Hubble diagram at higher redshifts: Model independent calibration of
  quasars
  In this paper, we present a model-independent approach to calibrate the
largest quasar sample. Calibrating quasar samples is essentially constraining
the parameters of the linear relation between the $\log$ of the ultraviolet
(UV) and X-ray luminosities. This calibration allows quasars to be used as
standardized candles. There is a strong correlation between the parameters
characterizing the quasar luminosity relation and the cosmological distances
inferred from using quasars as standardized candles. We break this degeneracy
by using Gaussian process regression to model-independently reconstruct the
expansion history of the Universe from the latest type Ia supernova
observations. Using the calibrated quasar dataset, we further reconstruct the
expansion history up to redshift of $z\sim 7.5$. Finally, we test the
consistency between the calibrated quasar sample and the standard
$\rm{\Lambda}CDM$ model based on the posterior probability distribution of the
GP hyperparameters. Our results show that the quasar sample is in good
agreement with the standard $\rm{\Lambda}CDM$ model in the redshift range of
the supernova, despite of mildly significant deviations taking place at higher
redshifts. Fitting the standard $\rm{\Lambda}CDM$ model to the calibrated
quasar sample, we obtain a high value of the matter density parameter $\Omega_m
= 0.382^{+0.045}_{-0.042}$, which is marginally consistent with the constraints
from other cosmological observations.
