  The axion is the most attractive candidate to solve the strong CP problem in
QCD. If it exists, the inflationary universe produces axion fluctuations which
are mixtures of isocurvature and adiabatic fluctuations in general. We
investigate how large isocurvature fluctuations are allowed or favored in order
to explain observations of the large scale structure of the present universe.
Generic flat universe models with mixed (isocurvature + adiabatic) density
fluctuations are studied. It is found that the observations are consistent with
the mixed fluctuation model if the ratio \alpha of the power spectrum of
isocurvature fluctuations to that of adiabatic fluctuations is less than \sim
0.1. In particular, the mixed fluctuation model with \alpha \sim 0.05, total
matter density \Omega_{0} =0.4, and Hubble parameter H_0=70 km/s/Mpc gives a
very good fit to the observational data. Since the height of the acoustic peak
in the angular power spectrum of the cosmic microwave background (CMB)
radiation decreases significantly when the isocurvature fluctuations are
present, the mixed fluctuation model can be tested in future satellite
experiments. Ratios of the amplitude at the peak location to that at the COBE
normalization scale for various models are given. Furthermore, we also obtain
the amplitude of isocurvature fluctuations as a function of axion parameters
and the Hubble parameter during the inflation. We discuss the axion
fluctuations in some realistic inflation models and find that a significant
amount of the isocurvature fluctuations are naturally produced.
