164
Relic neutrinos and axions
The thennally-averaged cross section is
( 1 I)
2
-2-2
(1 v abs = gaNNx m"
(5.104)
with
mN
(5.105)
gaNN ~ IPQ'
In the radiation-dominated era,
1/2
H = ( 8np )
= 211' (lI'g.)1/2 m1 .
(5.106)
3m~
3 5
x 2 mp
Putting all this together, we get
rabs(X) = 3gN
8g.
1/2 ( 312 1/2)2
( ~ )
mN mp
x-3/2e-x
H(x)
211'3
fpQm7r
10 1/2 (
mu
-3/2 -x
~ ( - )
)2 x e
(5.107)
g.
1.2 x 10- 3 eV
using (5.64). Above the quark-hadron phase transition, the processes (5.95)
dominate and r abs/ H scales as x, achieving its maximum value just after the
transition. Thus, we can estimate the final relic abundance as
Yu(O»)
Yu(oo) = Yu.eq 1 1 - ( 1 - Yu.eq
3
2!~ (8g.)
5 1/2 ( 3/2 1/2)2
]
x exp [ -
m;:'Q::
I (Xqlt)
I
= - -
0. 278 1 1 - (1 - 3.6g •• dec:Ya(O))
g •• dec
x exp [- G~)'" C.2 x ~~, ev)' I(X.')] I
(5.108)
where
I(Xqh) == 100
x,,,
x,-5/2e- x ' dx'
= - ~[x;:/2e-x,,, (2xqlt - 1) + 2,Jif(erf(Jxqlt) - 1)] (5.109)
and g •. dec is the value of g. at decoupling (freeze-out). The parameter Xqh "" 5 is
the value of x at the quark-hadron phase transition, so that I '" 1 0-4. Thus, if the