168
Relic neutrinos and axjons
topological configurations arise because the fundamental group of the manifold
SI associated with symmetry group U(I) of the ground state is non-trivial:
:11' 1 (Si) = z.
(5.128)
As we traverse any closed path threaded by a vortex, the phase of the condensate
varies continuously and changes by an integral multiple of 2:11' when we return to
the starting point By shrinking the size of the closed path, it is clear that there
is a linear vortex on which the phase of the order parameter is undefined. In a
real superftuid, there is a cylindrical core region, centred on this line, in which the
magnitude of the order parameter varies, approaching zero on the line.
Similar considerations apply to our U{I)pQ symmetry. In the early universe
when the PQ symmetry is broken, we expect the axion field to vary spatially, since
it is uncorrelated beyond the horizon. These topological considerations (S.128)
indicate that a random 'axion string' network will. therefore, be formed [33] just
as vortex configurations are formed in superftuid 4He. The thickness of the core
region is ~ ..... f G - I • Roughly, there are two types of string: long strings, spanning
the horizon, and small string loops. The loops oscillate and radiate axions, and this
is the dominant energy-loss mechanism [11,34]. The axions are massless when
they are emitted and the emission continues until they acquire a mass via instanton
effects when the temperature drops to T ..... AQCD. A numerical simulation of a
random network of (global) axion strings has been performed recently [3S, 36].
This shows that, after a short initial period of relaxation, the network evolves to
a 'scaling' regime, in which the large-scale behaviour of the network scales with
the Hubble radius and the energy density is given by
striq
~IL
(S.129)
PG
= 12
where ~ is a constant and IL is the string tension per unit length. Such behaviour
was predicted theoretically by Albrecht and Turok [37].The radiated axions have
a momentum spectrum g{k) which is peaked around wavelengths of order of
the horizon scale {k- I '" (41r H)-I) and which decays exponentially for shorter
wavelengths. The contribution Q~I to the current fractional relic axion energy
density is calculated as follows:
m
-1.18
-2
G
)
Q~g ~ (0.39 ± 0.26)h (IO-S eV
(S.130)
which is somewhat larger than, but comparable with, the value obtained from
the misalignment mechanism if we take the rms value for li;. So applying the
measured matter density bound (S.22) requires the axion mass to be greater than
about lO-s eV. as before. The numerical simulation was performed on a 256 3
lattice but it has been noted [38] that this might not be sufficient to observe
logarithmic corrections, proportional to ,-2 In t, to the scaling behaviour (S.129).
Such corrections would have the effect of enhancing axion production at later
Relic neutrinos and axjons
topological configurations arise because the fundamental group of the manifold
SI associated with symmetry group U(I) of the ground state is non-trivial:
:11' 1 (Si) = z.
(5.128)
As we traverse any closed path threaded by a vortex, the phase of the condensate
varies continuously and changes by an integral multiple of 2:11' when we return to
the starting point By shrinking the size of the closed path, it is clear that there
is a linear vortex on which the phase of the order parameter is undefined. In a
real superftuid, there is a cylindrical core region, centred on this line, in which the
magnitude of the order parameter varies, approaching zero on the line.
Similar considerations apply to our U{I)pQ symmetry. In the early universe
when the PQ symmetry is broken, we expect the axion field to vary spatially, since
it is uncorrelated beyond the horizon. These topological considerations (S.128)
indicate that a random 'axion string' network will. therefore, be formed [33] just
as vortex configurations are formed in superftuid 4He. The thickness of the core
region is ~ ..... f G - I • Roughly, there are two types of string: long strings, spanning
the horizon, and small string loops. The loops oscillate and radiate axions, and this
is the dominant energy-loss mechanism [11,34]. The axions are massless when
they are emitted and the emission continues until they acquire a mass via instanton
effects when the temperature drops to T ..... AQCD. A numerical simulation of a
random network of (global) axion strings has been performed recently [3S, 36].
This shows that, after a short initial period of relaxation, the network evolves to
a 'scaling' regime, in which the large-scale behaviour of the network scales with
the Hubble radius and the energy density is given by
striq
~IL
(S.129)
PG
= 12
where ~ is a constant and IL is the string tension per unit length. Such behaviour
was predicted theoretically by Albrecht and Turok [37].The radiated axions have
a momentum spectrum g{k) which is peaked around wavelengths of order of
the horizon scale {k- I '" (41r H)-I) and which decays exponentially for shorter
wavelengths. The contribution Q~I to the current fractional relic axion energy
density is calculated as follows:
m
-1.18
-2
G
)
Q~g ~ (0.39 ± 0.26)h (IO-S eV
(S.130)
which is somewhat larger than, but comparable with, the value obtained from
the misalignment mechanism if we take the rms value for li;. So applying the
measured matter density bound (S.22) requires the axion mass to be greater than
about lO-s eV. as before. The numerical simulation was performed on a 256 3
lattice but it has been noted [38] that this might not be sufficient to observe
logarithmic corrections, proportional to ,-2 In t, to the scaling behaviour (S.129).
Such corrections would have the effect of enhancing axion production at later