Early and Extended Helium Reionization Over More Than 600 Million Years of Cosmic Time

We measure the effective optical depth of HeII Ly\alpha\ absorption \tau${}_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{e}\mathsf{I}\mathsf{I}}$ at 2.3<z<3.5 in 17 UV-transmitting quasars observed with UV spectrographs on the Hubble Space Telescope (HST). The median \tau${}_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{e}\mathsf{I}\mathsf{I}}$ values increase gradually from 1.95 at z=2.7 to 5.17 at z=3.4, but with a strong sightline-to-sightline variance. Many $\simeq$35 comoving Mpc regions of the z>3 intergalactic medium (IGM) remain transmissive (\tau${}_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{e}\mathsf{I}\mathsf{I}}$<4), and the gradual trend with redshift appears consistent with density evolution of a fully reionized IGM. These modest optical depths imply average HeII fractions of x${}_{\mathsf{H}\mathsf{e}\mathsf{I}\mathsf{I}}$<0.01 and HeII ionizing photon mean free paths of $\simeq$50 comoving Mpc at z$\simeq$3.4, thus requiring that a substantial volume of the helium in the Universe was already doubly ionized at early times; this stands in conflict with current models of HeII reionization driven by luminous quasars. Along 10 sightlines we measure the coeval HI Ly\alpha\ effective optical depths, allowing us to study the density dependence of \tau${}_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{e}\mathsf{I}\mathsf{I}}$ at z$\sim$3. We establish that the dependence of \tau${}_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{e}\mathsf{I}\mathsf{I}}$ on increasing \tau${}_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{I}}$ is significantly shallower than expected from simple models of an IGM reionized in HeII. This requires higher HeII photoionization rates in overdense regions or underdense regions being not in photoionization equilibrium. Moreover, there are very large fluctuations in \tau${}_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{e}\mathsf{I}\mathsf{I}}$ at all ${\tau }_{\mathsf{e}\mathsf{f}\mathsf{f},\mathsf{H}\mathsf{I}}$, which greatly exceed the expectations from these simple models. These data present a distinct challenge to scenarios of HeII reionization - an IGM where HeII appears to be predominantly ionized at z$\simeq$3.4, and with a radiation field strength that may be correlated with the density field, but exhibits large fluctuations at all densities.