Sources of $H_0$-tensions in dark energy scenarios

By focusing on the simple $w\neq-1$ extension to $\Lambda$CDM, we assess which epoch(s) possibly source the $H_0$-tension. We consider Cosmic Microwave Background (CMB) data in three possible ways: $i)$ complete CMB data; $ii)$ excluding the $l<30$ temperature and polarization likelihoods; $iii)$ imposing early universe priors, that disentangle early and late time physics. Through a joint analysis with low-redshift supernovae type-Ia and gravitationally lensed time delay datasets, {and neglecting galaxy clustering Baryonic Acoustic Oscillation (BAO) data}, we find that the inclusion of early universe CMB priors is consistent with the local estimate of $H_0$ while excluding the low-$l$+lowE likelihoods mildly relaxes the tension. This is in contrast to joint analyses with the complete CMB data. Our simple implementation of contrasting the effect of different CMB priors on the $H_0$ estimate shows that the early universe information from the CMB data when decoupled from late-times physics could be in agreement with a higher value of $H_0$. {We also find no evidence for the early dark energy model using only the early universe physics within the CMB data. Finally using the BAO data in different redshift ranges to perform inverse distance ladder analysis, we find that the early universe modifications, while being perfectly capable of alleviating the $H_0$-tension when including the BAO galaxy clustering data, would be at odds with the Ly-$\alpha$ BAO data due to the difference in $r_{\rm d}\, vs.\, H_0$ correlation between the two BAO datasets.} We therefore infer and speculate that source for the $H_0$-tension between CMB and local estimates could possibly originate in the modeling of late-time physics within the CMB analysis. This in turn recasts the $H_0$-tension as an effect of late-time physics in CMB, instead of the current early-time CMB vs. local late-time physics perspective.