Heavier $W$-boson, dark matter and gravitational waves from strings in
  an SO(10) axion model
  Inspired by the recent determination of the $W$-boson mass by the CDF
collaboration, we revisit an SO(10) axion model in which a scalar $SU(2)_L$
triplet field with zero hypercharge is known to acquire a non-zero VEV through
its mixing with the Standard Model Higgs doublet. The triplet VEV provides a
sizable contribution to the W mass, which helps in significantly lowering the
$7\sigma$ discrepancy between the SM prediction and the higher CDF value for
$m_W$. We show that the relatively light triplet mass ($\sim (1-50)$ TeV) is
compatible with gauge coupling unification and observable proton decay. An
unbroken $Z_2$ gauge symmetry, coupled with the presence of two fermionic
$10$-plets required to resolve the axion domain wall problem, means that both
axions and a stable intermediate mass ($\sim 10^9-10^{10}$ GeV) fermion are
plausible dark matter candidates. We also display the gravitational wave
spectrum from the intermediate scale topologically stable cosmic strings
predicted by the model.
