  We present a formalism for computing time-dependent fluctuations in the
cosmological photoionizing radiation background, extending background
fluctuations models beyond the steady-state approximation. We apply this
formalism to estimate fluctuations in the HI LyA flux redshift space power
spectrum and its spatial correlation function at redshifts 2<z<4, assuming the
photoionization background is dominated by Quasi-stellar Objects (QSOs) and/or
galaxies. We show the shot noise in the power spectrum due to discrete sources
is strongly suppressed relative to the steady-state value at low wavenumbers by
a factor proportional to the lifetime of the sources, and that this suppression
may be used to constrain QSO lifetimes. The total HI LyA power spectrum
including shot noise is affected at tens of percent on short scales, and by as
much as an order of magnitude or more on scales exceeding the mean free path.
The spatial correlation function is similarly found to be sensitive to the shot
noise, although moderately insensitive to the effects of time-dependence on the
non-shotnoise contribution. Photoionization rate fluctuations substantially
modify the shape of the Baryonic Acoustic Oscillation peak in the correlation
function, including a small increase in its position that must be accounted for
to avoid biasing estimates of cosmological parameters based on the peak
position. We briefly investigate solving the full frequency dependent equation,
finding that it agrees with the frequency-independent to better than percent
accuracy. Simple formulas are provided for the power spectrum of fluctuations
in the photoionization rate that approximate the full computations.
