Cosmological implications of $n_s\approx 1$ in light of the Hubble
  tension
  Recently, a low-$z$ measurement of the Hubble constant, $H_0 = 73.04 \pm 1.04
{\rm ~km/s/Mpc}$, was reported by the SH0ES Team. The long-standing Hubble
tension, i.e. the difference between the Hubble constant from the local
measurements and that inferred from the cosmic microwave background data based
on the $\Lambda$CDM model, was further strengthened. There are many
cosmological models modifying the cosmology after and around the recombination
era to alleviate this tension. In fact, some of the models alter the
small-scale fluctuation amplitude relative to larger scales, and thus require a
significant modification of the primordial density perturbation, especially the
scalar spectral index, $n_s$. In certain promising models, $n_s$ is favored to
be larger than the $\Lambda$CDM prediction, and even the scale-invariant one,
$n_s=1$, is allowed. In this Letter, we focus on the very early Universe models
to study the implication of such unusual $n_s$. In particular, we find that an
axiverse with an axion in the equilibrium distribution during inflation can be
easily consistent with $n_s = 1$. This is because the axion behaves as a
curvaton with mass much smaller than the inflationary Hubble parameter. We also
discuss other explanations of $n_s$ different from that obtained based on the
$\Lambda$CDM.
