The impact of the locally measured Hubble parameter on the mass of Sterile neutrino

We present a precise analysis to test hypothetical models involving sterile neutrinos beyond the standard flat-$\mathrm{\Lambda }$CDM cosmology with the CMB observations from the $Planck$ mission and BAO measurements. This analysis shows that adding the locally measured Hubble parameter $H_0$ = $73.00\pm 1.75$ km ${\text{s}}^{-1}$ Mp${\text{c}}^{-1}$ to the data removes the need for the informative physical $m_{sterile}^{thermal}$ prior in CMB constraints of $m_{\nu ,sterile}^{eff}$. Under the constraints from the data containing the locally measured $H_0$ we obtain an upper limit $ m_{\nu,sterile}^{eff} < 0.306 $eVscalemassforthemassivesterileneutrino,andanupperlimit$\Sigma m_{\nu} < 0.214$ eV scale mass for the three degenerate massive neutrino (95 per cent confidence level). We also obtain the value ${\sigma }_8$ = $0.81_{-0.06}^{+0.05}$ (95 per cent confidence level), which is in compatibility with the constraints from $Planck$ 2015 CMB data at the 1$\sigma$ level. We find that introducing parameter $m_{\nu ,sterile}^{eff}$ to the model of cosmology reduces the ${\sigma }_8$ value and moves it closer to the obtained value for this parameter from the KiDS-450 analysis. Our results show that the locally measured Hubble parameter can increase constraints on ${\sigma }_8$ values.