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 $z>1$.