Weighing Neutrinos in $f(R)$ gravity

Abstract: We constrain the neutrino properties in $f(R)$ gravity using the latest observations from cosmic microwave background(CMB) and baryon acoustic oscillation(BAO) measurements. We first constrain separately the total mass of neutrinos $\sum m_\nu$ and the effective number of neutrino species $N_{\rm eff}$. Then we constrain $N_{\rm eff}$ and $\sum m_\nu$ simultaneously. We find $\sum m_\nu<0.462 {\rm eV}$ at a $95\%$ confidence level for the combination of Planck CMB data, WMAP CMB polarization data, BAO data and high-$l$ data from the Atacama Cosmology Telescope and the South Pole Telescope. We also find $N_{\rm eff}=3.32^{+0.54}_{-0.51}$ at a $95\%$ confidence level for the same data set. When constraining $N_{\rm eff}$ and $\sum m_\nu$ simultaneously, we find $N_{\rm eff}=3.58^{+0.72}_{-0.69}$ and $\sum m_\nu<0.860{\rm eV}$ at a $95\%$ confidence level, respectively.