Do JWST reionization (optical depth) puzzle, cosmological tensions, and CMB anomalies imply Harrison-Zel'dovich spectrum?

The James Webb Space Telescope (JWST) has observed massive galaxies at high redshifts, which implies an earlier epoch of reionization (EoR) compared with the cosmic microwave background (CMB) results. In this paper, based on Planck 2020 (NPIPE release), ACT DR4 and SPT-3G data, if assumed a Harrison-Zel'dovich (HZ) primordial power spectrum in the standard cosmological model, we show that the redshift or optical depth of reionization is larger than the case of a power-law (PL) primordial power spectrum. In HZ-ΛCDM model, the redshift of reionization is zreio = 9.11 ± 0.61, which is consistent with the JWST result that zreio ≈ 8.9. Moreover, the cosmological tensions, i.e. Hubble (H0) tension and S8 tension are alleviated in HZ-ΛCDM case. The Hubble constant is H0 = 70.38 ± 0.35 km/s/Mpc and the structure growth parameter is S8 = 0.7645 ± 0.0094 in HZ-ΛCDM model. We also consider two extensions of ΛCDM, including ΛCDM+AL and ΛCDM+Ωk models. But the extensions of ΛCDM with a HZ spectrum meet more serious CMB anomalies, i.e. lensing anomaly and spatial curvature anomaly as compared with the extensions of ΛCDM with a PL spectrum. We discuss that these two CMB anomalies may come from the degeneracy of cosmological parameters.