A consistent and robust measurement of the thermal state of the IGM at
  $2 \leq z \leq 4$ from a large sample of Ly$\alpha$ forest spectra: Evidence
  for late and rapid HeII reionization
  We characterise the thermal state of the intergalactic medium (IGM) in ten
redshift bins in the redshift range $2 \leq z \leq 4$ with a sample of 103 high
resolution, high S/N Ly$\alpha$ forest spectra using four different flux
distribution statistics. Our measurements are calibrated with mock spectra from
a large suite of hydrodynamical simulations post-processed with our thermal IGM
evolution code CITE, finely sampling amplitude and slope of the expected
temperature-density relation. The thermal parameters inferred from our
measurements of the flux power spectrum, Doppler parameter distribution, as
well as wavelet and curvature statistics agree well within their respective
errors and all clearly show the peak in temperature and minimum in slope of the
temperature density relation expected from HeII reionization. Combining our
measurements from the different flux statistics gives $T_0=(14750 \pm 1322)$K
for the peak temperature at mean density and a corresponding minimum slope
$\gamma = 1.225 \pm 0.120$. The peak in the temperature evolution occurs at $z
\approx 3$, in agreement with previous measurements that had suggested the
presence of such a peak, albeit with a large scatter. Using CITE, we also
calculate the thermal state of the IGM predicted by five widely used (spatially
homogeneous) UV-background models. The rather rapid thermal evolution inferred
by our measurements is well reproduced by two of the models, if we assume
(physically well motivated) non-equilibrium evolution with photo-heating rates
that are reduced by a moderate factor of $\sim 0.7-0.8$. The other three models
predict HeII reionization to be more extended with a somewhat earlier as well
as higher temperature peak than our measurements suggest.
