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.
