f(T) cosmology against the cosmographic method: A new study using mock and observational data

In this paper, we study the power-law $f(T)$ model using Hubble diagrams of type Ia supernovae (SNIa), quasars (QSOs), Gamma Ray Bursts (GRBs) and the measurements from baryonic acoustic oscillations (BAO) in the framework of the cosmographic method. Using mock data for SNIa, QSOs and GRBs generated based on the power-law $f(T)$ model, we show whether different cosmographic methods are suitable to reconstruct the distance modulus or not. In particular, we investigate the rational PADE polynomials $(3,2)$ and $(2,2)$ in addition to the fourth- and fifth- order Taylor series. We show that PADE $(3,2)$ is the best approximation that can be used in the cosmographic method to reconstruct the distance modulus at both low and high redshifts. In the context of PADE $(3,2)$ cosmographic method, we show that the power-law $f(T)$ model is well consistent with the real observational data from the Hubble diagrams of SNIa, QSOs and GRBs. Moreover, we find that the combination of the Hubble diagram of SNIa and the BAO observation leads to better consistency between the model-independent cosmographic method and the power-law $f(T)$ model. Finally, our observational constraints on the parameter of the effective equation of state of DE, described by the power-law $f(T)$ model, show the phantom-like behavior, especially when the BAO observations are included in our analysis.