Recovering $\Lambda$CDM Model From a Cosmographic Study

Using the mathematical definitions of deceleration and jerk parameters we obtain a general differential equation for squared Hubble parameter. For a constant jerk, this differential equation leads to an exact function for Hubble parameter. By the aid of this exact Hubble function we can exactly reconstruct any other cosmographic parameters. We also obtained a general function for transition redshift as well as spacetime curvature. Our derived functions clearly impose a lower limit on the jerk parameter which is $j_{min}\geq-0.125$. Moreover, we found that the jerk parameter indicates the geometry of the spacetime i.e any deviation from $j=1$ imply to a non-flat spacetime. In other word $j\neq 1$ reefers to a dynamical, time varying, dark energy. From obtained Hubble function we recover the analogue of $\Lambda$CDM model. To constrain cosmographic parameters as well as transition redshift and spacetime curvature of the recovered $\Lambda$CDM model, we used Metropolis-Hasting algorithm to perform Monte Carlo Markov Chain analysis by using observational Hubble data obtained from cosmic chronometric (CC) technique, BAO data, Pantheon compilation of Supernovae type Ia, and their joint combination. The only free parameters are $H$, $A(\Omega_{m})$ and $j$. From joint analysis we obtained $H_{0}=69.9\pm 1.7$, $A(\sim\Omega_{0m})=0.279^{+0.013}_{-0.017}$, $B(\sim\Omega_{0X})=0.721^{+0.017}_{-0.013}$, $j_{0}=1.038^{+0.061}_{-0.023}$ and $z_{t}=0.706^{+0.031}_{-0.034}$.