x*) with *x* the
coupling parameter of the theory
[180].
If this is allowed to take (possibly different) constant values during
the radiation and matter-dominated eras, then

(122) |

Here *t*_{eq} refers to the epoch of matter-radiation
equality when
_{r} =
_{m}.
Standard cosmology is recovered in the limit
*x* 0. The most
natural situation is that in which the value
of *x* stays constant, so that *x*_{r} =
*x*_{m}. However, since observational
constraints on *x* are in general different for the radiation and
matter-dominated eras, the most conservative limits on the theory are
obtained by letting *x*_{r} and *x*_{m} take
different values. Physically,
this would correspond to a phase transition or sudden change in the
expansion rate /
*R* of the Universe at *t* = *t*_{eq}.

With Eqs. (121) and (122), the conservation equation (120) reduces to

(123) |

where overdots denote derivatives with respect to time. Integration gives

(124) |

where _{v} is a
constant. The cosmological term
is thus an
inverse power-law function of the scale factor *R*, a scenario that has
received wide attention also in models where vacuum energy is not
proportional to that of radiation
[113].
Eq. (124) shows that the conserved quantity in this theory has a form
intermediate between that of ordinary radiation entropy
(*R*^{ 4}_{r}) and particle number
(*R*^{ 3}_{m}) when 0 < *x* < 1/4.

The fact that
_{r}
_{v}
*R*^{-4(1-x)}
places an immediate upper limit of 1/4 on *x* (in both eras),
since higher values would erase the dynamical distinction between radiation
and matter. With
*x* 1/4 it then
follows from (122) that
_{v}
1/3
_{r}.
This is consistent with Sec. 5.1, where we noted that
a vacuum component whose density climbs more steeply than
*R*^{-3} in the past
direction cannot have an energy density greater than that of radiation
at present. Freese
[180]
set a stronger bound by showing that
*x* 0.07 if the
baryon-to-photon ratio
is to be consistent with both primordial nucleosynthesis and present-day
CMB observations. (This argument assumes that *x* =
*x*_{r} = *x*_{m}.)
As a guideline in what follows, then, we will allow *x*_{r}
and *x*_{m}
to take values between zero and 0.07, and consider in addition the
theoretical possibility that *x*_{m} could increase to 0.25
in the matter-dominated era.

With _{m}(*R*) specified by (121),
_{r}
related to
_{v}
by (122) and
_{v}(*R*) given by (124), we can solve
for all three components as functions of time if the scale factor
*R*(*t*) is known. This comes as usual from the field
equations (113). Since these are the same as Eqs. (106) for standard
cosmology, they lead to the same result, Eq. (22):

(125) |

Here we have used Eqs. (118) to replace
_{}
with
_{v} +
_{c}
and (119) to replace
_{eff}
with
_{m} +
_{r}.
We have also set *k* = 0 since observations indicate that these
components together make up very nearly the critical density
(Sec. 4).

Eq. (125) can be solved analytically in the three cases which are of
greatest physical interest: (1) the *radiation-dominated regime*,
for which *t* < *t*_{eq} and
_{r} +
_{v}
>>
_{m} +
_{c};
(2) the *matter-dominated regime*, which has
*t*
*t*_{eq} and
_{r} +
_{v}
<<
_{m}
(if _{c}
= 0); and (3) the *vacuum-dominated regime*, for which
*t* *t*_{eq}
and _{r} +
_{v}
<<
_{m} +
_{c}.
The distinction between
regimes 2 and 3 allows us to model both matter-only universes like EdS
and vacuum-dominated cosmologies like
CDM or
BDM
(Table 2). The definitions of these
terms should
be amended slightly for this section, since we now consider flat models
containing not only matter and a cosmological constant, but radiation and
a decaying-vacuum component as well. The densities of the latter two
components are, however, at least four orders of magnitude below that
of matter at present. Thus models with
_{c} =
0, for example, have
_{m,0} = 1
to four-figure precision or better and are dynamically
indistinguishable from EdS during all but the first fraction (of order
10^{-4} or less) of their lifetimes. For definiteness, we will use
the terms "EdS,"
"CDM" and
"BDM" in this
section to refer to flat models in which
_{m,0} = 1,
0.3 and 0.03 respectively.
In all cases, the present dark-energy density (if any) comes almost
entirely from its constant-density component.

Eqs. (121), (122), (124) and (125) can be solved analytically for
*R*,
_{m},
_{r}
and _{v}
in terms of *R*_{ 0},
_{m,0},
_{r,0},
*x*_{r} and *x*_{m} (see
[1]
for details). The normalized scale factor is found to read

(126) |

The dark-energy density is given by

(127) |

where
=
3/(32 *G*) = 4.47 ×
10^{5} g cm^{-2} s^{2}.
The densities of radiation and matter are

(128) | |||

Here we have applied
_{m,0}
= _{m,0}
_{crit,0}
and
_{r,0}
= _{r,0}
_{crit,0}
as boundary conditions. The function
_{m}(*t*) is
defined as

(129) |

where _{0}
2 / [3*H*_{0}
(1 - _{m,0})^{1/2}]. The age of of the Universe
is

(130) |

where _{0}
[(1 -
_{m,0}) /
_{m,0}]^{1/2} and we have used
Eq. (56). Corrections from the radiation-dominated era can be
ignored since
*t*_{0} >> *t*_{eq} in all cases.

The parameter
*t*_{eq} is obtained as in standard cosmology by setting
_{r}
(*t*_{eq}) =
_{m}
(*t*_{eq}) in Eqs. (128). This leads to

(131) |

The epoch of matter-radiation equality plays a crucial role because it is
at about this time that the Universe became transparent to radiation (the
two events are not simultaneous but the difference between them is minor
for our purposes). Decay photons created before
*t*_{eq} would simply
have been thermalized by the primordial plasma and eventually re-emitted
as part of the CMB. It is the decay photons emitted *after* this
time which can contribute to the extragalactic background radiation,
and whose contributions we wish to calculate.
The quantity
*t*_{eq} is thus analogous to the galaxy formation time
*t*_{f} in previous sections.

The densities
_{m}(*t*),
_{r}(*t*) and
_{v}(*t*) are plotted as functions
of time in Fig. 22.
The left-hand panel (a) shows the effects of varying the parameters
*x*_{r} and *x*_{m} within a given
cosmological model (here,
CDM).
Raising the value of *x*_{m} leads to a proportionate
increase in
_{v}
and a modest drop in
_{r}. It also flattens the slope of both
components. The change in slope (relative to that of the matter component)
pushes the epoch of equality back toward the big bang (vertical lines).
Such an effect could in principle allow more time for structure to form
during the early matter-dominated era
[180],
although the "compression" of the radiation-dominated era rapidly
becomes unrealistic for values of *x*_{m} close to
1/4. Thus Fig. 22(a)
shows that the value of
*t*_{eq} is reduced by a factor of over 100 in going
from a model with
*x*_{m} = 10^{-4} to one with *x*_{m}
= 0.07. In the limit
*x*_{m}
1/4, the radiation-dominated era disappears
altogether, as remarked above and as shown explicitly by Eqs. (131).

Fig. 22(b) shows the effects of changes in
cosmological model for fixed values of *x*_{r} and
*x*_{m} (here both set to 10^{-4}).
Moving from the matter-filled EdS model toward vacuum-dominated ones
such as CDM and
BDM does three
things. The first is to increase
the age (*t*_{0}) of the Universe. This increases the
density of
radiation at any given time, since the latter is fixed at present and
climbs at the same rate in the past direction. Based on our experience
with the galactic EBL in previous sections, we may expect that this should
lead to significantly higher levels of background radiation when
integrated over time. However, there is a second effect in the present
theory which acts in the opposite direction: smaller values of
_{m,0}
boost the value of *t*_{eq} as well as
*t*_{0}, thus delaying the onset of
the matter-dominated era (vertical lines). As we will see, these two
changes all but cancel each other out as far as dark-energy contributions
to the background are concerned. The third consequence of
vacuum-dominated cosmologies is "late-time inflation," the sharp
increase in the expansion rate at recent times
(Fig. 18).
This translates in Fig. 22(b) into the drop-off
in the densities
of all three components at the right-hand edge of the figure for the
CDM and
BDM models.

In order to make use of the formalism we have developed in
Secs. 2
and 3, we need to define discrete "sources"
of radiation from
dark-energy decay, analogous to the galaxies of previous sections.
For this purpose we carve up the Universe into hypothetical regions of
arbitrary comoving volume *V*_{0}. The comoving number
density of these source regions is just

(132) |

These regions are introduced for convenience, and are not physically
significant since dark energy decays uniformly throughout space. We
therefore expect that the parameter *V*_{0} will not appear
in our final results.

The next step is to identify the "source luminosity." There are at
least two ways to approach this question
[199].
One could simply regard the source region as a ball of physical volume
*V*(*t*) =
^{3}(*t*)
*V*_{0} filled with fluctuating dark energy. As the
density of this energy drops by -
*d*_{v} during time *dt*, the ball loses
energy at a rate
- *d*_{v} / *dt*. If some fraction
of this energy
flux goes into photons, then the luminosity of the ball is

(133) |

This is the definition of vacuum luminosity which has been assumed
implicitly by workers such as Pavón
[194],
who investigated the thermodynamical stability of the vacuum decay process
by requiring that fluctuations in
_{v} not grow larger than the mean value of
_{v} with time. For convenience we will refer
to (133) as the *thermodynamical definition* of vacuum
luminosity (*L*_{th}).

A second approach is to treat this as a problem involving spherical symmetry
within general relativity. The assumption of spherical symmetry
allows the total mass-energy (*Mc*^{ 2}) of a localized
region of perfect
fluid to be identified unambiguously. Luminosity can then be related to
the *time rate of change* of this mass-energy. Assuming once
again that the two are related by a factor
, one has

(134) |

Application of Einstein's field equations leads to the following expression
[201]
for the rate of change of mass-energy in terms of the
pressure *p*_{v} at the region's surface:

(135) |

where *r*(*t*) =
(*t*)
*r*_{0} is the region's physical radius. Taking
*V* = 4/3
*r*^{3}, applying
the vacuum equation of state
*p*_{v} = - _{v}
*c*^{ 2} and substituting (135) into (134),
we find that the latter can be written in the form

(136) |

This is just as appealing dimensionally as Eq. (133),
and shifts the emphasis physically from fluctuations in the material
content of the source region toward changes in its geometry.
We will refer to (136) for convenience as the
*relativistic definition* of vacuum luminosity
(*L*_{rel}).

It is not obvious which of the two definitions (133) and (136) more correctly describes the luminosity of decaying dark energy; this is a conceptual issue. Before choosing between them, let us inquire whether the two expressions might not be equivalent. We can do this by taking the ratio

(137) |

Differentiating Eqs. (126) and (127) with respect to time, we find

(138) | |||

(139) |

The ratio
*L*_{th} / *L*_{rel} is therefore constant
[= 4(1 - *x*_{m}) / 3], taking values between
4/3 (in the limit
*x*_{m}
0 where standard cosmology is recovered) and 1 (in the opposite limit
where *x*_{m} takes its maximum theoretical value of 1/4).
There is thus little difference between the two scenarios in practice,
at least where this model of decaying dark energy is concerned.
We will proceed using the relativistic definition (136)
which gives lower intensities and hence more conservative limits on the
theory. At the end of the section it will be a small matter to calculate
the corresponding intensity for the thermodynamical case (133)
by multiplying through by 4/3 (1 - *x*_{m}).

We now turn to the question of the branching ratio , or fraction of decaying dark energy which goes into photons as opposed to other forms of radiation such as massless neutrinos. This is model-dependent in general. If the vacuum-decay radiation reaches equilibrium with that already present, however, then we may reasonably set this equal to the ratio of photon-to-total radiation energy densities in the CMB:

(140) |

The density parameter
_{} of CMB
photons is given in terms of
their blackbody temperature *T* by Stefan's law.
Using the COBE value
*T*_{cmb} = 2.728 K
[196],
we get

(141) |

The total radiation density
_{r,0} =
_{} +
_{}
is harder to determine,
since there is little prospect of detecting the neutrino component directly.
What is done in standard cosmology is to calculate the size of neutrino
contributions to
_{r,0} under
the assumption of entropy conservation.
With three light neutrino species, this leads to

(142) |

where
*T*_{}
is the blackbody temperature of the relic neutrinos and the
factor of 7/8 arises from the fact that these particles obey Fermi rather
than Bose-Einstein statistics
[200].
During the early stages of
the radiation-dominated era, neutrinos were in thermal equilibrium with
photons so that
*T*_{}
= *T*. They dropped out of equilibrium, however,
when the temperature of the expanding fireball dropped below about
*kT* ~ 1 MeV (the energy scale of weak interactions). Shortly
thereafter, when the temperature dropped to
*kT* ~ *m*_{e} *c*^{2} = 0.5 MeV,
electrons and positrons began to annihilate,
transferring their entropy to the remaining photons in
the plasma. This raised the photon temperature by a factor of
(1 + 2 × 7/8 = 11/4)^{1/3}
relative to that of the neutrinos. In standard cosmology, the ratio of
*T*_{}
/ *T* has remained at
(4/11)^{1/3} down to the present day, so that (142) gives

(143) |

Using (140) for , this would imply:

(144) |

We will take these as our "standard values" of
_{r,0} and
in what follows. They are conservative ones, in the sense that most
alternative lines of argument would imply higher values of
.
Birkel and Sarkar
[181],
for instance, have argued that
vacuum decay (with a constant value of *x*_{r}) would be
easier to reconcile
with processes such as electron-positron annihilation if the vacuum coupled
to photons but not neutrinos. This would complicate the theory, breaking
the radiation density
_{r}
in (120) into a photon part
_{} and a
neutrino part
_{} with different
dependencies on *R*.
One need not solve this equation, however, in order to appreciate
the main impact that such a modification would have.
Decay into photons alone would pump entropy into the photon component
relative to the neutrino component in an effectively *ongoing
version* of the electron-positron annihilation argument described
above. The neutrino temperature
*T*_{}
(and density
_{}) would continue
to be driven down relative to *T* (and
_{})
throughout the radiation-dominated era and into the matter-dominated one.
In the limit
*T*_{}
/ *T* 0 one sees
from (140) and (142) that such a scenario would lead to

(145) |

In other words, the present energy density of radiation would be lower,
but it would effectively *all* be in the form of photons.
Insofar as the decrease in
_{r,0} is
precisely offset by the increase
in , these
changes cancel each other out. The drop in
_{r,0},
however, has an added consequence which is not cancelled: it pushes
*t*_{eq} farther into the past, increasing the length of
time over which decaying dark energy has been contributing to the
background. This raises the
latter's intensity, particularly at longer wavelengths. The effect can be
significant, and we will return to this possibility at the end of the
section. For the most part, however, we will stay with the values of
_{r,0} and
given by
Eqs. (143) and (144).

Armed with a definition for vacuum luminosity, Eq. (136), and a value for
, Eq. (144),
we are in a position to calculate the luminosity of decaying dark energy.
Noting that
= 3(*R* /
*R*_{0})^{3}
( / *R*)
*V*_{0} and substituting
Eqs. (127) and (138) into (136), we find that

(146) |

The first of these solutions corresponds to models with
_{m,0} = 1
while the the second holds for the general case (0 <
_{m,0}
< 1). Both results
reduce at the present time *t* = *t*_{0} to

(147) |

where
_{v, 0} is the
*comoving luminosity density of decaying dark energy*

(148) |

Numerically, we find for example that

(149) |

In principle, dark-energy decay can produce a background
10 or even 50 times more luminous than that of galaxies, as given by
(20). Raising the value of the branching ratio
to 1
instead of 0.595 does not affect these results, since this must be
accompanied by a proportionate drop in the value of
_{r,0} as
argued above. The numbers in (149) do go up if one replaces the relativistic
definition (136) of vacuum luminosity with the thermodynamical
one (133) but the change is modest, raising
_{v, 0} by no
more than a factor of 1.2 (for *x*_{m} = 0.07). The primary
reason for the
high luminosity of the decaying vacuum lies in the fact that it converts
nearly 60% of its energy density into photons. By comparison, less
than 1% of the rest energy of ordinary luminous matter has gone into
photons so far in the history of the Universe.

We showed in Sec. 2 that the bolometric
intensity of an arbitrary distribution of sources with comoving number
density *n*(*t*) and luminosity
*L*(*t*) could be expressed as an integral over time by
(12). Let us
apply this result here to regions of decaying dark energy, for which
*n*_{v}(*t*) and *L*_{v}(*t*) are
given by (132) and (146)
respectively. Putting these equations into (12) along with
(126) for the scale factor, we find that

(150) |

The first of these integrals corresponds to models with
_{m,0} = 1 while
the second holds for the general case
(0 < _{m,0}
< 1). The latter may be simplified with a change of variables to
*y* [sinh(*t*
/ _{0})]^{8xm/3}. Using the facts
that sinh(*t*_{0} /
_{0}) = [(1 -
_{m,0}) /
_{m,0}]^{1/2} and cosh(*t*_{0} /
_{0}) = 1 /
[_{m,0}]^{1/2}
along with the definition (131)
of *t*_{eq}, both integrals reduce to the same formula:

(151) |

Here *Q*_{v} is found with the help of (148) as

(152) |

There are several points to note about this result. First, it does
not depend on *V*_{0}, as expected. There is also no
dependence on the uncertainty *h*_{0} in Hubble's constant,
since the two factors of *h*_{0}
in *H*_{0}^{ 2} are cancelled out by those in
_{r,0}.
In the limit
*x*_{m}
0 one sees that
*Q* 0 as expected.
In the opposite limit where
*x*_{m} 1/4,
decaying dark energy attains a maximum possible bolometric intensity of
*Q*
*Q*_{v} = 0.013 erg cm^{-2} s^{-1}. This is
50 times the bolometric intensity due to galaxies, as given by (21).

The matter density
_{m,0}
enters only weakly into this result, and plays no role at all in the limit
*x*_{m}
1/4. Based on our experience with the EBL
due to galaxies, we might have expected that *Q* would rise
significantly in models with smaller values of
_{m,0} since
these have longer ages,
giving more time for the Universe to fill up with light.
What is happening here, however, is that the larger values of
*t*_{0} are offset by larger values of
*t*_{eq} (which follow from the fact that
smaller values of
_{m,0} imply
smaller ratios of
_{m,0} /
_{r,0}).
This removes contributions from the early matter-dominated era and thereby
*reduces* the value of *Q*. In the limit
*x*_{m}
1/4 these two effects cancel each other out.
For smaller values of *x*_{m}, the
*t*_{eq}-effect proves to be the stronger
of the two, and one finds an overall decrease in *Q* for these cases.
With *x*_{m} = 0.07, for instance, the value of *Q*
drops by 2% when moving from the EdS model to
CDM, and by
another 6% when moving
from CDM to
BDM.

**5.8. Spectral energy distribution**

To obtain limits on the parameter *x*_{m}, we would like to
calculate
the spectral intensity of the background due to dark-energy decay, just as
we did for galaxies in Sec. 3. For this we
need to know the
spectral energy distribution (SED) of the decay photons. As discussed in
Sec. 5.1, theories in which the these photons are
distributed
with a *non*-thermal spectrum can be strongly constrained by means
of distortions in the CMB. We therefore restrict ourselves to the
case of a blackbody SED, as given by Eq. (78):

(153) |

where *T*(*t*) is the blackbody temperature. The function
*C*(*t*) is found as usual by normalization,
Eq. (57). Changing integration variables from
to
= *c* /
, we find

(154) |

Inserting our result (146) for *L*_{v}(*t*) and using
the facts that
(4) = 3! = 6 and
(4) =
^{4} / 90, we then
obtain for *C*(*t*):

(155) |

Here the upper expression refers as usual to the EdS case
(_{m,0} = 1),
while the lower applies to the general case (0 <
_{m,0} <
1). The temperature of the photons can be specified if we assume thermal
equilibrium between those created by vacuum decay and those already
present. Stefan's law then relates *T*(*t*) to the radiation
energy density
_{r}
(*t*) *c*^{ 2} as follows:

(156) |

Putting Eq. (128) into this expression and expanding the Stefan-Boltzmann constant, we find that

(157) |

where the constant
_{v} is given by

(158) |

This value of
_{v} tells us
that the peak of the observed spectrum of
decay radiation lies in the microwave region as expected, near that of the
CMB (_{cmb} =
0.11 cm). Putting (157) back into (155), we obtain

(159) |

These two expressions refer to models with
_{m,0} = 1
and 0 <
_{m,0} < 1
respectively. This specifies the SED (153) of decaying dark energy.

The spectral intensity of an arbitrary distribution of sources with
comoving number density *n*(*t*) and an SED
*F*(, *t*)
is expressed as an integral over time by Eq. (61). Putting
Eqs. (126), (132) and (153) into this equation, we obtain

(160) |

Here we have used integration variables
*t* /
*t*_{0} in the first case
(_{m,0} = 1)
and
*t* /
_{0} in the second
(0 < _{m,0}
< 1). The dimensional content of both integrals is contained
in the prefactor
*I*_{v}(_{0}), which reads

(161) |

We have divided this quantity through by photon energy
*hc* / _{0}
so as to express the results in continuum units (CUs) as usual, where
1 CU 1 photon
s^{-1} cm^{-2} Å^{-1} ster^{-1}.
We use CUs throughout this review, for the sake of uniformity as
well as the fact that these units carry several advantages from the
theoretical point of view (Sec. 3.2).
The reader who
consults the literature, however, will soon find that each
part of the electromagnetic spectrum has its own "dialect" of
preferred units. In the microwave region intensities are commonly
reported in terms of
*I*_{},
the integral of flux per unit frequency
*over* frequency, and usually expressed in units of
nW m^{-2} ster^{-1} = 10^{-6} erg s^{-1}
cm^{-2} ster^{-1}. To translate a given value of
*I*_{}
(in these units) into CUs, one need only multiply by a factor of
10^{-6} / (*hc*) = 50.34 erg^{-1}
Å^{-1}. The Jansky (Jy) is also often encountered, with
1 Jy = 10^{-23} erg s^{-1} cm^{-2} Hz^{-1}.
To convert a given value of
*I*_{}
from Jy ster^{-1} into CUs, one multiplies by a factor of
10^{-23} / *h*
= (1509 Hz
erg^{-1}) /
with in Å.

Eq. (160) gives the combined intensity of decay photons which have
been emitted at many wavelengths and redshifted by various amounts,
but reach us in a waveband centered on
_{0}. The
arbitrary volume *V*_{0} has dropped out of the integral as
expected, and this result is also independent of the uncertainty
*h*_{0} in Hubble's constant since
there is a factor of *h*_{0} in both
_{v, 0} and
*H*_{0}. Results are plotted in
Fig. 23
over the waveband 0.01-1 cm, together with existing observational data
in this part of the spectrum. The most celebrated of these is the
COBE detection of the CMB
[196]
which we have shown
as a heavy solid line (F96). The experimental uncertainties in this
measurement are far smaller than the thickness of the line. The other
observational limits shown in Fig. 23 have been
obtained in the far infrared (FIR) region, also from analysis of data
from the COBE
satellite. These are indicated with heavy dotted lines (F98
[202])
and open triangles (H98
[27] and L00
[203]).

Fig. 23(a) shows the spectral intensity of
background radiation
from vacuum decay under our standard assumptions, including the
relativistic definition (136) of vacuum luminosity and the values of
_{r,0} and
given by
(143) and (144)
respectively. Five groups of curves are shown, corresponding to values
of *x*_{m} between
3 × 10^{-5} and the theoretical maximum of 0.25.
For each value of *x*_{m} three curves are plotted: one
each for the EdS,
CDM and
BDM
cosmologies. As noted above in connection with the
bolometric intensity *Q*, the choice of cosmological model is less
important in determining the background due to vacuum decay than the
background due to galaxies. In fact, the intensities here are actually
slightly *lower* in vacuum-dominated models. The reason for this,
as before, is that these models have smaller values of
_{m,0} /
_{r,0}
and hence larger values of
*t*_{eq}, reducing the size of contributions
from the early matter-dominated era when *L*_{v} was large.

In Fig. 23(b), we have exchanged the
relativistic definition of vacuum luminosity for the thermodynamical one
(133),
and set = 1
instead of 0.595. As discussed in Sec. 5.6,
the increase in
is partly
offset by a drop in
_{r,0}.
There is a net increase in intensity, however, because smaller values of
_{r,0} push
*t*_{eq} back into the past, leading to
additional contributions from the early matter-dominated era. These
contributions particularly push up the long-wavelength part of
the spectrum in Fig. 23(b) relative to
Fig. 23(a), as seen most clearly in the case
*x*_{m} = 0.25. Overall, intensities in
Fig. 23(b) are higher than those in
Fig. 23(a) by about a factor of four.

These figures show that *the decaying-vacuum hypothesis is strongly
constrained by observations of the microwave background.* The parameter
*x*_{m} cannot be larger than 0.06 or the intensity of the
decaying vacuum
would exceed that of the CMB itself under the most conservative assumptions,
as represented by Fig. 23(a). This limit
tightens to
*x*_{m} 0.015
if different assumptions are made about the
luminosity of the vacuum, as shown by Fig. 23(b).
These numbers are comparable to the limit of
*x* 0.07
obtained from entropy conservation under the assumption that
*x* = *x*_{r} = *x*_{m}
[180].
And insofar as the CMB is usually
attributed entirely to relic radiation from the big bang, the real limit
on *x*_{m} is probably several orders of magnitude smaller
than this.

With these upper bounds on *x*_{m}, we can finally inquire
about the
potential of the decaying vacuum as a dark-energy candidate. Since
its density is given by (122) as a fraction *x* / (1 - *x*) of
that of radiation, we infer that its present density parameter
(_{v,0})
satisfies:

(162) |

Here, (a) and (b) refer to the scenarios represented by
Figs. 23(a) and 23(b),
with the corresponding values of
_{r,0} as
defined by Eqs. (143) and (145) respectively. We have assumed that
*h*_{0} 0.6 as
usual. It is clear from the
limits (162) that a decaying vacuum of the kind we have
considered here does not contribute significantly to the density
of the dark energy.

It should be recalled, however, that there are good reasons from quantum theory for expecting some kind of instability for the vacuum in a universe which progressively cools. (Equivalently, there are good reasons for believing that the cosmological "constant" is not.) Our conclusion is that if the vacuum decays, it either does so very slowly, or in a manner that does not upset the isotropy of the cosmic microwave background.