Etude de bruit de fond induit par les muons dans l'expérience ...

tel-00724955, version 1 - 23 Aug 2012
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26 The Dark Matter problem
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Figure 1.16: Evolution of the WIMP’s commoving number density as function of
temperature (in units of the WIMP mass m) in the early Universe. The solid curve
represents the equilibrium abundance, while the dashed lines correspond to actual
abundances for various choices of velocity-weighted annihilation cross section. The
temperature of the freeze out occurs when the reaction rate drops below the expansion
rate. Figure from [21].
where g is the number of internal degrees of freedom of the particle and f( p) is
the Fermi-Dirac or Bose-Einstein distribution. At high temperatures (T ≫ mχ),
neq χ ∝ T 3 , so that the number of photons and WIMPs is roughly the same. As the
Universe expands and cools down to a temperature below mχ, (T ≪ mχ), the
WIMP density is neq χ ∝ g(mχT/2π) 3/2 exp(−mχ/T ), that is Boltzmann suppressed.
At T ∼ mχ, the number density of WIMPs falls exponentially, and the rate for
annihilation χ’s (Γ = 〈σAv〉nχ) drops below the expansion rate, Γ H. At this
moment, WIMPs can no longer annihilate. The interactions, which maintained the
thermal equilibrium, freeze out, forming a relic cosmological abundance that remains
at present times [58].
The quantitative way of describing this process is done by using the Boltzmann
equation, which describes the time evolution of the number density nχ(t) of WIMPs:
dnχ
dt + 3Hnχ = −〈σAv〉 (nχ) 2 − (n eq
χ ) 2
(1.31)
where H = ˙a/a is the Hubble expansion rate, and a is the scale factor of the Universe.
The second term on the left-hand side accounts for the expansion of the Universe.
The first term in brackets on the right-hand side accounts for depletion of WIMPs
due to annihilation, and the second term arises from creation of WIMPs from the
inverse reaction. This equation can be derived by imposing that, in equilibrium, the
rate for annihilation and creation of WIMPs is equal. Accurate calculations require
a numerical solution of the Boltzmann equation, but an approximate solution for