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

Abstract

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$σ$ 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 $m_{ν,{\rm{sterile}}}^{\rm{eff}}<0.62$ eV (2$σ$) and the effective number of relativistic species $N_{\rm eff}<3.90$ (2$σ$) in the $f(R)$ model. As a comparison, we also obtain $m_{ν,{\rm{sterile}}}^{\rm{eff}}<0.56$ eV (2$σ$) and $N_{\rm eff}<3.92$ (2$σ$) in the standard $Λ$CDM model.

Figures (3)

• Figure 1: The scale-dependent linear growth rate $f_m(k,z)$ at $z=0$ for $B_0=10^{-4}$ (pink dashed), $10^{-5}$ (red dot), and $10^{-6}$ (violet dashed dot) of $f(R)$ model, and $m_{\nu,{\rm{sterile}}}^{\rm{eff}}=0.5$ 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 $\Lambda$CDM model. Note that we fix $\Omega_m$ at the same values for all the curves, and $N_{\rm eff}=4.046$ for the curves of sterile neutrino. It is clear that $\Lambda$CDM model gives a scale-independent growth rate, while both $f(R)$ and sterile neutrino predict scale-dependent growth rates. Besides, a larger $B_0$ tends to enhance the growth rate more, while $m_{\nu,{\rm{sterile}}}^{\rm{eff}}$ affects the growth rate in the opposite way.
• Figure 2: 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.
• Figure 3: The one-dimensional posterior distributions and two-dimensional marginalized contours (68% and 95% CL) for the $f(R)$+$m_{\nu,{\rm{sterile}}}^{\rm{eff}}+N_{\rm eff}$ model from the CMB+BAO+PV and CMB+BAO+PV+RSD data combinations.