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The spectral index $n_s$ of scalar perturbation is the significant initialcondition set by inflation theory for our observable Universe. According toPlanck results, current constraint is $n_s = 0.965\pm 0.004$, while an exactscale-invaiant Harrison-Zeldovich spectrum, i.e. $n_s=1$, has been ruled out at$8.4\sigma$ significance level. However, it is well-known that the standard$\Lambda$CDM model is suffering from the Hubble tension, which is at $\sim5\sigma$ significance level. This inconsistency likely indicates that thecomoving sound horizon at last scattering surface is actually lower thanexpected, which so seems to be calling for the return of $n_s=1$. Here, inlight of recent observations we find strong evidence for a $n_s=1$ Universe.And we show that if so, it would be confirmed conclusively by CMB-S4experiment.

Return of Harrison-Zeldovich spectrum in light of recent cosmological tensions