A bias-free cosmological analysis with quasars alleviating $H_0$ tension

Cosmological models and their parameters are widely debated because of theoretical and observational mismatches of the standard cosmological model, especially the current discrepancy between the value of the Hubble constant, $H_{0}$, obtained by Type Ia supernovae (SNe Ia), and the Cosmic Microwave Background Radiation (CMB). Thus, considering high-redshift probes like quasars (QSOs), having intermediate redshifts between SNe Ia and CMB, is a necessary step. In this work, we use SNe Ia and the most updated QSO sample, reaching redshifts up to $z\sim7.5$, applying the Risaliti-Lusso QSO relation based on a non-linear relation between ultraviolet and X-ray luminosities. We consider this relation both in its original form and corrected for selection biases and evolution in redshift through a reliable statistical method also accounting for the circularity problem. We also explore two approaches: with and without calibration on SNe Ia. We then investigate flat and non-flat standard cosmological models and a flat $w$CDM model, with a constant dark energy equation of state parameter $w$. Remarkably, when correcting for the evolution as a function of cosmology, we obtain closed constraints on $\Omega_M$ using only non-calibrated QSOs. We find that considering non-calibrated QSOs combined with SNe Ia and accounting for the same correction, our results are compatible with a flat $\Lambda$CDM model with $\Omega_M = 0.3$ and $H_0 = 70 \, \mathrm{km\,s^{-1}\,Mpc^{-1}}$. Intriguingly, the $H_0$ values obtained place halfway between the one from SNe Ia and CMB, paving the way for new insights into the $H_0$ tension.