Predictions for measuring the cross power spectrum of the HI 21-cm signal and the Lyman-$\alpha$ forest using OWFA

We have studied the possibility of measuring the cross-correlation of the redshifted HI 21-cm signal and the Lyman-$\alpha$ forest using an upcoming radio-interferometric array OWFA and an spectroscopic observation like SDSS-IV. Our results shows that it is possible to have a $6\sigma$ detection of the cross-correlation signal with OWFA PII using an observing time of $200$ hrs each in $N_p=25$ independent fields-of-view. However, not much can be done beyond this using the cross-correlation signal for $z_c=3.35$ and $B = 30 , {\rm MHz}$. Apart from this, we have also envisaged a situation where observations are carried out at $z_c=3.05$ and $2.55$ which lie closer to the peak of the quasar distribution at $z=2.25$ and with a larger bandwidth of $B = 60 , {\rm MHz}$. We see that the SNR of the cross-correlation detection can be significantly enhanced to $\sim 17$ for $z_c=2.55$ and $B = 60 , {\rm MHz}$\mathrm{.}Itisthenpossibletomeasure$\beta_T$and$\beta_F$ individually with an $\mathrm{S}\mathrm{N}\mathrm{R}\ge 5$ by combining the cross-correlation with the HI 21-cm auto-correlation measurements. We further find that a measurement of the binned cross-correlation power spectrum with $\mathrm{S}\mathrm{N}\mathrm{R}\ge 5$ is also possible in several bins at $k\le 0.3{\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$.