Cosmology based on $f(R)$ gravity with ${\cal O}(1)$ eV sterile neutrino
  We address the cosmological role of an additional ${\cal O}(1)$ eV sterile neutrino in modified gravity models. We confront the present cosmological data with predictions of the FLRW cosmological model based on a variant of $f(R)$ modified gravity proposed by one of the authors previously. This viable cosmological model which deviation from general relativity with a cosmological constant $\Lambda$ decreases as $R^{-2n}$ for large, but not too large values of the Ricci scalar $R$ provides an alternative explanation of present dark energy and the accelerated expansion of the Universe. Various up-to-date cosmological data sets exploited include Planck CMB anisotropy, CMB lensing potential, BAO, cluster mass function and Hubble constant measurements. We find that the CMB+BAO constraints strongly the sum of neutrino masses from above. This excludes values $\lambda\sim 1$ for which distinctive cosmological features of the model are mostly pronounced as compared to the $\Lambda$CDM model, since then free streaming damping of perturbations due to neutrino rest masses is not sufficient to compensate their extra growth occurring in $f(R)$ gravity. Thus, we obtain $\lambda>8.2$ ($2\sigma$) with cluster systematics and $\lambda>9.4$ ($2\sigma$) without that. In the latter case we find for the sterile neutrino mass $0.47\,\,\rm{eV}$$\,<\,$$m_{\nu,\,\rm{sterile}}$$\,<\,$$1\,\,\rm{eV}$ ($2\sigma$) assuming the active neutrinos are massless, not significantly larger than in the standard $\Lambda$CDM with the same data set: $0.45\,\,\rm{eV}$$\,<\,$$m_{\nu,\,\rm{sterile}}$$\,<\,$$0.92\,\,\rm{eV}$ ($2\sigma$). However, a possible discovery of a sterile neutrino with the mass $m_{\nu,\,\rm{sterile}} \approx 1.5\,$eV motivated by various anomalies in neutrino oscillation experiments would favor cosmology based on $f(R)$ gravity rather than the $\Lambda$CDM model. 
A. S. Chudaykin, D. S. Gorbunov, A. A. Starobinsky and R. A. Burenin
astro-ph.CO gr-qc hep-th