ESTIMATES OF THE COSMOLOGICAL AXION DENSITY

In the scenario where the temperature in the early universe is high enough to restore the Peccei-Quinn symmetry, previous estimates of the axion mass density are reconsidered. The axion contribution to the density parameter is of the form OMEGA(a) = K(f(a)/10(12) GeV)1.2 where f(a)/square-root 2 is the vacuum value of the magnitude of the Peccei-Quinn field. If axions radiated by strings are ignored, or if they have a flat spectrum as advocated by Harari and Sikivie, K approximately k-gamma(7/6), where gamma(-1) < 1 is the string spacing in Hubble units and k congruent-to 1-10. If their spectrum is peaked at frequency gamma/t as advocated by Davis, K approximately 10k-gamma(7/6). If gamma approximately 1, the first estimate is in essential agreement with earlier ones but the second is a factor of 7 smaller, which widens the allowed window for f(a). These estimates include only the axions coming from oscillations of the axion field between the walls. In addition there are axions radiated by the walls, whose relative contribution to K is of order r3/2 in the first case and 0.1 r3/2 in the second case, r being the time of wall annihilation divided by the time of wall formation. It could dominate, and close the axion window, if the walls survive for many Hubble times.