Weighing neutrinos in $f(R)$ gravity in light of BICEP2

Abstract: We constrain the neutrino mass in $f(R)$ gravity using the latest observations from the Planck, BAO and BICEP2 data. We find that the measurement on the B-modes can break the degeneracy between the massive neutrinos and the $f(R)$ gravity. We find a non-zero value of the Compton wavelengths $B_{0}$ at a $68\%$ confidence level for the $f(R)$ model in the presence of massive neutrinos when the BICEP2 data is used. Furthermore, the tension on the tensor-to-scalar ratios between the measured values from Plank and BICEP2 is significantly reconciled in our model.