Constraining warm dark matter power spectrum using the cross-correlation of HI 21 cm signal and the Lyman-$\alpha$ forest

We have considered the prospects for measuring the cross Warm Dark Matter (WDM) power spectrum of the redshifted HI 21-cm signal and the Lyman-$\alpha$ forest and thereby constraining WDM mass using observations with upcoming radio-interferometers - the Ooty Wide Field Array (OWFA) and SKA1-mid, and a spectroscopic survey of the quasars. We have considered a quasar survey with a mean observed quasar number density of ${\bar{\mathrm{n}}}_{\mathrm{Q}}=48$ deg${}^{-2}$ over a collecting area of 14455 deg${}^2$, and a mean spectroscopic SNR = 5. Our analysis with OWFA shows that it is possible to measure the WDM power spectrum in several $k$-bins at $k<0.4$ Mpc${}^{-1}$ with $\mathrm{S}\mathrm{N}\mathrm{R}>5$ using an observation of 200 hours each in 100 different fields-of-view for $m_{\rm WDM} = 0.25$ keV. Considering the possibility of the joint measurement of the parameters, the warm dark matter density parameter ${\mathrm{\Omega }}_{\mathrm{W}\mathrm{D}\mathrm{M}}$, and the dark energy density parameter ${\mathrm{\Omega }}_{\mathrm{\Lambda }0}$, we find that the relative error on the $1-\sigma$ measurement of the parameter $\Omega_{{\rm WDM}}$is$\sim 0.8$forafiducial$m_{\rm WDM} = 0.25$ keV. We further find that it is possible to have a measurement of the suppression of power from the Cold Dark Matter (CDM) power spectrum at a confidence level of $\sim 7.2-\sigma$and$\sim 2.7-\sigma$intwodifferent$k$-binsoverthe$k$-range$0.1 \leq k \leq 3.13$Mpc$^{-1}$for$m_{\rm WDM} = 0.15$ keV. Considering the analysis with SKA1-mid, we find that for a fiducial $m_{\mathrm{W}\mathrm{D}\mathrm{M}}=0.25$ keV, the suppression in the cross power spectrum can be measured at $\sim 10 - \sigma$around$k \sim 0.2 \rm{Mpc}^{-1}$foratotalobservingtimeof$20000$hrsdistributeduniformlyover$50$independentpointingswheretheavailable$k$-rangeisbinnedas$\Delta k = k/5$.