Numerous and narrow absorption lines in quasar spectra, the so-called Lyman
alpha forest, can be a unique probe to explore the early universe. In addi
tion, several detections of an absorption trough in the spectra have placed
stringent constraints on the physical properties of the intergalactic medi
um at high redshifts. They would play an important role in relation to the
evolution of the UV background radiation field. Focusing on the mutual rela
tions between the quasar absorption system and the UV background radiation
field, we examine the evolution of minihalos, gas clouds confined by the gr
avity of cold dark matter, and explore reasonable possibilities for minihal
os to reproduce the redshift distribution of Ly alpha clouds at 0 less than
or similar to z less than or similar to 4. We can find that the redshift d
istribution of minihalos shows good agreement with the observations over th
e whale range of redshifts 0 less than or similar to, z less than or simila
r to, 4 under two alternative cases: first, the UV background radiation has
a peak intensity at z similar to 1; second, the Lya forest has a comoving
number density that decreases exponentially with a characteristic timescale
, tau similar to a few Gyr.