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 $\sigma_{8}$) 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.


I. Introduction
The base 6-parameter ΛCDM (Λ-Cold-Dark-Matter) model is the most popular one to interpret the accelerating expansion of universe. This model is favored by most "observational probes", though it exists the fine-tune problem and the coincidence problem in theory. However, some tensions were found recently between the cosmic observations when one fitted observational data to this model. For example, the tension is found for estimating the values of H 0 : a lower value of H 0 = 67.3 ± 1.0 is provided by Planck-CMB experiment with an indirect estimate on H 0 [1], but a higher value of H 0 = 74.3 ± 2.1 is obtained by SST direct measurements of H 0 [2]; this tension also exists between the Planck-CMB experiment and the rich cluster counts, as they provide the the inconsistent value of σ 8 [1, 3].
The studies on these tensions are important, since any evidence of a tension may be useful to search new physics.
One possible interpretation to above tension is that the base 6-parameter ΛCDM model is incorrect or should be extended. Ref. [1] shows that, introducing m ν or introducing N ef f solely in ΛCDM model can not resolve the above tensions, but the tensions could be solved in the ΛCDM with including both m ν and N ef f or with including the massive sterile neutrinos m sterile ν,ef f . Here m ν denotes the total mass of three species of degenerate massive active neutrinos, and N ef f denotes the effective number of relativistic degrees of freedom, which relates to the neutrinos and the extra massless species. Combined analysis of cosmic data in other references also indicate the existence of the massive neutrinos, for examples, joint analysis from CMB and BAO (baryon acoustic oscillation) [4,5], from solar and atmospheric experiments [6][7][8], or from the reactor neutrino oscillation anomalies [9,10], etc..
Investigating other scenarios to solve the above tensions and restricting the mass of neutrinos in different scenarios are significative. Ref. [11] shows that possible discovery of sterile neutrino with mass m ef f ν,sterile ≈ 1.5eV , motivated by various anomalies in neutrino oscillation experiments, would favor cosmology based on f (R) gravity rather than the standard ΛCDM. In addition, one knows that plenties of functions f (R) of Ricci scalar R [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] are presented to modify the Einstein's gravity theory, in order to solve puzzles in general relativity. But several forms of f (R) are then found to be nonphysical, since they can not describe the expansion of universe in matter-dominated time [29,30].
So, studies on observationally viable f (R) theories are necessary. One of the viable f (R) theories has been studied in Refs. [31,32], where the f (R) theory can realize the most popular ΛCDM universe at background-dynamics level, while the effects of large scale structure with the cosmological perturbation theory in this f (R) model are different from that in the ΛCDM. In this paper, we investigate the behaviors of massive neutrinos in observationally viable f (R) theories with producing the ΛCDM background expansion history.
II. Viable f(R) gravity theory producing ΛCDM-background expansion The action of f (R) modified gravity theory is written as (1) L u is the Lagrangian density of universal matter including the radiation and the pressureless matter (baryon matter plus cold dark matter). Using the variation principle, one gets f R = df (R) dR , R µν and T µν denote the Ricci tensor and the energy-momentum tensor of universal matter, respectively. For a universe described by metric g µν = diag(−1, a(t) 2 , a(t) 2 , a(t) 2 ), the dynamical evolutionary equations of universe in f (R) theory are As shown in Refs. [32], the viable f (R) theory which realize the popular ΛCDM universe at background-dynamics level does not have an analytical expression of f (R) to describe a physical universe from the radiation-dominate epoch to the late-time acceleration, but it really has the analytical solutions of f (R) in different evolutional epochs of the universe. Concretely, Ref. [32] gives the forms of f (R) in two cases: one describes the evolution of the ΛCDM background from the radiation-dominate epoch to the matter-dominate epoch, and the other one represents the evolution of the ΛCDM background from the matter-dominate era to the future expansion. In this paper we focus on studying the f (R) function with producing ΛCDM background expansion from the matter-dominate epoch to the late-time acceleration 1 , which has the form as follows [31,32] where ̟ = and p + = 5+ √ 73 12 . D is the model parameter in this f (R) modified gravity, which can relate to the current value f R0 and the current value of the Compton wavelength B 0 by where the Compton wavelength is derived by B = fRR fR dR d ln a H dH/d ln a = ∂fR/∂ ln a fR H ∂H/∂ ln a . Obviously, Eq. (5) can partly realize the background expansion as that of the ΛCDM universe, while the cosmological perturbation behaviors in this f (R) model are different from that in ΛCDM model. Given that it is not natural by using two f (R) functions to mimic one total ΛCDM universe, in this paper we consider our universe including two stages: the early universe a < 0.02 (including the radiation-dominate epoch and the early stage of the matterdominate era) is described by the ΛCDM, and the universe a ≥ 0.02 (including the deep matter-dominate epoch and the late-time acceleration) is depicted by the above viable f (R) model.

III. Cosmological perturbations in viable f(R) gravity theory producing ΛCDM-background expansion
The line element with the perturbation reads where γ ij is the three-dimensional spatial metric in the spherical coordinate denotes the conformal 3-space curvature perturbation. The perturbed modified Einstein equations in f (R) theory can be derived as follows [33] The source term of the CMB temperature anisotropy is described by where g = −εe −ε = an e σ T e −ε is the visibility function and ε is the optical depth. ζ is given by ζ = ( 3 4 I 2 + 9 2 E 2 ), where I 2 ,E 2 indicate the quadrupole of the photon intensity and the E-like polarization, respectively.

A. Used data
In this section, we apply the cosmic data to constrain the above viable f (R) model. The used data are as follows.
(1) The CMB temperature and polarization information released by Planck 2015 [1]: the high−l C T T l likelihood (PlikTT), the high−l C EE l likelihood (PlikEE), the high−l C T E l likelihood (PlikTE), the low−l data and the lensing data.
(3) The BAO data: the 6dFGS [43], the SDSS-MGS [44], the BOSSLOWZ BAO measurements of D V = r drag [44] and the CMASS-DR11 anisotropic BAO measurements [44]. Since the WiggleZ volume partially overlaps that of the BOSSCMASS sample, we do not use the WiggleZ results in this paper. 6dFGS denotes the six-degree-Field Galaxy survey (6dFGS) at z ef f = 0.106 [43], SDSS-MGS denotes the SDSS Main Galaxy Sample (MGS) at z ef f = 0.15 [44], BOSSLOWZ denotes the Baryon Oscillation Spectroscopic Survey (BOSS) "LOWZ" at z ef f = 0.32 [44], and CMASS-DR11 denotes the BOSS CMASS at z ef f = 0.57 [44]. The recent analysis of latter two BAO data use peculiar velocity field reconstructions to sharpen the BAO feature and reduce the errors on D V = r drag . The point labelled BOSS CMASS at z ef f = 0.57 shows D V = r drag from the analysis of [45], updating the BOSS-DR9 analysis.
The prior value of Hubble constant H 0 = 100h km s −1 Mpc −1 is usually taken in cosmic analysis, though there are hundreds of measurement value of H 0 and lots of them are mutually inconsistent 2 . Ref. [57] points out that the prior value of the H 0 affects cosmological parameter estimation, but not very significantly. Here we take the HST prior, Constraints on neutrino mass in ΛCDM model or in dynamical dark energy models or in f (R) theory have been discussed in some references [1, [59][60][61][62][63][64]. Given that the constraints on Σm ν (or m sterile ν,ef f ) are model-dependent, we fit the cosmic data to limit the mass of neutrinos in above viable f (R) model by using the MCMC method [65][66][67][68][69][70].
Obviously, extra parameters f R0 and Σm ν (or m sterile ν,ef f with the required N ef f ) are added, relative to the base ΛCDM   [62], and m sterile ν,ef f < 0.61 with N ef f < 3.95 in f (R) model [62]. Obviously, a higher upper limit on m sterile ν,ef f and a lower limit on N ef f are obtained in our study. Some inconsistent results on sterile neutrino mass can also be found, for example, the sterile neutrino mass 0.47eV < m ef f ν,sterile < 1eV (2σ) is given in a f (R) model and 0.45eV < m ef f ν,sterile < 0.92eV is given in ΛCDM model [63], or the active neutrino mass m ν = 0.35 ± 0.10 is presented in ΛCDM model [4]. The constraint results on model parameter in viable f (R) theory are f R0 × 10 −6 > −1.89 for active neutrino case and f R0 × 10 −6 > −2.02 for sterile neutrino case at %95 limit. Though the fitting results on f R0 are affected by the additional parameters Σm ν (or m ef f ν,sterile with N ef f ), for using the Planck 2015 data in this paper it has the more stringent constraint than result given by Ref. [71]: f R0 × 10 −6 = −2.58 +2.14 −0.58 in 1σ regions. Table I also lists the values of six basic cosmological parameters. Ω b h 2 is the current baryon density, Ω c h 2 is the cold dark matter density at present, θ MC denotes the approximation to r * /D A , τ presents the Thomson scattering optical depth due to reionization, ln(10 10 A s ) is the Log power of the primordial curvature perturbations, and n s is the scalar spectrum power-law index. From table I and figure 2, it can be seen that the neutrino properties much more affect the fitting value of cold dark matter density than fitting values of other parameters. This results could be interpreted as follow. Since the massive neutrinos are considered as one kind of dark matter in universe, the mass of neutrino (active or sterile) would directly affect the dimensionless energy density of dark matter. According to the 3 Constraint on total mass of active neutrino are also investigated with including an additional free parameter N ef f in theoretical model, for example, Σmν < 0.826 with N ef f = 3.49 +0. 71 −0.73 [64] and Σmν = 0.533 +0.254 −0.411 with N ef f = 3.78 +0.84 −0.64 [61] are given in the f (R) models.  constraint results on Ω c h 2 and Ω ν h 2 , one can see that the larger uncertainty of Ω c h 2 value is caused by the looser constraint on the dimensionless energy density of sterile neutrino Ω ν h 2 , which maybe reflects the less information on sterile neutrino from cosmic observations. However, the constraint on Ω c h 2 is more strict for the active-neutrino case, since the constraint on the dimensionless energy density of active neutrino Ω ν h 2 is tighter than the case of sterile neutrino, which maybe reflects the more information on the active neutrino from cosmic observations. Except Ω c h 2 , other basic parameters in density, z re is the redshift at which universe is half reionized, t 0 denotes the age of the universe today (in Gyr), z * denotes the redshift for which the optical depth equals unity, r * denotes the comoving size of the sound horizon at z = z * , θ * denotes the angular size of sound horizon at z = z * (r * /D A ), z drag denotes the redshift at which baryondrag optical depth equals unity, r drag denotes the comoving size of the sound horizon at z = z drag , k D denotes the characteristic damping comoving wavenumber (Mpc −1 ), z eq denotes the redshift of matter-radiation equality, Ω ν h 2 is the neutrino density, Y p denotes the fraction of baryonic mass in helium. Obviously, from  [59]. For these constraint results on H 0 , it is also shown that the tension between direct and CMB measurements of H 0 gets slightly weaker in our considered model than that in the base ΛCDM model, where H 0 = 67.6 ± 0.6 is given by Ref.
[1]. In addition, it is found from Fig. 3 that the neutrino properties much affect the fitting value of parameters: z eq , k eq , 100θ s,eq , Ω ν h 2 and Y p , which could be partly explained by the dependency of the parameters on the cold dark matter density and might be useful for testing the neutrino properties in experiments. The values of σ 8 in viable f (R) model are almost the same for the cases of different-species neutrino, and the same result is also suitable for the parameters: f σ 8 , A s e −2τ and θ * , as exhibited in Fig. 3 and Fig. 4.

V. Conclusion
Tensions between several observations were found recently. The studies on tensions are important, since they are useful to search new physics. The massive neutrinos are introduced in cosmological models to solve the tensions concerning the inconsistent values of H 0 (or σ 8 ). Investigating other scenarios to solve these tensions and restricting the mass of neutrinos in different scenarios are significative. Given that several forms of f (R) are found to be nonphysical, we study the viable f (R) gravity with the massive neutrinos 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 and m ef f ν,sterile < 0.757 eV with N ef f < 3.22 for the sterile neutrino case, which are comparable with some other results. For the effects by the mass of neutrinos, the constraint results on model parameter become f R0 × 10 −6 > −1.89 and f R0 × 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 Ω c h 2 , the cosmological quantities at matter-radiation equality: z eq , k eq and 100θ s,eq , the neutrino density Ω ν h 2 and the fraction of baryonic mass in helium Y p . 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 ΛCDM model.
is supported by the National Natural Science Foundation of China (11645003,11475143,11575075). [