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A comment on power-law inflation with a dark radiation component

Tram et al. 2016 recently pointed out that power-law inflation in presence of a dark radiation component may relieve the 3.3 sigma tension which exists within standard LCDM between the determination of the local value of the Hubble constant by Riess et al. (2016) and the value derived from CMB anisotropy data by the Planck collaboration. In this comment, we simply point out that this interesting proposal does not help in solving the $σ_8$ tension between the Planck data and, e.g., the weak lensing measurements. Moreover, when the latest constraints on the reionization optical depth obtained from Planck HFI data are included in the analysis, the $H_0$ tension reappears and this scenario looses appeal.