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Relic neutrinos and axions
The decay width r a of the axion is tiny, so we may safely ignore it henceforth.
Initially, at high temperatures T » AQCD, the axion is massless and we assume
that a = O. Then the field is constant:
a(t) = aj
(5.114)
where aj is the initial 'misaligned' value of the field. As the temperature falls
m;(T) increases and the equation of motion is
a + 3Hiz +m~(T)a = 0
(5.115)
where, as usual, H == RI R is the Hubble parameter. Eventually, the temperature
reaches T; at which
ma(T;) = 3H(T;)
(5.116)
and, thereafter, a(t) oscillates with frequency ma(T). The energy density
associated with the axion field is
Pa = }a 2 + }m;a 2
(5.117)
so, using (5.115),
•
•
2
3H·
Pa = mamaa - a.
(5.118)
Averaging over one oscillation
(a 2 ) = m; (a 2 )
(5.119)
and then (5.118) gives
(Pa) = (:: - 3H) (Pa)
(5.120)
whose solution is
(Pa)R 3 (t) ex ma(T).
(5.121)
Thus, the axion number density na = (Pa)/ma(T) scales as R- 3 (t), even though
the axion mass is varying. The entropy density s also scales in this way, so
assuming that there has been no entropy production since the axion field began
to oscillate, their ratio is conserved. When the temperature T = T;, given by
(5.116),
Pa = lm~(T;)ar
(5.122)
since initially iz = 0, and
na I 45ma(T;)ar 45al
(5.123)
-; T; = 41r 2g.1:3 = 2"/s1rg.T;mp·
The present (misaligned) axion energy density is given as a fraction of the closure
energy density by
n:m = p;gs = na I ma so .
(5.124)
Pr:
S T;
Pr: