On using large scale correlation of the Ly-$\alpha$ forest and
  redshifted 21-cm signal to probe HI distribution during the post reionization
  era
  We investigate the possibility of detecting the 3D cross correlation power
spectrum of the Ly-$\alpha$ forest and HI 21 cm signal from the post
reionization epoch. The cross-correlation signal is directly dependent on the
dark matter power spectrum and is sensitive to the 21-cm brightness temperature
and Ly-$\alpha$ forest biases. These bias parameters dictate the strength of
anisotropy in redshift space. We find that the cross-correlation power spectrum
can be detected using $400 ~ \, \rm hrs$ observation with SKA-mid (phase 1) and
a futuristic BOSS like experiment with a quasar (QSO) density of $30 ~ \rm
deg^{-2}$ at a peak SNR of $15$ for a single field experiment at redshift $z =
2.5$. We also study the possibility of constraining various bias parameters
using the cross power spectrum. We find that with the same experiment $1
\sigma$ conditional errors on the 21-cm linear redshift space distortion
parameter $\beta_T$ and $\beta_{F}$ corresponding to the Ly-$\alpha $ forest
are $\sim 2.7\%$ and $\sim 1.4 \%$ respectively for $10$ independent pointings
of the SKA-mid (phase 1). This prediction indicates a significant improvement
over existing measurements. We claim that the detection of the 3D cross
correlation power spectrum will not only ascertain the cosmological origin of
the signal in presence of astrophysical foregrounds but will also provide
stringent constraints on large scale HI biases. This provides an independent
probe towards understanding cosmological structure formation.
