Cosmography by orthogonalized logarithmic polynomials
  Cosmography is a powerful tool to investigate the Universe kinematic and then
to reconstruct dynamics in a model-independent way. However, recent new
measurements of supernovae Ia and quasars have populated the Hubble diagram up
to high redshifts ($z \sim 7.5$) and the application of the traditional
cosmographic approach has become less straightforward due to the large
redshifts implied. Here we investigate this issue through an expansion of the
luminosity distance-redshift relation in terms of "orthogonal" logarithmic
polynomials. In particular we point out the advantages of a new procedure of
"orthogonalization" and we show that such an expansion provides a very good fit
in the whole $z=0\div 7.5$ range to both real and mock data obtained assuming
various cosmological models. Moreover, despite of the fact that the
cosmographic series is tested well beyond its convergence radius, the
parameters obtained expanding the luminosity distance - redshift relation for
the $\Lambda$CDM model are broadly consistent with the results from a fit of
mock data obtained with the same cosmological model. This provides a method to
test the reliability of a cosmographic function to study cosmological models at
high redshifts and it demonstrates that the logarithmic polynomial series can
be used to test the consistency of the $\Lambda$CDM model with the current
Hubble diagram of quasars and supernovae Ia. We confirm a strong tension (at
$>4\sigma$) between the concordance cosmological model and the Hubble diagram
at $z>1.5$. Such a tension is dominated by the contribution of quasars at $z>2$
and starts to be present also in the few supernovae Ia observed at $z>1$.
