SU(2)$_{\tiny\mbox{CMB}}$ at high redshifts and the value of $H_0$

Abstract: We investigate a high-$z$ cosmological model to compute the co-moving sound horizon $r_s$ at baryon-velocity freeze-out towards the end of hydrogen recombination. This model assumes a replacement of the conventional CMB photon gas by deconfining SU(2) Yang-Mills thermodynamics, three flavours of massless neutrinos ($N_\nu=3$), and a {\sl purely baryonic} matter sector (no cold dark-matter (CDM)). The according SU(2) temperature-redshift relation of the CMB is contrasted with recent measurements appealing to the thermal Sunyaev-Zel'dovich effect and CMB-photon absorption by molecular rotations bands or atomic hyperfine levels. Relying on a realistic simulation of the ionization history throughout recombination, we obtain $z_=1693.55\pm 6.98$ and $z_{\rm drag}=1812.66\pm 7.01$. Due to considerable widths of the visibility functions in the solutions to the associated Boltzmann hierarchy and Euler equation we conclude that $z_$ and $z_{\rm drag}$ over-estimate the redshifts for the respective photon and baryon-velocity freeze-out. Realistic decoupling values turn out to be $z_{{\rm lf},*}=1554.89\pm 5.18$ and $z_{\rm lf,drag}=1659.30\pm 5.48$. With $r_s(z_{\rm lf,drag})=(137.19\pm 0.45)\,$Mpc and the essentially model independent extraction of $r_s\cdot H_0=\mbox{const}$ from low-$z$ data in arXiv:1607.05617 we obtain a good match with the value $H_0=(73.24\pm 1.74)\,$km\,s$^{-1}$\,Mpc$^{-1}$ extracted in arXiv:1604.01424 by appealing to Cepheid calibrated SNe~Ia, new parallax measurements, stronger constraints on the Hubble flow, and a refined computation of distance to NGC4258 from maser data. We briefly comment on a possible interpolation of our high-$z$ model, invoking percolated and unpercolated U(1) topological solitons of a Planck-scale axion field, to the phenomenologically successful low-$z$ $\Lambda$CDM cosmology.