Implication of the Hubble tension for the primordial Universe in light of recent cosmological data

In prerecombination resolutions of the Hubble tension, such as early dark energy, new physics before recombination shifts values relevant cosmological parameters so that models can fit with cosmic microwave background and baryon acoustic oscillations observations well $\Lambda$CDM does. this paper, we clarify how parameter are related $\delta H_0$, particularly find shift primordial scalar spectral index scales ${\delta n_s}\simeq 0.4{\delta H_0\over H_0}$ by performing Monte Carlo Markov chain analysis Planck2018+BAO+Pantheon+R19+Keck Array/BICEP dataset. A novel point our result is if current $H_0$ measured locally correct, complete resolution tension seems to be pointing a scale invariant Harrison-Zeldovich spectrum, i.e. $n_s= 1$ for $H_0\sim 73$km/s/Mpc.