Neutralino dark matter
183
with the present neutralino abundance YlC.o given by
3.785
I
(mx )
YlCo=
-
(6.55)
•
.jg •• Tdl:cmpmlC Tdec a + 3(b - a/4)Td«/m x
and the freeze-out temperature satisfying
mlC = In[0.0764mp(a + 6bTd«/m lC)c(2 + c)mlC(g •• Tdl:cmlC/Td«)-1/2] (6.56)
Tdec
which can be solved iteratively; c is a numerical constant of order unity
determining when the early- and late-time solutions are matched: c :::: i typically
gives a 5-10% precision. In special circumstances this estimate can be wrong
by factors of two or more. These are (i) when the annihilation occurs near an
s-channel pole; (ii) when the annihilation occurs near a mass threshold; and (iii)
when there is 'co-annihilation', i.e. when there is another particle (X') (e.g. a
squark) with a mass that only slightly exceeds m l C , and the X can be converted to
a X' via scattering from standard model particles. If the annihilation cross section
for the X's is larger than that of the xs, then the abundance of both is controlled
by the annihilation of the heavier and more strongly interacting particle. These
special cases are important in practice and allow certain regions of the MSSM
parameter space that would otherwise be forbidden.
The coefficients a and b in (6.53) are bounded above by partial-wave
unitarity arguments, with the bounds being of order m x' essentially on
dimensional grounds. This leads to a model-independent lower bound on the
relic abundance [21]
m
)2
(6.57)
f2 l C .o ~ (200;eV .
Using the upper bound in (5.22) then gives mlC $ 100 TeV. Of course,
in the MSSM models with which we are concerned, the cross sections are
proportional to a;m, so the largest cosmologically acceptable WIMP mass will
be reduced by a factor of aem ...... 10- 2 from this most conservative bound.
Thus, in supersymmetric models, we expect m l C $ I TeV to be required by the
cosmological constraint.
The calculation of the annihilation cross section in the MSSM is
straightforward in principle but quite complicated in practice and we shaH only
comment on the salient features. The most important channels for neutralino
annihilation are those that appear in lowest order (tree-level) perturbation theory,
see figure 6.1. These are annihilation into a pair of fermions
x?x? -+ ji (/=q,l,v)
(6.58)
and into a pair of bosons
X?X? -+ W+W-, ZOZO, W±H~, ZOAo, ZOHo, ZOho, H+H(6.59)
183
with the present neutralino abundance YlC.o given by
3.785
I
(mx )
YlCo=
-
(6.55)
•
.jg •• Tdl:cmpmlC Tdec a + 3(b - a/4)Td«/m x
and the freeze-out temperature satisfying
mlC = In[0.0764mp(a + 6bTd«/m lC)c(2 + c)mlC(g •• Tdl:cmlC/Td«)-1/2] (6.56)
Tdec
which can be solved iteratively; c is a numerical constant of order unity
determining when the early- and late-time solutions are matched: c :::: i typically
gives a 5-10% precision. In special circumstances this estimate can be wrong
by factors of two or more. These are (i) when the annihilation occurs near an
s-channel pole; (ii) when the annihilation occurs near a mass threshold; and (iii)
when there is 'co-annihilation', i.e. when there is another particle (X') (e.g. a
squark) with a mass that only slightly exceeds m l C , and the X can be converted to
a X' via scattering from standard model particles. If the annihilation cross section
for the X's is larger than that of the xs, then the abundance of both is controlled
by the annihilation of the heavier and more strongly interacting particle. These
special cases are important in practice and allow certain regions of the MSSM
parameter space that would otherwise be forbidden.
The coefficients a and b in (6.53) are bounded above by partial-wave
unitarity arguments, with the bounds being of order m x' essentially on
dimensional grounds. This leads to a model-independent lower bound on the
relic abundance [21]
m
)2
(6.57)
f2 l C .o ~ (200;eV .
Using the upper bound in (5.22) then gives mlC $ 100 TeV. Of course,
in the MSSM models with which we are concerned, the cross sections are
proportional to a;m, so the largest cosmologically acceptable WIMP mass will
be reduced by a factor of aem ...... 10- 2 from this most conservative bound.
Thus, in supersymmetric models, we expect m l C $ I TeV to be required by the
cosmological constraint.
The calculation of the annihilation cross section in the MSSM is
straightforward in principle but quite complicated in practice and we shaH only
comment on the salient features. The most important channels for neutralino
annihilation are those that appear in lowest order (tree-level) perturbation theory,
see figure 6.1. These are annihilation into a pair of fermions
x?x? -+ ji (/=q,l,v)
(6.58)
and into a pair of bosons
X?X? -+ W+W-, ZOZO, W±H~, ZOAo, ZOHo, ZOho, H+H(6.59)