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Cosmic constraint on massive neutrinos in viable f(R) gravity with producing LCDM background expansion

Tensions between several cosmic observations were found recently, such as the inconsistent values of $H_{0}$ (or $σ_{8}$) were indicated by the different cosmic observations. Introducing the massive neutrinos in $Λ$CDM could potentially solve the tensions. Viable $f(R)$ gravity producing $Λ$CDM background expansion with massive neutrinos is investigated in this paper. We fit the current observational data: Planck-2015 CMB, RSD, BAO and SNIa to constrain the mass of neutrinos in viable $f(R)$ theory. The constraint results at 95\% confidence level are: $Σm_ν<0.202$ eV for the active neutrino case, $m_{ν, sterile}^{eff}<0.757$ eV with $N_{eff}<3.22$ for the sterile neutrino case. For the effects by the mass of neutrinos, the constraint results on model parameter at 95\% confidence level become $f_{R0}\times 10^{-6}> -1.89$ and $f_{R0}\times 10^{-6}> -2.02$ for two cases, respectively. It is also shown that the fitting values of several parameters much depend on the neutrino properties, such as the cold dark matter density, the cosmological quantities at matter-radiation equality, the neutrino density and the fraction of baryonic mass in helium. At last, the constraint result shows that the tension between direct and CMB measurements of $H_0$ gets slightly weaker in the viable $f(R)$ model than that in the base $Λ$CDM model.