Dark energy constraints from quasar observations

Recent measurements of the parameters of the Concordance Cosmology Model ($\Lambda$CDM) done in the low-redshift Universe with Supernovae Ia/Cepheids, and in the distant Universe done with Cosmic Microwave Background (CMB) imply different values for the Hubble constant (67.4 $\pm$ 0.5 km s$^{-1}$ Mpc$^{-1}$ from Planck vs 74.03 $\pm$ 1.42 km s$^{-1}$ Mpc$^{-1}$, Riess et al. 2019). This Hubble constant tension implies that either the systematic errors are underestimated, or the $\Lambda$CDM does not represent well the observed expansion of the Universe. Since quasars - active galactic nuclei - can be observed in the nearby Universe up to redshift z $\sim$ 7.5, they are suitable to estimate the cosmological properties in a large redshift range. Our group develops two methods based on the observations of quasars in the late Universe up to redshift z$\sim $4.5, with the objective to determine the expansion rate of the Universe. These methods do not yet provide an independent measurement of the Hubble constant since they do not have firm absolute calibration but they allow to test the $\Lambda$CDM model, and so far no departures from this model were found.