Constraining the ratio of median pixel optical depth profile around z ~
  4 quasars using the longitudinal proximity effect
  We present a detailed study of the longitudinal proximity effect using a
sample of 85 quasars spanning an emission redshift range of $3.5 \leq z_{em}
\leq 4.5$ and Lyman continuum luminosity ($L_{912}$) ranging from 1.06$\times
10^{31}$ to 2.24$\times 10^{32}$ erg s$^{-1}$ Hz$^{-1}$. We use the
high-quality spectra of these quasars obtained at a spectral resolution of
$R\sim$ 5100 and S/N $\sim$ 30 using X-SHOOTER spectrograph mounted on the Very
Large Telescope (VLT). In our analysis, we compared the transmitted flux and
pixel optical depth of the Ly$\alpha$ absorption originating from the vicinity
of quasars to those from the general intergalactic medium by using a redshift
matched control sample. The longitudinal proximity effect is found up to $r
\leq 12$ Mpc (proper) from quasars. By appropriately scaling up the pixel
optical depth in the vicinity of quasars to account for the excess ionization
by quasars, we constrain the ratio of median HI optical depth in the vicinity
of the quasar to that of the IGM ($R_\tau(r)$). The $R_\tau (r)$ is found to be
significantly higher than unity up to 6 Mpc from the quasar with a typical
radial profile of the form $R_\tau(r) = 1+A \times exp(-r/r_0)$ with $A=9.16\pm
0.68$ and $r_0= 1.27\pm 0.08$ Mpc. The integrated value of the scaled pixel
optical depth over the radial bin of 0-6 Mpc is found to be higher by a factor
of $2.55 \pm 0.17$ than the corresponding integrated value of the median pixel
optical depth of the IGM. We also found $R_\tau (r)$ to be luminosity
dependent.
