Light dark photon and fermionic dark radiation for the Hubble constant
  and the structure formation
  Motivated by the tensions in the Hubble constant $H_0$ and the structure
growth $\sigma_8$ between $Planck$ results and other low redshift measurements,
we discuss some cosmological effects of a dark sector model in which dark
matter (DM) interacts with fermionic dark radiation (DR) through a light gauge
boson (dark photon). Such kind of models are very generic in particle physics
with a dark sector with dark gauge symmetries. The effective number of
neutrinos is increased by $\delta N_{eff} \sim 0.5$ due to light dark photon
and fermionic DR, thereby resolving the conflicts in $H_0$. The elastic
scattering between DM and DR induces suppression for DM's density perturbation,
but without acoustic oscillations. For weakly-interacting DM around $100$GeV,
the new gauge coupling should be $\sim 10^{-4}$ to have sizable effect on
matter power spectrum in order to relax the tension in $\sigma_8$.
