Now let us consider the cosmological implications [3-5]. PQ symmetry is unbroken at temperatures T ^ F. When the symmetry breaks, the initial value of the phase, that is ea/F, is random beyond the then-current particle horizon scale. One can analyze the fate of these fluctuations by solving the equations for a scalar field in an expanding universe. The only unusual feature is that the effective mass of the axion field depends on temperature. The axion mass is very small for T ^ A, even relative to its zero-temperature value, because the non-perturbative QCD effects that generate it involve coherent gluon field fluctuations (instantons) which are suppressed at high temperature. It saturates, of course, for T ^ A. The full temperature dependence of the mass can be pretty reliably estimated, although the necessary calculations are technically demanding.

From standard treatments of scalar fields in an expanding universe, we learn that there is an effective cosmic viscosity, which keeps the field frozen so long as the expansion parameter is large compared to the mass, H = R/R ยป m. In the opposite limit, H ^ m, the field undergoes lightly damped oscillations, which result in an energy density that decays as p a 1/R3. At intermediate times there is a period of quasi-adiabatic damping. This damping has a consequence that is very important for the present discussion, namely that the final mass density, normalized to the ambient T3, varies roughly proportional to FQ2. The qualitative feature, that the final density decreases with decreasing F, may appear paradoxical, since the axions are getting heavier, but it is not hard to understand heuristically. For smaller values of F, corresponding to larger mass, the temperature at which the axion field begins to feel the effect of cosmic viscosity sets in earlier, and there are more damping cycles. However, the initial energy density depends only on the mismatch angle Q and is independent of F. The time-oscillating field can be interpreted as pressureless matter or dust (note that spatial inhomogeneities on small scales, which would provide pressure, begin to be damped as they enter the horizon). In simple words, we can say that the initial misalignment in the axion field, compared with what later turns out to be the favoured value, relaxes by emission of axions in a very cold coherent state, or Bose-Einstein condensate. It is not in thermal equilibrium with ordinary matter; the interactions are far too weak to enforce that equilibrium.

If we ignore the possibility of inflation, then - for the large values of F of interest - the horizon scale at the PQ transition at T & F corresponds to a spatial region today that is negligibly small on cosmological scales. Thus, in calculating the axion density we are justified in performing an average over the initial mismatch angle. This allows us to calculate a unique prediction for the density, given the microscopic model. The result of the calculation is usually quoted in the following form:

where pdark is the dark energy density. In this way, we would deduce that axions form a good dark matter candidate for F ~ 1012 GeV and that larger values of F are forbidden. These conclusions are unchanged if we allow for the possibility that an epoch of inflation preceded the PQ transition.

Things are very different, however, if inflation occurs after the PQ transition [12].2 For then, the effective Universe accessible to present-day observation, instead of containing many horizon-volumes from the time of the PQ transition, is contained well within just one. It is therefore not appropriate to average over the initial mismatch angle. We have to restore it as a contingent universal constant. That is, it is a pure number that characterizes the observable Universe as a whole, but which clearly cannot be determined from any more basic quantities, even in principle - indeed it is a different number elsewhere in the multiverse!

In that case, it is appropriate to replace Eq. (9.1) by:

This differs from the earlier form in that I have normalized the axion density relative to baryon density rather than dark matter density. This change is completely trivial at a numerical level, of course. (For concreteness, I have taken pdark/pbaryon = 6.) It reflects, however, two important ideas. First, changes in the mismatch angle 9 do not significantly affect baryogenesis, so that the baryon density is a fixed proportion of the photon density at high temperature, and provides an appropriate gauge for measuring the

2 The logical possibility of axion cosmology based on that large F and small initial mismatch in inflationary cosmology has been known since the publication of ref. [3]. The present discussion extends a portion of ref. [12].

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