DE models with combined $H_0 \cdot r_d $ from BAO and CMB dataset and friends

Abstract: It has been theorized that Dynamical Dark Energy (DDE) could be a possible solution to the Hubble tension. To avoid the degeneracy between the Hubble parameter $H_0$ and the sound horizon scale $r_d$, in this article we use their multiplication as one parameter $c/\left(H_0 r_d\right)$ and we use it to infer cosmological parameters for 6 different models - $\Lambda$CDM and 5 DDE parametrizations -- the Chevallier-Polarski-Linder (CPL), the Barboza-Alcaniz (BA), the Low correlation (LC), the Jassal-Bagla-Padmanabhan (JBP) and the Feng-Shen-Li-Li model. We choose a dataset that treats this combination as one parameter, that includes the Baryon Acoustic Oscillation (BAO) data $0.11 \le z \le 2.40$ and additional points from the Cosmic Microwave Background (CMB) Peaks ($z \approx 1090$). To them, we add the marginalized Pantehon dataset and GRB dataset. We see that the tension is moved from $H_0$ and $r_d$ to $c/\left(H_0 r_d\right)$ and $\Omega_m$. There is only one model that satisfies the Planck 2018 constraints on both parameters and this is LC with a huge error. The rest cannot fit into both constraints. $\Lambda$CDM is preferred with respect to the statistical measures.