No evidence for dynamical dark energy in two models

To investigate whether the dark energy evolves over time, we propose two null tests and constrain them using the data combination of cosmic microwave background radiation, baryonic acoustic oscillations, Type Ia supernovae, Planck-2015 lensing and cosmic chronometers. We find that, for these two models, there is no evidence of the dynamical dark energy at the $1.2\sigma$ confidence level. Interestingly, both models could slightly alleviate (i) the current Hubble constant ($H_0$) tension between the global fitting derivation by the Planck collaboration and the local observation by Riess {\it et al.}; (ii) the root-mean-square density fluctuations ($\sigma_8$) tension between the Planck-2015 data and several low-redshift large scale structure probes.