74
5 Appendix
where K 1 and K 2 are the Bessel function of second kind. It is usually computed after
expanding
σ χχ→ f f v = =s 0 + s 1 v
2
+ O(v
4
)
(5.7)
with s 0 the cross section in the s-wave and s 1 the first correction in the p-wave. The
leading term is s 0 for the dark sector Dirac fermions interacting through the dark
photon, in both the s- and t-channel.
Equation (5.5) is usually re-written in terms of the function Y (t) = n(t)/T
3 and
the variable x = m χ /T =
2t H(T = m χ ) as
dY
dx
= −
λ(x)
x 2
Y
2
(x) − Y
2
eq
(5.8)
with
λ(x) =
m
3
χ σ χχ→ f f v
H (T = m χ )
,
(5.9)
and in this form numerically solved.
Equation (5.8) can be solved analytically by dropping the second term Y
2
eq (x)—
which is small because decreasing like e
−x —approximating
σ χχ→ f f v = σ χχ→ f f v + O(v
2
) ,
(5.10)
where v =
√
2/x and writing
λ(x) =
√
180 m Pl m χ
π
√
g ∗ x
σ χχ→ f f
(5.11)
by means of
H (T = m χ ) =
π
g ∗ (T = m χ )
90
m
2
χ
m Pl
.
(5.12)
The solution for x
larger than decoupling temperature x d is
Y (x
) =
x d
λ
.
(5.13)
This quantity is related to the relic density
ρ χ = m χ n(x
) = m
4
χ
Y (x
)
28x d
(5.14)
or, in terms of the normalized quantity χ = ρ χ /ρ c as