  We study for what specific values of the theoretical parameters the axion can
form the totality of cold dark matter. We examine the allowed axion parameter
region in the light of recent data collected by the WMAP5 mission plus baryon
acoustic oscillations and supernovae \cite{komatsu}, and assume an inflationary
scenario and standard cosmology. We also upgrade the treatment of
anharmonicities in the axion potential, which we find important in certain
cases. If the Peccei-Quinn symmetry is restored after inflation, we recover the
usual relation between axion mass and density, so that an axion mass $m_a =(85
\pm 3){\rm \mu eV}$ makes the axion 100% of the cold dark matter. If the
Peccei-Quinn symmetry is broken during inflation, the axion can instead be 100%
of the cold dark matter for $m_a < 15{\rm meV}$ provided a specific value of
the initial misalignment angle $\theta_i$ is chosen in correspondence to a
given value of its mass $m_a$. Large values of the Peccei-Quinn symmetry
breaking scale correspond to small, perhaps uncomfortably small, values of the
initial misalignment angle $\theta_i$.
