Constraints on kinematic parameters at $z\ne0$

Abstract

The standard cosmographic approach consists in performing a series expansion of a cosmological observable around $z=0$ and then using the data to constrain the cosmographic (or kinematic) parameters at present time. Such a procedure works well if applied to redshift ranges inside the $z$-series convergence radius ($z<1$), but can be problematic if we want to cover redshift intervals that fall outside the $z-$series convergence radius. This problem can be circumvented if we work with the $y-$redshift, $y=z/(1+z)$, or the scale factor, $a=1/(1+z)=1-y$, for example. In this paper, we use the scale factor $a$ as the variable of expansion. We expand the luminosity distance and the Hubble parameter around an arbitrary $\tilde{a}$ and use the Supernovae Ia (SNe Ia) and the Hubble parameter data to estimate $H$, $q$, $j$ and $s$ at $z\ne0$ ($\tilde{a}\neq1$). We show that the last relevant term for both expansions is the third. Since the third order expansion of $d_L(z)$ has one parameter less than the third order expansion of $H(z)$, we also consider, for completeness, a fourth order expansion of $d_L(z)$. For the third order expansions, the results obtained from both SNe Ia and $H(z)$ data are incompatible with the $Λ$CDM model at $2σ$ confidence level, but also incompatible with each other. When the fourth order expansion of $d_L(z)$ is taken into account, the results obtained from SNe Ia data are compatible with the $Λ$CDM model at $2σ$ confidence level, but still remains incompatible with results obtained from $H(z)$ data. These conflicting results may indicate a tension between the current SNe Ia and $H(z)$ data sets.