The impact of relativistic effects on the 3D Quasar-Lyman-α cross-correlation

We study the impact of relativistic effects in the 3-dimensional cross-correlation between Lyman-$\alpha$ forest and quasars. Apart from the relativistic effects, which are dominated by the Doppler contribution, several systematic effects are also included in our analysis (intervening metals, unidentified high column density systems, transverse proximity effect and effect of the UV fluctuations). We compute the signal-to-noise ratio for the Baryonic Oscillation Spectroscopic Survey (BOSS), the extended Baryonic Oscillation Spectroscopic Survey (eBOSS) and the Dark Energy Spectroscopic Instrument (DESI) surveys, showing that DESI will be able to detect the Doppler contribution in a Large Scale Structure (LSS) survey for the first time, with a S/N $>7$ for $r_{\rm min} > 10$ Mpc$/h$, where r$_{\rm min}$ denotes the minimum comoving separation between sources. We demonstrate that several physical parameters, introduced to provide a full modelling of the cross-correlation function, are affected by the Doppler contribution. By using a Fisher matrix approach, we establish that if the Doppler contribution is neglected in the data analysis, the derived parameters will be shifted by a non-negligible amount for the upcoming surveys.