Quantitative analysis of the thermal damping of coherent axion oscillations

Unruh and Wald have recently discussed a new mechanism for damping coherent axion oscillations, ‘‘thermal damping,’’ which occurs due to the temperature dependence of the axion mass and neutrino viscosity. We investigate the effect quantitatively and find that the present energy density in axions can be written as ${\rho }_a$=${\rho }_{a0}$/(1+$J_{\mathrm{UW})}$, where ${\rho }_{a0}$ is what the axion energy density would be in the absence of the thermal-damping effect and $J_{\mathrm{UW}}$ is an integral whose integrand depends upon (${\mathrm{dm}}_a$/dT${)}^2$. As a function of $f_{\mathrm{PQ}(\mathrm{\equiv }}$Peccei-Quinn symmetry-breaking scale) $J_{\mathrm{UW}}$ achieves its maximum value for $f_{\mathrm{PQ}\mathrm{\simeq }3\times {10}^{12}}$ GeV; unless the axion mass turn-on is very sudden, ‖(T/$m_a$)(${\mathrm{dm}}_a$/dT)‖≫1, $J_{\mathrm{UW}}$ is ≪1, implying that this damping mechanism is not significant.