Ax;ons
165
mass of the axion were greater than about 0.1 eV, the relic abundance would be
near to the equilibrium abundance. In fact. masses this large are already excluded
by the data, so the relic abundance depends upon the initial value. For example,
with ma saturating the SN 1987A bound (5.94), we estimate that
Ya(oo) '" 0.007 + 0.993 Ya(O).
(5.110)
Ya.eq
Ya.eq
At high temperatures, it is plausible to assume that there are no axions, so that
Ya (0) = 0, in which case the relic abundance is very far from thermal. In any
event, it is clear that thermal axions cannot provide anything like the measured
matter density. As in (5.16), we define 0: to be the fraction of the closure energy
density provided by thermal axions:
oth =
a - -
P:
(5.11l)
Pc
Then, analogously to (5.35), we find that
m = g •• dec Othh2(130 eV).
(5.112)
a
to
a
Saturating the measured value (5.22) of the current mass density would require
ma ..... 18 eV for closure, a value which is clearly excluded by the observational
bounds already obtained.
In all of the foregoing discussion, it was tacitly assumed that the classical
axion field had a constant value. in fact the value (5.61) needed to ensure that
the strong 9"-term vanishes. However, in the early universe, when the temperature
T ..... fa » AQCD, the U(l)PQ symmetry is broken and massless axions are
created. The potential which gives the axions a mass arises from non-perturbative
instanton effects only when the temperature drops to T '" AQCD. Thus, at high
temperatures, there is no reason why the axion should have the preferred value
for which 9" = O. When instantons generate a potential for the axion field. it
will roll towards the preferred value, so the foregoing assumption that the field
is a constant is not true during this era. This 'misalignment' of the field with its
ground-state value means that there is a non-zero axion field energy density which
we shall now calculate.
We assume that the axion field is spatially homogeneous and depends only
on time. Then. from (5.63), the effective axion action is
S = f d 4 x Ji(!a 2 - !m~a2 + raa)
= f d 4 x R3(t)(!ti 2 - !m~a2 + raa)
where R(t) is the cosmological scale factor and we have retained only the
quadratic (mass) term in the axion potential. The equation of motion is
dt
d [R 3 (a + r a)) + R 3 m;(T)a = O.
(5. 113)
165
mass of the axion were greater than about 0.1 eV, the relic abundance would be
near to the equilibrium abundance. In fact. masses this large are already excluded
by the data, so the relic abundance depends upon the initial value. For example,
with ma saturating the SN 1987A bound (5.94), we estimate that
Ya(oo) '" 0.007 + 0.993 Ya(O).
(5.110)
Ya.eq
Ya.eq
At high temperatures, it is plausible to assume that there are no axions, so that
Ya (0) = 0, in which case the relic abundance is very far from thermal. In any
event, it is clear that thermal axions cannot provide anything like the measured
matter density. As in (5.16), we define 0: to be the fraction of the closure energy
density provided by thermal axions:
oth =
a - -
P:
(5.11l)
Pc
Then, analogously to (5.35), we find that
m = g •• dec Othh2(130 eV).
(5.112)
a
to
a
Saturating the measured value (5.22) of the current mass density would require
ma ..... 18 eV for closure, a value which is clearly excluded by the observational
bounds already obtained.
In all of the foregoing discussion, it was tacitly assumed that the classical
axion field had a constant value. in fact the value (5.61) needed to ensure that
the strong 9"-term vanishes. However, in the early universe, when the temperature
T ..... fa » AQCD, the U(l)PQ symmetry is broken and massless axions are
created. The potential which gives the axions a mass arises from non-perturbative
instanton effects only when the temperature drops to T '" AQCD. Thus, at high
temperatures, there is no reason why the axion should have the preferred value
for which 9" = O. When instantons generate a potential for the axion field. it
will roll towards the preferred value, so the foregoing assumption that the field
is a constant is not true during this era. This 'misalignment' of the field with its
ground-state value means that there is a non-zero axion field energy density which
we shall now calculate.
We assume that the axion field is spatially homogeneous and depends only
on time. Then. from (5.63), the effective axion action is
S = f d 4 x Ji(!a 2 - !m~a2 + raa)
= f d 4 x R3(t)(!ti 2 - !m~a2 + raa)
where R(t) is the cosmological scale factor and we have retained only the
quadratic (mass) term in the axion potential. The equation of motion is
dt
d [R 3 (a + r a)) + R 3 m;(T)a = O.
(5. 113)