  The radiation background above the ionization edge of HeII varies strongly
during and after helium reionization, because the attenuation length of such
photons is relatively short (<40 Mpc) and because the ionizing sources
(quasars) are rare. Here we construct analytic and Monte Carlo models to
examine these fluctuations, including, for the first time, those during the
reionization era itself. In agreement with detailed numerical simulations, our
analytic model for the post-reionization Universe predicts order-of-magnitude
fluctuations in the HeII ionization rate. Observations of the hardness ratio
between HeII/HI show even larger fluctuations, which may be due to more
complicated radiative transfer effects. During reionization, the fluctuations
are even stronger. In contrast to hydrogen reionization, our model predicts
that regions with strong HeII Lyman-alpha forest transmission should be
reasonably common even during the beginning stages of reionization, because of
strong illumination from bright quasars. Partly due to this, the mean ionizing
background does not evolve strongly during and after helium reionization; it is
roughly proportional to the filling fraction of HeIII regions. On the other
hand, regions full of HeII and also "fossil" ionized regions that contain no
(or few) active sources appear as strong IGM absorbers. Their presence
exaggerates the evolution of the hardness ratio, making it evolve faster than
naively expected during the reionization era.
