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