  In this paper, we consider a possibility that the temperature anisotropy of
cosmic microwave background (CMB) is dominantly generated by the primordial
fluctuations of QCD-axion like particles under a circumstance that inflaton's
perturbation is too small to explain the CMB anisotropy. Since the axion
potential is generated by acquiring its energy from radiation, the primordial
fluctuations of the axion field generated in the inflation era are correlated
with the CMB anisotropies. Consequently, the observations stringently constrain
a model of the axion and the early universe scenario. The following conditions
must be satisfied: (i) sufficient amplitudes of the CMB anisotropy (ii)
consistency with the axion isocurvature constraint and (iii) the
non-Gaussianity constraint. To satisfy these conditions, a large energy
fraction $\Omega_A^{}$ of the axion is necessary at the QCD scale when the
axion-potential is generated, but simultaneously, it must become tiny at the
present era due to the isocurvature constraint. Thus an additional scenario of
the early universe, such as low scale thermal inflation, is inevitable to
dilute the axions after the QCD scale. We investigate such a model and obtain
its allowed parameter region.
