  The misalignment mechanism for axion production depends on the
temperature-dependent axion mass. The latter has recently been determined
within the interacting instanton liquid model (IILM), and provides for the
first time a well-motivated axion mass for all temperatures. We reexamine the
constraints placed on the axion parameter space in the light of this new mass
function. We find an accurate and updated constraint $ f_a \le 2.8(\pm2)\times
10^{11}\units{GeV}$ or $m_a \ge 21(\pm2) \units{\mu eV}$ from the misalignment
mechanism in the classic axion window (thermal scenario). However, this is
superseded by axion string radiation which leads to $ f_a \lesssim
3.2^{+4}_{-2} \times 10^{10} \units{GeV}$ or $m_a \gtrsim 0.20 ^{+0.2}_{-0.1}
\units{meV}$. In this analysis, we take care to precisely compute the effective
degrees of freedom and, to fill a gap in the literature, we present accurate
fitting formulas. We solve the evolution equations exactly, and find that
analytic results used to date generally underestimate the full numerical
solution by a factor 2-3. In the inflationary scenario, axions induce
isocurvature fluctuations and constrain the allowed inflationary scale $H_I$.
Taking anharmonic effects into account, we show that these bounds are actually
weaker than previously computed. Considering the fine-tuning issue of the
misalignment angle in the whole of the anthropic window, we derive new bounds
which open up the inflationary window near $\theta_a \to \pi$. In particular,
we find that inflationary dark matter axions can have masses as high as
0.01--1$\units{meV}$, covering the whole thermal axion range, with values of
$H_I$ up to $10^9$GeV. Quantum fluctuations during inflation exclude dominant
dark matter axions with masses above $m_a\lesssim 1$meV.
