The Evolution of the HeII-Ionizing Background at Redshifts 2.3<z<3.8
  Inferred from a Statistical Sample of 24 HST/COS HeII Ly$\alpha$ Absorption
  Spectra
  We present measurements of the large-scale (~40 comoving Mpc) effective
optical depth of HeII Ly$\alpha$ absorption, $\tau_{\rm eff}$, at 2.54<z<3.86
toward 16 HeII-transparent quasars observed with the Cosmic Origins
Spectrograph (COS) on the Hubble Space Telescope (HST), to characterize the
ionization state of helium in the intergalactic medium (IGM). We provide the
first statistical sample of $\tau_{\rm eff}$ measurements in six
signal-to-noise ratio >3 HeII sightlines at z>3.5, and study the redshift
evolution and sightline-to-sightline variance of $\tau_{\rm eff}$ in 24 HeII
sightlines. We confirm an increase of the median $\tau_{\rm eff}$ from ~2 at
z=2.7 to >5 at z>3, and a scatter in $\tau_{\rm eff}$ that increases with
redshift. The z>3.5 HeII absorption is predominantly saturated, but isolated
narrow ($\Delta v<650$ km/s) transmission spikes indicate patches of reionized
helium. We compare our measurements to predictions for a range of UV background
models applied to outputs of a large-volume (146 comoving Mpc)$^3$
hydrodynamical simulation by forward-modeling our sample's quality and size. At
z>2.74 the variance in $\tau_{\rm eff}$ significantly exceeds expectations for
a spatially uniform UV background, but is consistent with a fluctuating
radiation field sourced by variations in the quasar number density and the mean
free path in the post-reionization IGM. We develop a method to infer the
approximate median HeII photoionization rate $\Gamma_{\rm HeII}$ of a
fluctuating UV background from the median $\tau_{\rm eff}$, finding a factor ~5
decrease in $\Gamma_{\rm HeII}$ between z~2.6 and z~3.1. At z~3.1 a
$\Gamma_{\rm
HeII}=\left[9.1^{+1.1}_{-1.2}\,\mathrm{(stat.)}\,^{+2.4}_{-3.4}\,\mathrm{(sys.)}\right]\times10^{-16}\,\mathrm{s}^{-1}$
corresponds to a median HeII fraction of ~2.5%, indicating that our data probe
the tail end of HeII reionization.
