The function q(z) as a consistency check for cosmological parameters

In the Friedmann cosmology the deceleration of the expansion q plays a fundamental role. We derive the deceleration as a function of redshift $q(z)$ in two scenarios: $\Lambda$ CDM model and modified Chaplygin gas (MCG) model, without assuming spatial flatness. The expression relating the transition redshift from decelerated to accelerated expansion $z_t$ to the cosmological parameters is obtained; it is seen that the curvature parameter does not appear in this expression. Of course the numerical value of $z_t$ depends on $\Omega_k$, since $\Omega_L + \Omega_M + \Omega_k = 1$. The exact function $q(z)$ allows the calculation of an infection point at $z \approx 0.1$ for the flat $\Lambda$ CDM model. This inflection point is visible in Fig.5 of the work of Daly and Djorgovski, Ap.J. 612,652 (2004), where the deceleration was plotted directly from observational data. We also consider the possibility of a small contribution of the curvature, namely, $\Omega_k = - 0.1 \Omega_L$, and calculate the transition redshift to be $z_t = 0.56$, which falls inside the interval $0.46 \pm 0.13$ obtained by Riess et al., Ap.J. 607, 665 (2004). The calculated transition redshift for the MCG model is considerably lower $z_t \approx 0.2$, but the observational data do not exclude a pure Chaplygin gas.