Towards a Solution to the H0 Tension: the Price to Pay

The tension between current expansion rate H$_0$ using Planck data and direct model-independent measurements in the local universe has reached a tension above 5$\sigma$ in the context of the $\Lambda$CDM model. The growing tension among early time and local measurements of H$_0$ has not ameliorated and remains a crucial and open question in cosmology. Solutions to understand this tension are possible hidden sources of systematic error in the observable measurements or modifications to the concordance $\Lambda$CDM model. In this work, we investigate a solution to the H$_0$ tension by modifying $\Lambda$CDM and we add at early times extra relativistic energy density $\rho_{ex}$. In some context this $\rho_{ex}$ corresponds to EDE or Bound Dark Energy. We implement our model in CAMB and perform a full COSMO-MC (MCMC) allowing to simultaneously fit the latest data from CMB anisotropies and the value of H$_0 = 74.03\pm 1.42,\mathrm{km\, s^{-1}Mpc^{-1}}$ from A. Riess [R-19] \ci{Riess:2019cxk}. The inclusion of $\rho_{ex}$ ameliorates the tension between early and late time measurements only slightly and we obtain a value H$_0 = (68.70\pm 0.45$\ksm) still in conflict with local measurements [R-19]. We follow up our analysis by proposing two forecasting standard deviation $\sigma_H=1$ and $\sigma_H=0.5$ (in units of \ksm) for local distance measurements H$_0 = (74.03\pm \sigma_H)$ \ksm. From MCMC we obtain a value of H$_0 =(72.83\pm 0.47)$ \ksm\, at 68\% confidence level for $\sigma_H =0.5$, fully consistent with [R-19], while the price to pay is a percentage increase of $0.12\%$ in CMB $\chi_{cmb}^2$. Finally, the extra energy density $\rho_{ex}$ leaves distinctive imprints in the matter power spectrum at scales $k \sim k_c$ with $k_c=a_cH(a_c)$ and in the CMB power spectrum, allowing for independent verification of our analysis.