Probing $f(R)$ cosmology with sterile neutrinos via measurements of scale-dependent growth rate of structure

In this paper, we constrain the dimensionless Compton wavelength parameter $B_0$ of $f(R)$ gravity as well as the mass of sterile neutrino by using the cosmic microwave background observations, the baryon acoustic oscillation surveys, and the linear growth rate measurements. Since both the $f(R)$ model and the sterile neutrino generally predict scale-dependent growth rates, we utilize the growth rate data measured in different wavenumber bins with the theoretical growth rate approximatively scale-independent in each bin. The employed growth rate data come from the peculiar velocity measurements at $z=0$ in five wavenumber bins, and the redshift space distortions measurements at $z=0.25$ and $z=0.37$ in one wavenumber bin. By constraining the $f(R)$ model alone, we get a tight 95\% upper limit of $\log_{10}B_0<-4.1$. This result is slightly weakened to $\log_{10}B_0<-3.8$ (at 2$\sigma$ level) once we simultaneously constrain the $f(R)$ model and the sterile neutrino mass, due to the degeneracy between the parameters of the two. For the massive sterile neutrino parameters, we get the effective sterile neutrino mass $m_{\nu,{\rm{sterile}}}^{\rm{eff}}<0.62$ eV (2$\sigma$) and the effective number of relativistic species $N_{\rm eff}<3.90$ (2$\sigma$) in the $f(R)$ model. As a comparison, we also obtain $m_{\nu,{\rm{sterile}}}^{\rm{eff}}<0.56$ eV (2$\sigma$) and $N_{\rm eff}<3.92$ (2$\sigma$) in the standard $\Lambda$CDM model.

One of the most profound puzzles in contemporary physics is the cause for the current cosmic acceleration [1,2].Although the mainstream for explaining this latetime cosmic acceleration is by introducing an extra dark energy fluid under the framework of general relativity (a typical dark energy candidate is cosmological constant Λ, corresponding to a ΛCDM cosmology) [3][4][5], other possibility like the modification to general relativity on large scales also attracts more and more attention.A simple modified gravity scenario is the so-called f (R) theory which generates acceleration via adding a function of the Ricci scalar R to the Einstein-Hilbert action [6,7].For recent reviews of f (R) theory, see, e.g., Refs.[8][9][10][11][12].
Generally, f (R) theory introduces an extra scalar degree of freedom f R ≡ df /dR characterized by a squared Compton wavelength proportional to f RR ≡ d 2 f /dR 2 , or by a dimensionless quantity [13] where H = ȧ/a is the Hubble parameter, a is the scale factor, and a dot denotes the derivative with respect to the cosmic time t.Due to the existence of this extra scalar degree of freedom, f (R) models tend to enhance the growth of the matter perturbations below the Compton wavelength scale.Thus f (R) models generally predict higher and scale-dependent growth rates compared with the standard ΛCDM model.By accurately measuring the growth rate of large-scale structure, we therefore have a powerful tool to constrain f (R) models.
A direct way of measuring the large-scale growth rate comes from redshift-space distortions (RSD) in galaxy surveys [14,15], which depict the distortions of the measured galaxy maps in redshift space.Such distortions arise from the peculiar velocities of galaxies, which actually relate to the evolution of matter perturbation δ m .Thus RSD provide information about the large-scale growth rate f m = d ln δ m /d ln a or the bias-independent growth rate f m σ 8 (z) with σ 8 (z) the root-mean-square mass fluctuation in spheres with radius 8h −1 Mpc at redshift z. 1 Recent galaxy surveys have measured f m σ 8 (z) in different scale ranges with precision up to 7% [16][17][18][19][20][21].Some works also have used these data to constrain some specific f (R) models [22,23].
Nevertheless, current RSD models used to extract f m from the power spectrum of galaxy survey always assume a scale-independent f m on all scales, which is true for the standard ΛCDM model but not valid for the f (R) cosmology.To see clearly, we plot the general k-dependent growth rate f m (k, z) at z = 0 in Fig. 1 for the standard ΛCDM and f (R) models with B 0 = 10 −4 , 10 −5 and 10 −6 , where k is the comoving wavenumber and the subscript "0" denotes the current value of the corresponding quantity.The plotting range for k covers most of current RSD surveys' scales.It is clear that the growth rate in the f (R) model significantly deviates from the scaleindependence.Particularly for the B 0 = 10 −4 case, the deviation arrives about 14% at k = 0.3h Mpc −1 , even larger than current RSD surveys' precisions.So it will 1 Note that, in practice, σ 8 (z) is computed in linear regime.Although the expression, , where r = 8h −1 Mpc and P (k, z) is the matter power spectrum, contains an integral on all scales, the top-hat window function, W (kr) = 3(sin kr−kr cos kr)/(kr) 3 , decays rapidly when k is beyond the linear regime, and thus the nonlinear modes contribute very little to the integral.
( 1 0 -4 , 0 ) ( 1 0 -5 , 0 ) ( 1 0 -6 , 0 ) ( 0 , 0 ) ( 0 , 0 .5 e V ) ( 0 , 1 .0 e V ) ( B 0 , m e f f ν,s t e r i l e ) FIG. 1: The scale-dependent linear growth rate fm(k, z) at z = 0 for B0 = 10 −4 (pink dashed), 10 −5 (red dot), and 10 −6 (violet dashed dot) of f (R) model, and m eff ν,sterile = 0.5 eV (olive dashed dot dot) and 1.0 eV (blue short dashed) of sterile neutrino.The black solid line denotes the growth rate for the standard ΛCDM model.Note that we fix Ωm at the same values for all the curves, and N eff = 4.046 for the curves of sterile neutrino.It is clear that ΛCDM model gives a scaleindependent growth rate, while both f (R) and sterile neutrino predict scale-dependent growth rates.Besides, a larger B0 tends to enhance the growth rate more, while m eff ν,sterile affects the growth rate in the opposite way.
induce substantial systematic errors for the fit results when we simply use all the RSD data to constrain the f (R) models without careful selection according to their measured scales.Actually, by performing N-body simulations, Ref. [24] illustrated that the measured growth rate from RSD surveys agrees with that predicted in f (R) model only for k < 0.06h Mpc −1 at z = 0.
To avoid such systematic errors, we constrain f (R) model using the scale-dependent growth rate data in this work.Recently, Ref. [25] tried to measure the scale-dependent growth rate by using the observations of peculiar velocities of galaxies from 6dF Galaxy Survey velocity sample in combination with a newly-compiled sample of low-redshift type Ia supernovae.The measurement obtained the scale-dependent growth rates in five k bins at z = 0: f m σ 8 (k) = 0.79 ± 0.21, 0.30 ± 0.14, 0.32 ± 0.19, 0.64 ± 0.17, and 0.48 ± 0.22, for k ∈ [0.005, 0.02], [0.02, 0.05], [0.05, 0.08], [0.08, 0.12], and [0.12, 0.15]h Mpc −1 , respectively. 2Hereafter, we use PV to denote these five data points.Besides, we also use the growth rate measurements f m σ 8 (z) = 0.35 ± 0.06 2 The original data are not gaussian.For the purpose of using these data to constrain f (R) model, we choose the upper limits of the data as the global errors.The growth rates are measured in a series of ∆k ∼ 0.03h Mpc −1 bins in Ref. [25].We assume that each data point is measured independently.We calculate the theoretical value of fmσ 8 (k i ) in each bin at k i = k cen i with k cen i the centre of the ith bin, and then compare it with the measurement value of the ith bin.
at z = 0.25 and f m σ 8 (z) = 0.46 ± 0.04 at z = 0.37 from SDSS DR7 [19].These two data points actually belong to the RSD measurement but are obtained from the power spectrum with k ∈ [0.005, 0.033]h Mpc −1 .In this scale range, the growth rate in f (R) model can be approximatively considered scale-independent from Fig. 1.So it is appropriate to use these two data to constrain the f (R) model.Hereafter, we use RSD to denote these two data points.
A successful f (R) model should give a stable acceleration solution at late times and also pass the local solar system tests.Several viable f (R) models that satisfy these conditions have been proposed in the literature [26][27][28][29][30][31][32][33][34].However, we only care about the large-scale perturbations of f (R) models in this paper.So we consider a parametrized f (R) model whose background is the same as that of ΛCDM and perturbations are parametrized by two functions [35,36], where the parameter λ 1 has dimension of length, which can be related to the current value of B via λ 2 1 = B 0 c 2 /(2H 2 0 ) with c the speed of light.The functions µ(k, a) and γ(k, a) are used in the public MGCAMB code [37,38] to quantify the modifications to the Poisson and anisotropy equations: (4) where Ψ and Φ are two metric perturbation potentials, ∆ is the comoving density perturbation, and σ denotes the anisotropic stress.It was showed that Eq. ( 2) together with Eqs. ( 3) and (4) can well model the large scale growth in f (R) models for B 0 1 [36].In our paper, we will constrain B 0 using above-mentioned growth rate data.Since the growth rate is very sensitive to the order of magnitude of B 0 , we will take log 10 B 0 as a free parameter.
Besides the value of B 0 of f (R) model, another interest of this work is the mass of massive neutrinos.The total mass of massive active neutrinos in a f (R) universe has been widely reported in previous works, e.g., Refs.[38][39][40][41][42].So we here focus on the mass of massive sterile neutrinos.Recently, the extra massive sterile neutrino species has attracted much attention because of its magic power of simultaneously alleviating the tensions between different observations [43][44][45][46][47][48][49][50][51][52].To describe the mass of sterile neutrino, we need two extra free parameters, the effective sterile neutrino mass m eff ν,sterile and the effective number of relativistic species N eff .With these two parameters, the true mass of a thermally-distributed sterile neutrino reads m thermal sterile = (N eff − 3.046) −3/4 m eff ν,sterile .In order to avoid a negative m thermal sterile , N eff must be larger than 3.046 in a universe with sterile neutrinos.The mass of sterile neutrino also corresponds to a characteristic scale, below which sterile neutrino is free-streaming and tends to suppress the growth of structure.Thus sterile neutrino can also lead to a scale-dependent growth rate (see Fig. 1).There might exist a degeneracy between B 0 and the mass of sterile neutrino when we use the growth rate data to simultaneously constrain the two.We will constrain the parameters of sterile neutrino in the f (R) model and verify whether this degeneracy exists or not.Our calculation is based on the public Markov-Chain Monte-Carlo package Cos-moMC [53].
The free parameter vector is: {ω b , ω c , θ MC , τ, n s , ln(10 10 A s ), log 10 B 0 , m eff ν,sterile , N eff }, where ω b , ω c , and H 0 denote the background physical baryon density, physical cold dark matter density, and Hubble constant today, respectively, θ MC is approximation to the angular size of the sound horizon at the time of last-scattering, τ denotes the optical depth to reionization, and ln(10 10 A s ) and n s are the amplitude and the spectral index of the primordial scalar perturbation power spectrum.Following Ref. [40], we set a prior [−10, 1] for log 10 B 0 .The priors of other free parameters are the same as those used by Planck Collaboration [54].For massive active neutrino species, we assume its total mass m ν = 0.06 eV.To constrain other cosmological parameters and break degeneracies between them, we also employ the observational data from cosmic microwave background (CMB) and baryon acoustic oscillation (BAO) observations.For the CMB data, we use the temperature power spectrum data C T T [54] and lensing data C φφ [55] from Planck 3 in combination with the WMAP 9-yr polarization (TE and EE) power spectrum data [56].For the BAO data, we use the measurements from 6dFGS (z = 0.1) [57], SDSS DR7 (z = 0.35) [58], WiggleZ (z = 0.44, 0.60, and 0.73) [59], and BOSS DR11 (z = 0.32 and 0.57) [60].Note that the late-time integrated Sachs-Wolfe (ISW) effect and the lensing signal from the CMB measurements can also give constraint on B 0 , since B 0 affects the evolutions of the metric perturbation potentials Φ and Ψ from Eqs. ( 2)-( 4), and the ISW effect and CMB lensing potential are directly related to Φ + Ψ and Φ + Ψ, respectively.
First, we constrain the f (R) parameter B 0 without considering the extra sterile neutrino species.The main fit results are shown in Table I and Fig. 2. By using the CMB+BAO data and the scale-dependent growth rate PV data, we obtain the 95% upper limit of log 10 B 0 < −2.7.Here note that the main contribution to the constraint result of B 0 comes from the PV data, since current CMB+BAO data (with the CMB lensing measurement) The one-dimensional posterior distributions and two-dimensional marginalized contours (68% and 95% CL) for the f (R) model from the CMB+BAO+PV and CMB+BAO+PV+RSD data combinations.
can only give a weak constraint on B 0 , e.g., B 0 < 0.12 at 2σ level [61].From Fig. 2, we can see a weak peak for log 10 B 0 at about −4 from the CMB+BAO+PV data.A peak with a nonzero B 0 means that a relatively large growth rate is more favored by the data.The PV data in the first and fourth k bins may dominate the appearance of this peak, since the growth rates measured in these two bins are noticeably large.However, this peak for log 10 B 0 disappears once we further add the RSD data into our analysis.Besides, the 95% upper limit of log 10 B 0 is improved to log 10 B 0 < −4.1 with the help of the RSD data.These changes result from the fact that the RSD data favor a lower growth rate and have higher precision than the PV data.
Our fit result for B 0 from the CMB+BAO+PV+RSD data is comparable to the limit, log 10 B 0 < −4.07 (at 2σ level), quoted in Ref. [40], where the WiggleZ galaxy power spectrum with k max = 0.2h Mpc −1 is used, and is also comparable to the limit from the recent Planck 2015 result, log 10 B 0 < −4.01 (at 2σ level), obtained by combining the RSD data at z = 0.57 [62].These results represent the overall levels of constraint on the parametrized f (R) model that can be achieved from the current information on linear scales.A tighter constraint on f (R) model can be obtained if the probes on smaller scales are utilized, such as the Planck Sunyaev-Zeldovich (PSZ) cluster counts [63], the strong lensing galaxy signals [64], the distance indicators [65], and the fifth-force effects on diffuse dwarf galaxy components [66,67].However, it is more difficult to control the possible systematic errors of these small-scale measurements.For example, the tension between the Planck temperature power spectrum measurement and the PSZ cluster counts may be a hint of some unknown systematic errors existing in the PSZ measurement.Besides, the physics of f (R) theory on nonlinear scales is also complicated.So we do not incorporate the information on small scales in our analysis.Next, we consider the cosmological constraints on f (R) parameter in a universe with sterile neutrinos.The fit results from the CMB+BAO+PV and CMB+BAO+PV+RSD data combinations are shown in Table I and Fig. 3. From Fig. 3, we also see a peak for log 10 B 0 at about −4 for the CMB+BAO+PV data.Comparing Fig. 3 with Fig. 2, we find that there indeed exists a slight degeneracy between the f (R) parameter and the sterile neutrino parameters.Due to this degeneracy, the 95% upper limits of log 10 B 0 are enlarged to log 10 B 0 < −1.8 and < −3.8 for the CMB+BAO+PV data and the CMB+BAO+PV+RSD data, respectively.For the sterile neutrino parameters, we get m eff ν,sterile < 0.61 eV and N eff < 3.95 (2σ) from the CMB+BAO+PV data, and m eff ν,sterile < 0.62 eV and N eff < 3.90 (2σ) from the CMB+BAO+PV+RSD data.To see how the f (R) gravity affects the fit value of the sterile neutrino mass, we also constrain the sterile neutrino parameters in the ΛCDM universe by using the same observational data.The main fit results are also shown in Table I.The CMB+BAO+PV data give m eff ν,sterile < 0.43 eV and N eff < 3.96 (2σ) in the ΛCDM universe.When the RSD data are also taken into account, the constraint results become m eff ν,sterile < 0.56 eV and N eff < 3.92 (2σ).We can find that the 95% upper limits of the effective sterile neutrino mass are also enlarged in the f (R) gravity, which also results from the degeneracy between the f (R) parameter and the sterile neutrino parameters.
Current cosmic acceleration can result from either a dark energy fluid or a modification to gravity on large scales.Among various modified gravity theories, the so-called f (R) theory is a simple one.The key feature of the f (R) theory that differs from the standard ΛCDM model is that f (R) models generally predict higher and scale-dependent growth rates.In this paper, we focus on a parametrized f (R) model, whose power of modification to the large-scale structure growth is described by a dimensionless Compton wavelength parameter B 0 .We constrain log 10 B 0 by using the CMB data,  the BAO data, and the linear growth rate data.To avoid the possible systematic errors when constraining the f (R) model from the growth rate data assumed to be scale-independent on all the survey scales, we utilize the scale-dependent growth rate data measured in several different wavenumber bins.They are the PV measurements of f m σ 8 (k, z = 0) in five wavenumber bins, k ∈ [0.005, 0.02], [0.02, 0.05], [0.05, 0.08], [0.08, 0.12] and [0.12, 0.150]h Mpc −1 .We also use the RSD measurements of f m σ 8 (z) at z = 0.25 and z = 0.37 in one wavenumber bin k ∈ [0.005, 0.033]h Mpc −1 .In each wavenumber bin, the theoretical growth rate predicted by the f (R) model can be approximatively considered scale-independent.By using the CMB+BAO+PV+RSD data, we get a tight 95% upper limit of log 10 B 0 < −4.1.
We also extend our discussions to a universe with sterile neutrino species.Unlike the case in f (R) gravity, the extra sterile neutrino tends to suppress the growth of structure below its free-streaming scale.We find a slight degeneracy between the f (R) parameter and the sterile neutrino parameters.Due to this degeneracy, the constraint on the f (R) parameter is slightly weakened to log 10 B 0 < −3.8 at 2σ level from the CMB+BAO+PV+RSD data.For the massive sterile neutrino parameters, we get m eff ν,sterile < 0.62 eV and N eff < 3.90 (2σ).As a comparison, we also obtain m eff ν,sterile < 0.56 eV and N eff < 3.92 (2σ) in the standard ΛCDM model.

TABLE I :
The mean values and 1σ errors for the main parameters of the f (R) model, the ΛCDM+m eff ν,sterile + N eff model, and the f (R)+m eff ν,sterile +N eff model from the CMB+BAO+PV and CMB+BAO+PV+RSD data combinations.For the parameters m eff ν,sterile , N eff , and log 10 B0, we show their 95% upper limits and mean values in brackets.Note that the mass of sterile neutrino m eff ν,sterile is in unit of eV and the Hubble constant H0 is in unit of km s −1 Mpc −1 .