Whether the late dark energy parameterization can reconcile the Hubble tension?

In this paper, we construct ten dark energy models to test whether they can reconcile the Hubble tension and how much is it affected by parameterization. To provide a fair test, the models are rich including fractional form, logarithm form, exponential form, inverse exponential form and several non-parameterized models. The dataset we use are the NPIPE pipeline of cosmic microwave background (CMB) power-spectrum data from \pl2020, Pantheon+ samples from Supernovae Type Ia, and baryon acoustic oscillation. They MCMC calculation implies the dark energy transferring from $w< -1$ to $w > -1$, for the four parameterized dark energy models. However, they are unable to reconcile the Hubble tension well. Importantly, we find that the phantom-like dark energy with $w < -1$ best can reduce the Hubble tension to $0.1808\sigma$. However, the AIC analysis indicates that this alleviation is at the cost of high AIC. We also investigate the effect of constructions on the derivative of equation of state $dw/da$, cosmic density parameter, CMB power spectrum $C^{TT}$ and matter spectra $P(k)$. We also find that Hubble tension may be related to the reionization process.