  We present a precise analysis to test hypothetical models involving sterile
neutrinos beyond the standard flat-$\Lambda$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\pm1.75
$ km $\textrm s^{-1}$ Mp$\textrm 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 $ eV scale mass for the massive sterile neutrino, and an upper limit
$\Sigma m_{\nu} < 0.214$ eV scale mass for the three degenerate massive
neutrino (95 per cent confidence level). We also obtain the value $\sigma_{8} $
= $0.81^{+0.05}_{-0.06} $ (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.
