The Net Advance of Physics: The Nature of Dark Matter, by Kim Griest -- Section 6C.
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The number density of any particle which was once in thermal
equilibrium in the Early Universe can be found by solving the
relevant set of Boltzmann equations. In most cases only one is
needed:
where H is the Hubble constant, n is the number density, t is time,
and
is the thermally averaged cross section times the relative
velocity of the interacting particles.
We are using to denote the
LSP. The first term on the right-hand side is the reduction in LSP
number density due to the Hubble expansion, the second term is
reduction due to self-annihilation, and the third term is the
increase due to particle production. The third term can be
simplified using the fact that ``ordinary" particles such as quarks
and electrons stay in thermal equilibrium throughout the period
during which the Wimp number density ``freezes out" (see Section
4). When thermal equilibrium obtains, creation equals
annihilation, so the second and third terms are equal. Therefore
one can eliminate the ``ordinary particle" cross sections and
number densities and find the usual equation
Starting at an early time when all particles were in equilibrium, one
integrates this equation either numerically or using the standard
``freeze-out" approximation [21], and obtains the number density
at t=0 (today). The relic abundance
is simply ,
where
is the mass of the LSP.
The difficult step in obtaining the current day density is usually the
calculation of the annihilation cross section of two LSPs into all
standard model particles. In order to perform this calculation, one
must first determine which particle is the LSP, and then evaluate all
the relevant Feynman diagrams. Going through the list of
supersymmetric particles, one finds, basically by process of
elimination, that only the sneutrino and neutralino are likely
candidates. In the vast majority of models, the neutralino is favored
over the sneutrino, so most work has concentrated on the
neutralino as dark matter candidate.
For the neutralino, several dozen Feynman diagrams contribute to
self-annihilation, including possible annihilation into quarks,
leptons, W, Z, and Higgs bosons, and involving most of the
super-partners as exchange particles. So in order to perform the
calculation one needs to first obtain the mass and couplings of all
the supersymmetric particles. Since supersymmetry is broken, the
mass terms are unknown, giving rise to many free parameters in the
most general supersymmetric model. Usually, in order to simplify
things, one considers the ``minimal" supersymmetric model,
the model with the fewest new particles, but still there are many
undetermined parameters. So to further simplify, several other
assumptions are usually made.
In minimal supergravity, some GUT scale assumptions can reduce
the number of parameters to just a few. In what follows, we use
some, but not all, of the supergravity assumptions, and have as a
result 5 free parameters [22]. The parameters are the gaugino mass
parameters and
,
the Higgsino mass parameter
, the
pseudoscalar Higgs mass ,
and the ratio of Higgs vacuum
expectation values .
For any set of parameters, one can
calculate all the masses, mixings, and couplings, and then the
annihilation cross section. Thus after a long calculation, one finally
obtains in terms of the five
parameters. If one obtains a relic
abundance in the range
, then that set of
parameters defines a potential dark matter candidate. However,
before deciding that this is a dark matter candidate one must ensure
that one of the many accelerator experiments that have searched
for supersymmetric particles has not already ruled out that model.