We present a measurement of the Lyman \alpha flux probability distribution function ( PDF ) obtained from a set of eight high resolution quasar spectra with emission redshifts at 3.3 \leq z \leq 3.8 .
We carefully study the effect of metal absorption lines on the shape of the PDF .
Metals have a larger impact on the PDF measurements at lower redshift , where there are relatively fewer Lyman \alpha absorption lines .
This may be explained by an increase in the number of metal lines which are blended with Lyman \alpha absorption lines toward higher redshift , but may also be due to the presence of fewer metals in the intergalactic medium at earlier times .
We also provide a new measurement of the redshift evolution of the effective optical depth , \tau _ { eff } , at 2.8 \leq z \leq 3.6 , and find no evidence for a deviation from a power law evolution in the \log ( \tau _ { eff } ) - \log ( 1 + z ) plane .
The flux PDF measurements are furthermore of interest for studies of the thermal state of the intergalactic medium ( IGM ) at z \simeq 3 .
By comparing the PDF to state-of-the-art cosmological hydrodynamical simulations , we place constraints on the temperature of the IGM and compare our results with previous measurements of the PDF at lower redshift .
At redshift z = 3 , our new PDF measurements are consistent with an isothermal temperature-density relation , T = T _ { 0 } \Delta ^ { \gamma - 1 } , with a temperature at the mean density of T _ { 0 } = 19250 \pm 4800 K and a slope \gamma = 0.90 \pm 0.21 ( 1 \sigma uncertainties ) .
In comparison , joint constraints with existing lower redshift PDF measurements at z < 3 favour an inverted temperature-density relation with T _ { 0 } = 17900 \pm 3500 K and \gamma = 0.70 \pm 0.12 , in broad agreement with previous analyses .