Abstract
Tensions between several cosmic observations were found recently, such as the inconsistent values of $H_{0}$ (or $\sigma_{8}$) were indicated by the different cosmic observations. Introducing the massive neutrinos in $\Lambda$CDM could potentially solve the tensions. Viable $f(R)$ gravity producing $\Lambda$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: $\Sigma m_\nu<0.202$ eV for the active neutrino case, $m_{\nu, sterile}^{eff}<0.757$ 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 $\Lambda$CDM model.
Highlights
H0 = 67.3 ± 1.0 is provided by Planck-CMB experiment with an indirect estimate on H0 [1], but the higher value of H0 = 74.3 ± 2.1 is obtained by SST direct measurements of H0 [2]; this tension exists between the Planck-CMB experiment and the rich cluster counts, as they provide the inconsistent value of σ8 [1,3].The studies on these tensions are important, since any evidence of a tension may be useful in the search for new physics
We investigate the behaviors of massive neutrinos in observationally viable f (R) theories with producing the CDM background expansion history
As shown in Ref. [32], the viable f (R) theory which realizes the popular CDM universe at background-dynamics level does not have an analytical expression of f (R) to describe a physical universe from the radiation-dominated epoch to the late-time acceleration, but it really has the analytical solutions of f (R) in different evolutionary epochs of the universe
Summary
[32], the viable f (R) theory which realizes the popular CDM universe at background-dynamics level does not have an analytical expression of f (R) to describe a physical universe from the radiation-dominated epoch to the late-time acceleration, but it really has the analytical solutions of f (R) in different evolutionary epochs of the universe. Ln a Obviously, Eq (5) can partly realize the background expansion as that of the CDM universe, while the cosmological perturbation behaviors in this f (R) model are different from that in the CDM model Given that it is not natural by using two f (R) functions to mimic one total CDM universe, in this paper we consider our universe including two stages: the early universe a < 0.02 (including the radiationdominated epoch and the early stage of the matter-dominated era) is described by the CDM, and the universe a ≥ 0.02. I2, E2 indicate the quadrupole of the photon intensity and the E-like polarization, respectively
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