Comparing phantom dark energy models with various diagnostic tools

In this paper, we apply various diagnostic tools to the phantom dark energy (DE) models, including the big rip (BR), the little sibling of the big rip (LSBR), and the little rip (LR). The evolutionary trajectories of the statefinder \(S_{n}(z)\), the growth rate \(\epsilon (z)\), and both Om(z) and \(\omega _{d}-\omega {}'_{d}\) diagnostics are plotted according to the best fit extracted from each model using a Markov Chain Monte Carlo method. In the high-redshift region, all these diagnostics show the degeneracy of the phantom DE models with each other and also with the \(\Lambda \)CDM model, even considering cases of different parameter values shows strong degeneracy. However, in the low-redshift region this degeneracy was broken by using \(S_{3}^{(1)}\) and \(S_{4}^{(1)}\) diagnostics, or otherwise by using the composite null diagnostic (CND). All diagnostic tools show the rich behavior of the LSBR model as it can be quintessence-like or phantom-like depending on the \(\Omega _{lsbr}\) parameter, in contrast to the LR model which is only phantom-like. A direct comparison of the phantom DE models in the \(\left\{ {S_{3}^{(1)}, S_{4}^{(1)}} \right\} \), \(\left\{ {S_{3}^{(1)}, S_{3}^{(2)}} \right\} \), and \(\omega _{d}-\omega {}'_{d}\) plane is also made. We further show that the separation between the models can be directly seen at the current state \(\left\{ S_{3}^{(1)}(t_{0}),S_{4}^{(1)}(t_{0}) \right\} \) of the models. We have reached that these diagnostic tools are rather robust in differentiating not only different DE models but also the same kind of models with different equation of state (EoS) which lead to different fates of the Universe.