Constraints on the Transition Redshift using Hubble Phase Space Portrait

One of the most significant discoveries in modern cosmology is that the universe is currently in a phase of accelerated expansion after a switch from a decelerated expansion. The redshift corresponding to this epoch is referred to as the transition redshift $z_t$. In this work we put constraints on the $z_t$ with both model-independent and model-dependent approaches. We consider 32 Hubble parameter measurements and the Pantheon sample of Type Ia Supernovae (SNe). In order to include the possible systematic effects in this analysis, we use the full covariance matrix of systematic uncertainties for the Hubble parameter measurements. We plot a Hubble Phase Space Portrait (HPSP) between $\dot{H}(z)$ and $H(z)$ in a model-independent way. From this HPSP diagram, we estimate the transition redshift as well as the current value of the equation of state parameter $\omega_0$ in a model-independent way. By considering H(z) measurements, we find the best fit value of $z_t=0.591^{+0.332}_{-0.332}$ and $\omega_0=-0.677^{+0.238}_{-0.238}$. We obtain the best fit value of $z_t=0.849^{+0.117}_{-0.117}$ and $\omega_0=-0.870^{+0.013}_{-0.013}$ using the Pantheon database. Further, we also use a model dependent approach to determine $z_t$. Here, we consider a non-flat $\Lambda$CDM model as a background cosmological model. We reconstruct the cosmic triangle plot among $\log(\Omega_{m0})$, $-\log(2\Omega_{\Lambda0})$ and $3\log(1+z_t)$ where the constraints of each parameter are determined by the location in this triangle plot. Using $\Omega_{m0}$ and $\Omega_{\Lambda0}$ values, we find the best value of the transition redshift $z_t=0.619^{+0.580}_{-0.758}$, which is in good agreement with the Planck 2018 results at $1\sigma$ confidence level. We also simulate the observed Hubble parameter measurements in the redshift range $0<z<2$ and perform the same analysis to estimate the transition redshift.