**8.1. The lightest supersymmetric particle**

Weakly interacting massive particles (WIMPs) are as-yet undiscovered particles whose rest masses far exceed those of baryons, but whose interaction strengths are comparable to those of neutrinos. The most widely-discussed examples arise in the context of supersymmetry (SUSY), which is motivated independently of the dark-matter problem as a theoretical framework for many attempts to unify the forces of nature. SUSY predicts that, for every known fermion in the standard model, there exists a new bosonic "superpartner" and vice versa (more than doubling the number of fundamental degrees of freedom in the simplest models; see [301] for a review). These superpartners were recognized as potential dark-matter candidates in the early 1980s by Cabibbo [302], Pagels and Primack [303], Weinberg [304] and others [305, 306, 307, 308, 309], with the generic term "WIMP" being coined in 1985 [310].

There is, as yet, no firm experimental evidence for SUSY WIMPs.
This means that their rest energies, if they exist, lie beyond the
range currently probed by accelerators (and in particular beyond the
rest energies of their standard-model counterparts). Supersymmetry is,
therefore, not an exact symmetry of nature. The masses of the
superpartners, like that of the axion
(Sec. 6), must
have been generated by a symmetry-breaking event in the early Universe.
Subsequently, as the temperature of the expanding fireball dropped
below their rest energies, heavier species would have dropped out of
equilibrium and begun to disappear by pair annihilation, leaving
progressively lighter ones behind. Eventually, only one massive
superpartner would have remained: the *lightest supersymmetric
particle* (LSP). It is this particle which plays the role of the
WIMP in SUSY theories. Calculations using the Boltzmann equation show
that the collective density of relic LSPs today lies within one or two
orders of magnitude of the required CDM density across much of the
parameter space of most SUSY theories
[311].
In this respect, SUSY WIMPs are more natural DM candidates than axions
(Sec. 6), whose cosmological density ranges
*a priori* over many orders of magnitude.

SUSY WIMPs contribute to the cosmic background radiation in at least
three ways. The first is by pair annihilation to photons. This process
occurs even in the simplest, or minimal SUSY model (MSSM), but is very
slow because it takes place via intermediate loops of charged particles
such as leptons and quarks and their antiparticles.
The underlying reason for the stability of the LSP in the MSSM is an
additional new symmetry of nature, known as R-parity, which is necessary
(among other things) to protect the proton from decaying via
intermediate SUSY states. The other two types of background contributions
occur in *non*-minimal SUSY theories, in which R-parity is not
conserved (and in which the proton can decay). In these theories,
LSPs can decay into photons directly via loop diagrams, and also
indirectly via tree-level decays to secondary particles which then
scatter off pre-existing background photons to produce a signal.

The first step in assessing the importance of each of these processes
is to choose an LSP. Early workers variously identified this as the
photino
()
[302],
the gravitino
()
[303],
the sneutrino ()
[307]
or the selectron ()
[308].
(SUSY superpartners are denoted by a tilde and take the same names
as their standard-model counterparts, with a prefix "s" for
superpartners of fermions and a suffix "ino" for those of bosons.)
In a landmark study, Ellis
[309]
showed in 1984 that most
of these possibilities are disfavoured, and that the LSP is in fact most
likely to be a *neutralino*
(), a linear
superposition of the photino
(),
the zino
() and two neutral
higgsinos (_{1}^{0} and
_{2}^{0}). (These are the SUSY
spin-1/2 counterparts of the photon, *Z*^{0} and Higgs bosons
respectively.) There are four neutralinos, each a mass eigenstate made
up of (in general) different amounts of photino, zino, etc., although
in special cases a neutralino could be "mostly photino," say,
or "pure zino." The LSP is by definition the lightest such
eigenstate. Accelerator searches place a lower limit on its
rest energy which currently stands at
*m*_{}*c*^{2} > 46 GeV
[312].

In minimal SUSY, the density of neutralinos drops only by way of the
(slow) pair-annihilation process, and it is quite possible for these
particles to "overclose" the Universe if their rest energy is too
high. This does not change the geometry of the Universe, but rather
speeds up its expansion rate, which is proportional to the square root
of the total matter density from Eq. (22). In such a situation,
the Universe would have reached its present size in too short a time.
Lower bounds on the age of the Universe thus impose an upper bound on
the neutralino rest energy which has been set at
*m*_{} *c*^{2}
3200 GeV
[313].
Detailed exploration of the parameter space of minimal
SUSY theory tightens this upper limit in most cases to
*m*_{} *c*^{2}
600 GeV
[314].
Much recent work is focused on
a slimmed-down version of the MSSM known as the *constrained minimal
SUSY model* (CMSSM), in which all existing experimental bounds and
cosmological requirements are comfortably met by neutralinos with
rest energies in the range 90 GeV
*m*_{} *c*^{2}
400 GeV
[315].

Even in its constrained minimal version, SUSY physics contains at least
five adjustable input parameters, making the neutralino a considerably
harder proposition to test than the axion or the massive neutrino.
Fortunately, there are several other ways (besides accelerator
searches) to look for these particles. Because their rest energies
are above the temperature at which they decoupled from the primordial
fireball, WIMPs have non-relativistic velocities and are found
predominantly in gravitational potential wells like those of our
own Galaxy. They will occasionally scatter against target nuclei in
terrestrial detectors as the Earth follows the Sun around the Milky Way.
Annual variations in this signal resulting from the Earth's orbital motion
through the Galactic dark-matter halo can be used to isolate a WIMP signal.
Just such a signal was reported by the DAMA team in 2000
using detectors in Italy's Gran Sasso mountains, with an implied WIMP
rest energy of
*m*_{} *c*^{2} = 52^{+10}_{-8}
GeV
[316].
However, subsequent
experiments under the Fréjus peak in France (EDELWEISS
[317])
and at Stanford University
[318]
and the Soudan mine in Minnesota
[319]
(CDMS) have not
been able to reproduce this result. New detectors such as
ZEPLIN in England's Boulby mine
[320]
and IGEX at Canfranc in Spain
[321] are rapidly
coming online to help with the search.

A second, indirect search strategy is to look for annihilation byproducts from neutralinos which have collected inside massive bodies. Most attention has been directed at the possibility of detecting antiprotons from the Galactic halo [322] or neutrinos from the Sun [323] or Earth [324]. The heat generated in the cores of gas giants like Jupiter or Uranus has also been considered as a potential annihilation signature [325]. The main challenge in each case lies in separating the signal from the background noise. In the case of the Earth, one can look for neutrino-induced muons which are distinguishable from the atmospheric background by the fact that they are travelling straight up. The AMANDA experiment, whose detectors are buried deep in the Antarctic ice, has recently reported upper limits on the density of terrestrial WIMPs based on this principle [326].

Pair annihilation into *photons* provides a complementary indirect
search technique. The photons so produced lie in the
-ray portion
of the spectrum for the range of WIMP rest energies considered here
(50 GeV
*m*_{} *c*^{2}
1000
GeV). Beginning with Sciama
[308],
Silk and Srednicki
[322],
many workers have studied the possibility of
-rays from
SUSY WIMP annihilations in the halo of the Milky Way, which gives the
strongest signal. Prognoses for detection have ranged from very optimistic
[327]
to very pessimistic
[328];
converging gradually to
the conclusion that neutralino-annihilation contributions would be at or
somewhat below the level of the Galactic background, and possibly
distinguishable from it by their spectral shape
[329,
330,
331].
Recent studies have focused on possible enhancements of the signal in
the presence of a high-density Galactic core
[332],
a flattened halo
[333],
a very extended singular halo
[334],
a massive central black hole
[335],
significant substructure
[336,
337,
338]
and adiabatic compression due to baryons
[339].
The current state of the art in this area is summarized in Ref.
[340]
with attention to prospects for detection by the
upcoming GLAST mission.

WIMPs at the higher end of the mass range (~ 1 TeV) would produce a weaker signal, but it has been argued that this might be more than made up for by the larger effective area of the atmospheric Cerenkov telescopes (ACTs) used to detect them [342]. Not all authors are as sanguine [343], but new observations of high-energy -rays from the Galactic center by the CANGAROO [344] and VERITAS collaborations [345] provide tantalizing examples of what might be possible with this technique. The Milagro extensive air-shower array is another experiment that has recently set upper limits on the density of ~ TeV WIMPs in the vicinity of the Sun [346]. Other teams have carried the search farther afield, toward objects like dwarf spheroidal galaxies [347], the Large Magellanic Cloud [348] and the giant elliptical galaxy M87 in Virgo [349].

The possibility of neutralino-annihilation contributions to the
diffuse extragalactic background, rather than the signal from
localized concentrations of dark matter, has received less attention.
First to apply the problem to SUSY WIMPs were Cabibbo
[302],
who however assumed a WIMP rest energy (10-30 eV) which we now know is
far too low. Like the decaying neutrino
(Sec. 7), this would
produce a background in the ultraviolet. It is excluded, however,
by an argument due to Lee and Weinberg, which restricts WIMPs to rest
energies above 2 GeV
[306].
EBL contributions from SUSY WIMPs
in this range were first estimated by Silk and Srednicki
[322].
Their conclusion, and those of most workers who have followed them
[350,
351,
352],
is that neutralino annihilations would be
responsible for no more than a small fraction of the observed
-ray
background. Here we review this argument, reversing our
usual procedure and attempting to set a reasonably conservative
*upper* limit on neutralino contributions to the EBL.

We concentrate on processes in which neutralino pairs annihilate
*directly* into photon pairs via intermediate loop diagrams
(Fig. 32), since these provide the most
distinctive signature
of new physics. Neutralino annihilations actually produce most of their
photons indirectly, via tree-level annihilations to hadrons (mostly pions)
which then decay to photons, electrons, positrons and neutrinos. (The
electrons and positrons add even more to the signal by inverse Compton
scattering off low-energy CMB photons.) However, the
energies of the photons produced in this way are broadly distributed,
resulting in a continuum
-ray
spectrum which is difficult if not
impossible to distinguish from the astrophysical background
[353].
By contrast, the one-loop annihilation processes in
Fig. 32
give rise to a photon spectrum that is essentially *monoenergetic*,
*E*_{}
*m*_{} *c*^{2} (subject only to modest Doppler
broadening due to galactic rotation). No conventional astrophysical
processes produce such a narrow peak, whose detection against the diffuse
extragalactic background would constitute compelling evidence
for dark matter.

We again take galactic dark-matter halos as our sources of background
radiation, with comoving number density *n*_{0}. Photon
wavelengths are
distributed normally about the peak wavelength in the galaxy rest frame:

(221) |

where
*m*_{10}
*m*_{} *c*^{2} / (10 GeV) is the neutralino
rest energy in units of 10 GeV. The standard deviation
_{} can be
related to the velocity dispersion of bound dark-matter particles as in
previous sections, so that
_{} =
2(*v*_{c} / *c*)
_{ann}. With
*v*_{c} ~ 220 km s^{-1} and
*m*_{10} ~ 1 this is of order
~ 10^{-9} Å. For convenience we specify this with the
dimensionless parameter
_{9}
_{} /
(10^{-9} Å).
The spectral energy distribution is then given by Eq. (75) as

(222) |

The luminosity due to neutralino annihilations is proportional to the
rest energy of the annihilating particles times the annihilation rate,
which in turn goes as the cross-section
(*v*) times the square
of the neutralino number density,
*n*_{}^{2}.
The resulting expression may be written

(223) |

where < *v*
>_{} is the photo-annihilation
cross-section (in cm^{3} s^{-1}),
*m*_{}^{-2}
_{}^{2}(*r*)
4 *r*^{2}
*dr* is the radial average of
*n*_{}^{2} over the halo
(in cm^{-6}) and
_{}(*r*) is the neutralino density
distribution (in g cm^{-3}). Berezinsky
[354]
have determined
< *v*
>_{}
*a*_{} for non-relativistic neutralinos as

(224) |

Here is the fine
structure constant,
*m*_{} is the mass of an intermediate sfermion,
*y*
(*Z*_{12} / *Z*_{11})
tan_{W},
_{W} is the weak
mixing angle and *Z*_{ij} are elements of the
real orthogonal matrix which diagonalizes the neutralino mass matrix.
In particular, the "pure photino" case is specified by
*Z*_{11} =
sin_{W},
*y* = 1 and the "pure zino" by *Z*_{11} =
cos_{W},
*y* = - tan^{2}
_{W}. Collecting
these expressions together and parametrizing the sfermion rest energy by
_{10}
*m*_{} / 10 GeV, we obtain:

(225) |

Here *f*_{} (=1 for photinos, 0.4 for zinos) is a dimensionless
quantity whose value parametrizes the makeup of the neutralino.

Since we attempt in this section to set an upper limit on EBL
contributions from neutralino annihilations, we take
*f*_{} 1
(the photino case). In the same spirit, we would like to use lower
limits for the sfermion mass
_{10}. It is
important to estimate this quantity accurately since the cross-section
goes as _{10}^{-4}. Giudice and Griest
[355]
have made a detailed study of
photino annihilations and find a lower limit on
_{10}
as a function of *m*_{10}, assuming that photinos provide
at least 0.025*h*_{0}^{-2} of the critical
density. Over the range
0.1 *m*_{10}
4, this lower limit is
empirically well fit by a function of the form
_{10}
4*m*_{10}^{ 0.3}.
If this holds over our broader range of masses, then we obtain an
upper limit on the neutralino annihilation cross-section of
< *v*
>_{}
(3 ×
10^{-29} cm^{3}s^{-1})
*m*_{10}^{0.8}. This expression gives results which
are about an order of magnitude higher than the cross-sections quoted by
Gao
[351].

For the WIMP density distribution
_{}(*r*) we adopt the
simple and widely-used *isothermal model*
[331]:

(226) |

Here _{} = 5 ×
10^{-25} g cm^{-3} is the approximate
dark-matter density in the solar vicinity, assuming a spherical halo
[260],
*r*_{} = 8
kpc is the distance of the Sun from the Galactic center
[356]
and *a* = (2 - 20) kpc is a core radius. To fix this latter
parameter, we can integrate (226) over volume to obtain total halo mass
*M*_{h}(*r*) inside radius *r*:

(227) |

Observations of the motions of Galactic satellites imply that the total
mass inside 50 kpc is about 5 × 10^{11}
*M*_{}
[75].
This in (227) implies *a* = 9 kpc, which we consequently adopt.
The maximum extent of the halo is not well-constrained observationally,
but can be specified if we take
*M*_{h} = (2 ± 1) × 10^{12}
*M*_{}
as in (175). Eq. (227) then gives a halo radius
*r*_{h} = (170 ± 80) kpc. The cosmological density
of WIMPs in galactic dark-matter halos adds up to
_{h} =
*n*_{0} *M*_{h} /
_{crit,0}
= (0.07 ± 0.04) *h*_{0}.

If there are no other sources of CDM, then the total matter density is
_{m,0} =
_{h} +
_{bar}
0.1*h*_{0} and the observed flatness of the
Universe (Sec. 4) implies a strongly
vacuum-dominated cosmology.
While we use this as a lower limit on WIMP contributions to the dark
matter in subsequent sections, it is quite possible that CDM also exists
in larger-scale regions such as galaxy clusters. To take this into account
in a general way, we define a *cosmological enhancement factor*
*f*_{c}
(_{m,0} -
_{bar}) /
_{h}
representing the added contributions
from WIMPs outside galactic halos (or perhaps in halos which extend
far enough to fill the space between galaxies).
This takes the value *f*_{c} = 1 for the most conservative
case just described, but rises to
*f*_{c} = (4 ± 2) *h*_{0}^{-1} in
the CDM model with
_{m,0} =
0.3, and (14 ± 7) *h*_{0}^{-1} in the EdS
model with
_{m,0} = 1.

With _{}(*r*) known, we are in a position to calculate
the quantity :

(228) |

Using the values for
_{},
*r*_{} and
*a* specified above and
setting *r*_{h} = 250 kpc to get an upper limit, we find that
(5 × 10^{65}
cm^{-3}) *m*_{10}^{-2}. Putting this result
along with the cross-section (225) into (223), we obtain:

(229) |

Inserting Giudice and Griest's
[355]
lower limit on the sfermion mass
_{10} (as
empirically fit above), we find that
(229) gives an upper limit on halo luminosity of
*L*_{h,ann}
(5 × 10^{35} erg s^{-1})
*f*_{} *m*_{10}^{-0.2}.
Higher estimates can be found in the literature
[357],
but these assume a singular halo whose density drops off as only
_{}(*r*)
*r*^{-1.8} and extends out to a very large
halo radius, *r*_{h} = 4.2
*h*_{0}^{-1} Mpc. For a standard isothermal
distribution of the form (226), our results confirm that
halo luminosity due to neutralino annihilations alone is very low,
amounting to less than 10^{-8} times the total bolometric
luminosity of the Milky Way.

The combined bolometric intensity of neutralino annihilations between
redshift *z*_{f} and the present is given by substituting
the comoving number density *n*_{0} and luminosity
*L*_{h,ann} into Eq. (15) to give

(230) |

where *Q*_{,ann} = (*cn*_{0}
*L*_{h,ann}*f*_{c}) / *H*_{0}
and we have assumed spatial flatness.
With values for all these parameters as specified above, we find

(231) |

Here we have set *z*_{f} = 30 (larger values do not
substantially increase
the value of *Q*) and used values of *f*_{c} = 1,
4*h*_{0}^{-1} and
20*h*_{0}^{-1}
respectively. The effects of a larger cosmological enhancement factor
*f*_{c}
are partially offset in (230) by the fact that a universe with
higher matter density
_{m,0} is
younger, and hence contains less
background light in general. Even the highest value of *Q* given
in (231) is negligible in comparison to the
intensity (21) of the EBL due to ordinary galaxies.

The total spectral intensity of annihilating neutralinos is found by substituting the SED (222) into (62) to give

(232) |

For a typical neutralino with
*m*_{10}
10 the annihilation spectrum peaks near
_{0}
10^{-7}
Å. The dimensional prefactor reads

(233) |

Here we have divided through by the photon energy *hc* /
_{0} to put
results into continuum units or CUs as usual
(Sec. 3.2).
Eq. (232) gives the combined intensity of radiation from
neutralino annihilations, emitted at various wavelengths and redshifted
by various amounts, but observed at wavelength
_{0}. Results are
plotted in Fig. 33 together with observational
constraints.

We turn next to *non*-minimal SUSY theories in which R-parity is
not necessarily conserved and the LSP (in this case the neutralino) can
decay. The cosmological consequences of R-parity breaking have been
reviewed by Bouquet and Salati
[358].
There is one *direct* decay mode into photons,
+
.
Feynman diagrams for this process are shown in
Fig. 34.
Because these decays occur via loop diagrams, they are again subdominant.
We consider theories in which R-parity breaking is accomplished
spontaneously. This means introducing a scalar sneutrino with a nonzero
vacuum expectation value
*v*_{R}
< _{R} >,
as discussed by Masiero and Valle
[359].
Neutralino decays into photons could be detectable if
*m*_{} and
*v*_{R} are large
[360].

The photons produced in this way are again monochromatic, with
*E*_{} = 1/2
*m*_{} *c*^{2}. In fact the SED here is the
same as (222) except that peak wavelength is doubled,
_{loop} =
2*hc* / *m*_{} *c*^{2} = (2.5 ×
10^{-6} Å) *m*_{10}^{-1}.
The only parameter that needs to be recalculated is the halo
luminosity *L*_{h}. For one-loop neutralino decays of lifetime
_{},
this takes the form:

(234) |

Here *N*_{} = *M*_{h} / *m*_{}
is the number of neutralinos in the halo and
*b*_{} is the *branching ratio*, or fraction
of neutralinos that decay into photons. This is estimated by
Berezinsky
[360] as

(235) |

where the new parameter *f*_{R}
*v*_{R} / (100 GeV). The
requirement that SUSY WIMPs not carry too much energy out of stellar
cores implies that
*f*_{R} is of order ten or more
[359].
We take *f*_{R} > 1 as
a lower limit.

We adopt *M*_{h} = (2 ± 1) × 10^{12}
*M*_{} as
usual, with *r*_{h} = (170 ± 80) kpc
from the discussion following (227).
As in the previous section, we parametrize our
lack of certainty about the distribution of neutralinos on larger scales
with the cosmological enhancement factor
*f*_{c}. Collecting these results
together and expressing the decay lifetime in dimensionless form as
*f*_{}
_{} /
(1 Gyr), we obtain for the luminosity of one-loop
neutralino decays in the halo:

(236) |

With *m*_{10} ~
*f*_{R} ~
*f*_{} ~ 1, Eq. (236) gives
*L*_{h,loop} ~ 2 × 10^{7}
*L*_{}. This
is considerably brighter than the
halo luminosity due to neutralino annihilations in minimal SUSY models,
but still amounts to less than 10^{-3} times the bolometric
luminosity of the Milky Way.

Combined bolometric intensity
is found as in the previous section, but with *L*_{h,ann}
in (230) replaced by *L*_{h,loop} so that

(237) |

This is again small. However, we see that massive (*m*_{10}
10)
neutralinos which provide close to the critical density
(_{m,0} ~ 1)
and decay on timescales of order 1 Gyr or less
(*f*_{}
1)
could in principle rival the intensity of the conventional EBL.

To obtain more quantitative constraints, we turn to spectral intensity.
This is given by Eq. (232) as before, except that the dimensional
prefactor
*I*_{,ann} must be replaced by

(238) |

Results are plotted in Fig. 35
for neutralino rest energies
1 *m*_{10}
100. While
their bolometric intensity is low, these particles are capable of
significant EBL contributions in narrow portions of the
-ray
background. To keep the diagram from becoming too cluttered, we have
assumed values of
*f*_{} such that the highest predicted intensity
in each case stays just below the EGRET limits. Numerically,
this corresponds to lower bounds on the decay lifetime
_{}
of between 100 Gyr (for
*m*_{} *c*^{2} = 10 GeV) and 10^{5}
Gyr (for
*m*_{} *c*^{2} = 300 GeV).
For rest energies at the upper end of this range, these limits are
probably optimistic because the decay photons are energetic enough to
undergo pair production on CMB photons. Some would not reach us from
cosmological distances, instead being re-processed into lower energies
along the way. As we show in the next section, however, stronger limits
arise from a different process in any case. We defer further discussion
of Fig. 35 to Sec. 8.6.

The dominant decay processes for the LSP neutralino in non-minimal SUSY
(assuming spontaneously broken R-parity) are *tree-level* decays to
leptons and neutrinos,
^{+} +
^{-} +
_{}.
Of particular interest is the case
= *e*; Feynman
diagrams for this process are shown in Fig. 36.
Although these processes do not contribute directly to the EBL, they do
so indirectly, because the high-energy electrons undergo inverse Compton
scattering (ICS) off the CMB photons via *e* +
_{cmb}
*e* +
. This
gives rise to a flux of
high-energy photons which can be at least as important as those from
the direct (one-loop) neutralino decays considered in the previous
subsection
[361].

The spectrum of photons produced in this way depends on the rest energy
of the original neutralino. We consider first the case
*m*_{10}
10,
which is more or less pure ICS. The input ("zero-generation") electrons
are monoenergetic, but after multiple scatterings they are distributed
like *E*^{-2}
[362].
From this the spectrum of outgoing photons can be calculated as
[363]

(239) |

where

Here *E*_{e} = 1/3
*m*_{} *c*^{2} = (3.3 GeV)
*m*_{10} is the energy
of the input electrons, *m*_{e} is their rest mass, and
*E*_{cmb} = 2.7*kT*_{cmb}
is the mean energy of the CMB photons. Using
*m*_{e} *c*^{2} = 0.51 MeV and
*T*_{cmb} = 2.7 K, and allowing for decays at arbitrary
redshift *z* (after Berezinsky
[363]),
we obtain the expression
*E*_{max}(*z*) = (36 keV)
*m*_{10}^{2}(1 + *z*)^{-1}.

The halo SED may be determined as a function of wavelength by setting
*F*()
*d* =
*EN*(*E*) *dE* where *E* = *hc* /
. Normalizing
the spectrum so that
_{0}^{}
*F*()
*d* =
*L*_{h,tree}, we find:

(240) |

where _{} =
*hc* / *E*_{max} = (0.34 Å)
*m*_{10}^{-2}(1 + *z*) and
*L*_{h,tree} is the halo luminosity due to tree-level decays.

In the case of more massive neutralinos with
*m*_{10}
10, the
situation is complicated by the fact that outgoing photons become
energetic enough to initiate pair production via
+
_{cmb}
*e*^{+}
+ *e*^{-}. This injects new electrons
into the ICS process, resulting in electromagnetic cascades.
For particles which decay at high redshifts (*z*
100),
other processes such as photon-photon scattering must also be
taken into account
[364].
Cascades on *non*-CMB background photons may also be important
[365].
A full treatment of these effects requires detailed numerical analysis
[366].
Here we simplify the problem by assuming that the LSP is stable enough
to survive into the late matter-dominated (or vacuum-dominated) era.
The primary effect of cascades is to steepen the decay spectrum at
high energies, so that
[363]

(241) |

where

Here *E*_{0} is a minimum absorption energy. We adopt the
numerical expressions
*E*_{x} = (1.8 × 10^{3} GeV)(1 +
*z*)^{-1} and
*E*_{c} = (4.5 × 10^{4} GeV)(1 +
*z*)^{-1} after Protheroe
[367].
Employing the relation
*F*()
*d* =
*EN*(*E*) *dE* and normalizing as before, we find:

(242) |

where the new parameters are
_{x} =
*hc* / *E*_{x} = (7 × 10^{-9} Å)(1 +
*z*) and
_{c} =
*hc* / *E*_{c} = (3 × 10^{-10} Å)(1
+ *z*).

The luminosity *L*_{h,tree} is given by

(243) |

where *b*_{e} is now the branching ratio for all processes
of the form
*e* + all and
*E*_{e} = 2/3
*m*_{} *c*^{2}
is the total energy lost to the electrons.
We assume that all of this eventually finds its way into the EBL.
Berezinsky
[360]
supply the following branching ratio:

(244) |

Here *f*_{} parametrizes the composition of the neutralino,
taking the value 0.4 for the pure higgsino case.
With the halo mass specified by (175) and
*f*_{}
_{} /
(1 Gyr) as usual, we obtain:

(245) |

This is approximately four orders of magnitude higher than the halo
luminosity due to one-loop decays, and provides for the first time the
possibility of significant EBL contributions. With all adjustable
parameters taking values of order unity, we find that
*L*_{h,tree} ~ 2 × 10^{10}
*L*_{}, which
is comparable to the bolometric luminosity of the Milky Way.

The combined bolometric intensity of all neutralino halos is computed
as in the previous two sections. Replacing
*L*_{h,loop} in (230) with
*L*_{h,tree} leads to

(246) |

These are of the same order as (or higher than) the bolometric intensity of the EBL from ordinary galaxies, Eq. (21).

To obtain the spectral intensity, we substitute the SEDs
*F*_{ics}()
and
*F*_{casc}()
into Eq. (62). The results can be written

(247) |

where the quantities
*I*_{,tree} and
(*z*) are defined as
follows. For neutralino rest energies *m*_{10}
10 (ICS):

(248) | |||

Conversely, for *m*_{10}
10 (cascades):

(249) | |||

Numerical integration of Eq. (247) leads to the plots in Fig. 37. Cascades (like the pair annihilations we have considered already) dominate the -ray part of the spectrum. The ICS process, however, is most important at lower energies, in the x-ray region. We discuss the observational limits and the constraints that can be drawn from them in more detail in Sec. 8.6.

Gravitinos () are
the SUSY spin-3/2
counterparts of gravitons. Although often mentioned along with
neutralinos, they are not favoured as dark-matter candidates
in the simplest SUSY theories. The reason for this, known as the
*gravitino problem*
[305],
boils down to the fact that
they interact *too* weakly, not only with other particles
but with themselves as well. Hence they annihilate slowly and
survive long enough to "overclose" the Universe unless some other
way is found to reduce their numbers. Decays are one possibility,
but not if the gravitino is a stable LSP. Gravitino decay products must
also not be allowed to interfere with processes such as primordial
nucleosynthesis
[304].
Inflation, followed by a judicious
period of reheating, can thin out their numbers to almost any desired
level. But the reheat temperature
*T*_{R} must satisfy
*kT*_{R}
10^{12}
GeV or gravitinos will once again become too numerous
[309].
Related arguments based on entropy
production, primordial nucleosynthesis and the CMB power spectrum
force this number down to
*kT*_{R}
(10^{9}
- 10^{10}) GeV
[368]
or even *kT*_{R}
(10^{6} -
10^{9}) GeV
[369].
These temperatures
are incompatible with the generation of baryon asymmetry in the Universe,
a process which is usually taken to require
*kT*_{R} ~ 10^{14}
GeV or higher
[96].

Recent developments are however beginning to loosen the baryogenesis
requirement
[370],
and there are alternative models in which
baryon asymmetry is generated at energies as low as ~ 10 TeV
[371]
or even 10 MeV - 1 GeV
[372].
With this in mind
we include a brief look at gravitinos here. There are two possibilities:
(1) If the gravitino is *not* the LSP, then it decays early in
the history of the Universe, well before the onset of the
matter-dominated era. In models where the gravitino decays both
radiatively and hadronically, for example, it can be "long-lived
for its mass" with a lifetime of
_{}
10^{6} s
[373].
Particles of this kind have important
consequences for nucleosynthesis, and might affect the shape of the
CMB if _{} were
to exceed ~ 10^{7} s. However, they are
irrelevant as far as the EBL is concerned. We therefore restrict our
attention to the case (2), in which the gravitino is the LSP.
In light of the results we have already obtained for the neutralino,
we disregard annihilations and consider only models in which the LSP
can decay.

The decay mode depends on the specific mechanism of R-parity violation.
We follow Berezinsky
[374]
and concentrate on dominant
*tree-level* processes. In particular we consider the decay
*e*^{+}
+ all , followed by ICS off the CMB,
as in Sec. 8.4. The spectrum of photons produced by this
process is identical to that in the neutralino case, except that the
mono-energetic electrons have energy
*E*_{e} = 1/2 *m*_{}
*c*^{2} = (5 GeV) *m*_{10}
[374],
where *m*_{} *c*^{2} is the rest energy of the
gravitino and
*m*_{10}
*m*_{} *c*^{2} / (10 GeV) as before.
This in turn implies that
*E*_{max} = (81 keV) *m*_{10}^{2}(1 +
*z*)^{-1} and
_{} =
*hc* / *E*_{max} = (0.15 Å)
*m*_{10}^{-2}(1 + *z*). The values of
_{x} and
_{c} are
unchanged.

The SED comprises Eqs. (240) for ICS and (242) for cascades, as before. Only the halo luminosity needs to be recalculated. This is similar to Eq. (243) for neutralinos, except that the factor of 2/3 becomes 1/2, and the branching ratio can be estimated at [374]

(250) |

Using our standard value for the halo mass *M*_{h}, and
parametrizing the gravitino decay lifetime by
*f*_{}
_{} / (1
Gyr) as before, we obtain the following halo luminosity due to gravitino
decays:

(251) |

This is higher than the luminosity due to neutralino decays, and
exceeds the luminosity of the Milky Way by several times if
*f*_{} ~ 1.

The bolometric intensity of all gravitino halos is computed exactly
as before. Replacing *L*_{h,tree} in (230) with
*L*_{h,grav}, we find:

(252) |

It is clear that gravitinos must decay on timescales longer than the
lifetime of the Universe (*f*_{}
16), or they
would produce a background brighter than that of the galaxies.

The spectral intensity is the same as before, Eq. (247),
but with the new numbers for
_{} and
*L*_{h}. This results in

(253) |

where the prefactor
*I*_{} is defined as follows. For *m*_{10}
10 (ICS):

(254) | |||

Conversely, for *m*_{10}
10 (cascades):

(255) | |||

The function
(*z*) has the same
form as in Eqs. (248) and (249) and does not need
to be redefined (requiring only the new value for the cutoff wavelength
_{}). Because
the branching ratio *b*_{e} in (250)
is independent of the gravitino rest mass, *m*_{10} appears
in these results only through
_{}. Thus the
ICS part of the spectrum goes as
*m*_{10}^{-1} while the cascade part does not
depend on *m*_{10} at all.
As with neutralinos, cascades dominate the
-ray part
of the spectrum, and the ICS process is most important in the x-ray region.
Numerical integration of Eq. (253) leads to the results
plotted in Fig. 38. We proceed in the next
section to discuss these, comparing them to our
previous results for neutralinos, and beginning with an overview of the
observational constraints.

**8.6. The x-ray and
-ray
backgrounds**

The experimental situation as regards EBL intensity in the x-ray and -ray regions is more settled than that in the optical and ultraviolet. Detections (as opposed to upper limits) have been made in both bands, and are consistent with expectations based on known astrophysical sources. The constraints that we derive here are thus conservative ones, in the sense that the EBL flux which could plausibly be due to decaying WIMPs is almost certainly smaller than the levels actually measured.

At the lowest or soft x-ray energies, which lie roughly between
0.1-3 keV (4-100 Å), new measurements have been reported from the
Chandra spacecraft
[375].
(Universal conventions have not been
established regarding the boundaries between different wavebands; we
follow most authors and define these according to the different detection
techniques that must be used in each region.) These data appear as a
small bowtie-shaped box near
_{0} ~ 10
Å in Fig. 1,
where it can be seen that they interpolate beautifully between previous
detections at shorter and longer wavelengths.
The small rectangle immediately to the right of the Chandra bowtie (near
_{0} ~ 100
Å) in Fig. 1 comes from
measurements by the EUVE satellite in 1993
[376].

The hard x-ray background (3-800 keV, or 0.02-4 Å) is crucial in constraining the decays of low-mass neutralinos and gravitinos via the ICS process, as can be seen in Figs. 37 and 38. We have plotted two compilations of observational data in the hard x-ray band, both by Gruber [377, 378]. The first (labelled "G92" in Figs. 37 and 38) is an empirical fit to various pre-1992 measurements, including those from the Kosmos and Apollo spacecraft, HEAO-1 and balloon experiments. The range of uncertainty in this data increases logarithmically from 2% at 3 keV to 60% at 3 MeV [377]. The second compilation (labelled "G99") is a revision of this fit in light of new data at higher energies, and has been extended deep into the -ray region. New results from XMM-Newton [379] and the Rossi X-ray Timing Explorer [380] confirm the accuracy of this revised fit at low energies ("L02" and "R03" respectively in Figs. 37 and 38). The prominent peak in the range 3-300 keV (0.04-4 Å) is widely attributed to integrated light from active galactic nuclei (AGN) [381].

In the low-energy -ray region (0.8-30 MeV, or 0.0004-0.02 Å) we have used results from the COMPTEL instrument on the Compton Gamma-Ray Observatory (CGRO), which was operational from 1990-2000 [382]. Four data points are plotted in Figs. 33, 35, 37 and 38, and two more (upper limits only) appear at low energies in Figs. 37 and 38. These experimental results, which interpolate smoothly between other data at both lower and higher energies, played a key role in the demise of the "MeV bump" (visible in Figs. 37 and 38 as a significant upturn in Gruber's fit to the pre-1992 data from about 0.002-0.02 Å). This apparent feature in the background had attracted a great deal of attention from theoretical cosmologists as a possible signature of new physics. Figs. 37 and 38 suggest that it could also have been interpreted as the signature of a long-lived non-minimal SUSY WIMP with a rest energy near 100 GeV. The MeV bump is, however, no longer believed to be real, as the new fit ("G99") makes clear. Most of the background in this region is now suspected to be due to Type Ia supernovae (SNIa) [383].

We have included two measurements in the high-energy
-ray band
(30 MeV-30 GeV, or 4 × 10^{-7} - 4 × 10^{-4}
Å): one from
the SAS-2 satellite which flew in 1972-3
[384]
and one from the EGRET instrument which was part of the
CGRO mission along with COMPTEL
[385].
As may be seen in
Figs. 33, 35,
37 and 38,
the new results essentially extend the old ones to 120 GeV
(_{0} =
10^{-7} Å), with error bars which have been reduced by
a factor of about ten. Most of this extragalactic background is
thought to arise from unresolved blazars, highly variable AGN whose
relativistic jets point in our direction
[386].
Some authors have recently argued that Galactic contributions to the
background were underestimated in the original EGRET
analysis
[387,
388];
if so, the true extragalactic background intensity would be lower than
that plotted here, strengthening the constraints we derive below.

Because the extragalactic component of the
-ray
background has not been reliably detected beyond 120 GeV, we have fallen
back on measurements of *total* flux in the very high-energy
(VHE) region (30 GeV-30 TeV, or 4 × 10^{-10} -
4 × 10^{-7} Å). These were obtained from a
series of balloon experiments by Nishimura in 1980
[389],
and appear in Figs. 33 and
35 as filled dots
(labelled "N80"). They constitute a very robust upper limit on EBL flux,
since much of this signal must have originated in the upper atmosphere.
At the very highest energies, in the ultra high-energy (UHE) region
(> 30 TeV), these data join smoothly to upper limits on the diffuse
-ray flux
from extensive air-shower arrays such as HEGRA
(20-100 TeV
[390])
and CASA-MIA (330 TeV-33 PeV
[391]).
Here we reach the edge of the EBL for practical purposes, since
-rays with
energies of ~ 10 - 100 PeV are attenuated by pair
production on CMB photons over scales ~ 30 kpc
[392].

Some comments are in order here about units. For experimental reasons,
measurements of x-ray and
-ray
backgrounds are often expressed in terms of integral flux
*EI*_{E}( >
*E*_{0}), or number of photons with energies
above *E*_{0}. This presents no difficulties since the
differential spectrum in this region is well approximated with a single
power-law component,
*I*_{E}(*E*_{0})
= *I*_{*}(*E*_{0} /
*E*_{*})^{-}.
The conversion to integral form is then given by

(256) |

The spectrum is specified in either case by its index
together with the values of *E*_{*} and
*I*_{*} (or *E*_{0} and
*EI*_{E} in the integral
case). Thus SAS-2 results were reported as
=
2.35^{+0.4}_{-0.3} with
*EI*_{E} =
(5.5 ± 1.3) × 10^{-5} s^{-1} cm^{-2}
ster^{-1} for *E*_{0} = 100 MeV
[384].
The EGRET spectrum is instead fit by
= 2.10 ± 0.03 with
*I*_{*} = (7.32 ± 0.34) ×
10^{-9} s^{-1} cm^{-2}
ster^{-1} MeV^{-1} for
*E*_{*} = 451 MeV
[385].
To convert a differential flux in these units to
*I*_{} in CUs, one multiplies by
*E*_{0}/_{0} = *E*_{0}^{2} / *hc*
= 80.66*E*_{0}^{2} where *E*_{0}
is photon energy in MeV.

We now discuss our results, beginning with the neutralino annihilation
fluxes plotted in Fig. 33.
These are at least four orders of
magnitude fainter than the background detected by EGRET
[385]
(and five orders of magnitude below the upper limit set by the data of
Nishimura
[389]
at shorter wavelengths). This agrees
with previous studies assuming a critical density of neutralinos
[322,
351].
Fig. 33 shows that EBL contributions
would drop by another order of magnitude in the favoured scenario with
_{m,0}
0.3, and by another
if neutralinos are confined to galaxy halos
(_{m,0}
0.1*h*_{0}). Because the annihilation rate
goes as the square of the WIMP density, it has been argued that modelling
WIMP halos with steep density cusps might raise their luminosity,
possibly enhancing their EBL contributions by a factor of as much as
~ 10^{4} -10^{5}
[393].
While such a scenario might in principle
bring WIMP annihilations back up to the brink of observability in the
diffuse background, density profiles with the required steepness
are not seen in either our own Galaxy or those nearby. More recent
assessments have reconfirmed the general outlook discussed above in
Sec. 8.2; namely, that the best place to look for
WIMP annihilations is in the direction of nearby concentrations of dark
matter such as the Galactic center and dwarf spheroidals in the Local
Group
[340,
341].
The same stability that makes minimal-SUSY
WIMPs so compelling as dark-matter candidates also makes them hard to
detect.

Fig. 35 shows the EBL contributions from
one-loop neutralino
decays in *non*-minimal SUSY. We have put *h*_{0} =
0.75, *z*_{f} = 30 and
*f*_{R} = 1. Depending on
their decay lifetime (here parametrized by
*f*_{}),
these particles are capable in principle of producing a backgound
comparable to (or even in excess of) the EGRET limits.
The plots in Fig. 35 correspond to the smallest
values of *f*_{} that are consistent with the data for
*m*_{10} = 1, 3, 10, 30 and 100.
Following the same procedure here as we did for axions in
Sec. 6,
we can repeat this calculation over more finely-spaced intervals in
neutralino rest mass, obtaining a lower limit on decay lifetime
_{}
as a function of
*m*_{}. Results are shown
in Fig. 39 (dotted lines).
The lower limit obtained in this way varies from 4 Gyr for the lightest
neutralinos (assumed to be confined to galaxy halos with a total matter
density of
_{m,0} =
0.1*h*_{0}) to 70,000 Gyr for the heaviest (which provide
enough CDM to put
_{m,0} = 1).

Fig. 37 is a plot of EBL flux from
*indirect* neutralino
decays via the tree-level, ICS and cascade processes described in
Sec. 8.4. These provide us with our strongest constraints
on non-minimal SUSY WIMPs. We have set *h*_{0} = 0.75,
*z*_{f} = 30 and
*f*_{} = *f*_{R}
= 1, and assumed values of *f*_{} such that the
highest predicted intensities lie just under observational limits, as
before. Neutralinos at the light end of the mass range are constrained by
x-ray data, while those at the heavy end run up against
the EGRET measurements. Both the shape and absolute
intensity of the ICS spectra depend on the neutralino rest mass, but the
cascade spectra depend on *m*_{10} through intensity alone
(via the prefactor
*I*_{,tree}).
When normalized to the observational upper bound, all
curves for *m*_{10} > 10 therefore overlap. Normalizing
across the full range of neutralino rest masses (as for one-loop decays)
gives the lower bound on lifetime
_{}
plotted in Fig. 39
(dashed lines). This varies from 60 Gyr to 6 × 10^{7} Gyr,
depending on the values of
*m*_{} *c*^{2} and
_{m,0}.

Fig. 38, finally, shows the EBL contributions
from tree-level
*gravitino* decays. These follow the same pattern as the neutralino
decays. Requiring that the predicted signal not exceed the x-ray and
-ray
observations, we obtain the lower limits on decay lifetime
plotted in Fig. 39 (solid lines). The flatness
of these curves (relative to the constraints on neutralinos) is a
consequence of the fact that the branching ratio (250) is independent of
*m*_{10}. Our lower limits on
_{} range
from 200 Gyr to 20, 000 Gyr.

Let us sum up our findings in this section. We have considered
neutralinos and gravitinos, either of which could be the LSP and hence
make up the dark matter. In the context of non-minimal SUSY theories,
these particles can decay and contribute to both the x-ray and
-ray
backgrounds. We have shown that any such decay must occur
on timescales longer than 10^{2} - 10^{8} Gyr (for
neutralinos) or 10^{2} - 10^{4} Gyr (for gravitinos),
depending on their rest masses
and various theoretical input parameters. These results confirm that,
whether it is a neutralino or gravitino, *the LSP in non-minimal
SUSY theories must be very nearly stable *. To the extent that an
"almost-stable" LSP would require that R-party conservation be
violated at improbably low levels, our constraints suggest that the
SUSY WIMP either exists in the context of minimal SUSY theories,
or not at all.