5.2 Boltzmann Equation and Relic Density
73
the product of the corresponding cross section σ times the number density of the
particles partaking n, times the their relative velocity v.
This process proceeds as long as the rate is larger than the Hubble constant
H (T ) =
π
√
g ∗ (T )
√
90
T
2
m Pl
,
(5.2)
where m Pl the Planck mass and g ∗ (T ) is the number of effective degrees of freedom
at the given temperature is given by
g ∗ (T ) =
bosons
g b
T b
T
4
+
7
8
fermions
g f
T f
T
4
,
(5.3)
where g b, f is the number of degrees of freedom of the corresponding particle. The
value of the function g ∗ (T ) goes from 106.5 above the EW phase transition to 3.38
at temperature around 0.1 MeV.
After < H , the particles are decoupled and their number density frozen.
The number density at the equilibrium at a given temperature T (for k B = 1) is
given by
n eq (T ) = g ∗
d
3 p
(2π) 3
1
e E/T ± 1
=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
g ∗
mT
2π
3/2
e
−m/T non-relativistic (T m)
ζ(3)
π 2 g ∗ T
3
relativistic bosons (T m)
3
4
ζ(3)
π 2 g ∗ T
3
relativistic fermions (T m) ,
(5.4)
where ζ(3) 1.2
The number density n(t) of a weakly interacting, massive particle χ at a certain
time t in the evolution of the Universe is computed by means of the Boltzmann
equation
˙
n(t) + 3 H (t)n(t) = −−σ χχ→ f f v
n
2
(t) − n
2
eq (t)
,
(5.5)
where H (t) is the Hubble constant and σ χχ→ f f v is the thermal average of the
cross section for a pair of the particles χ, with relative velocity v = (s − 4m
2
χ )/m
2
χ ,
to annihilate into SM fermions f ; this term depletes the density as the particles χ
turns into SM fermions. The thermal average is defined as
σ χχ→ f f v =
4m
2
χ
∞ ds
√
s(s − 4m
2
χ )K 1
√
s
T
σ χχ→ f f
8m 4
χ T
K 2
m χ
T
2
,
(5.6)
73
the product of the corresponding cross section σ times the number density of the
particles partaking n, times the their relative velocity v.
This process proceeds as long as the rate is larger than the Hubble constant
H (T ) =
π
√
g ∗ (T )
√
90
T
2
m Pl
,
(5.2)
where m Pl the Planck mass and g ∗ (T ) is the number of effective degrees of freedom
at the given temperature is given by
g ∗ (T ) =
bosons
g b
T b
T
4
+
7
8
fermions
g f
T f
T
4
,
(5.3)
where g b, f is the number of degrees of freedom of the corresponding particle. The
value of the function g ∗ (T ) goes from 106.5 above the EW phase transition to 3.38
at temperature around 0.1 MeV.
After < H , the particles are decoupled and their number density frozen.
The number density at the equilibrium at a given temperature T (for k B = 1) is
given by
n eq (T ) = g ∗
d
3 p
(2π) 3
1
e E/T ± 1
=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
g ∗
mT
2π
3/2
e
−m/T non-relativistic (T m)
ζ(3)
π 2 g ∗ T
3
relativistic bosons (T m)
3
4
ζ(3)
π 2 g ∗ T
3
relativistic fermions (T m) ,
(5.4)
where ζ(3) 1.2
The number density n(t) of a weakly interacting, massive particle χ at a certain
time t in the evolution of the Universe is computed by means of the Boltzmann
equation
˙
n(t) + 3 H (t)n(t) = −−σ χχ→ f f v
n
2
(t) − n
2
eq (t)
,
(5.5)
where H (t) is the Hubble constant and σ χχ→ f f v is the thermal average of the
cross section for a pair of the particles χ, with relative velocity v = (s − 4m
2
χ )/m
2
χ ,
to annihilate into SM fermions f ; this term depletes the density as the particles χ
turns into SM fermions. The thermal average is defined as
σ χχ→ f f v =
4m
2
χ
∞ ds
√
s(s − 4m
2
χ )K 1
√
s
T
σ χχ→ f f
8m 4
χ T
K 2
m χ
T
2
,
(5.6)