Asymmetric dark matter and effective number of neutrinos

We study the effect of the MeV-scale asymmetric dark matter annihilation on the effective number of neutrinos $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ at the epoch of the big bang nucleosynthesis. If the asymmetric dark matter $\chi$ couples more strongly to the neutrinos $\nu$ than to the photons $\gamma$ and electrons $e^{-}$, ${\mathrm{\Gamma }}_{\chi \gamma ,\chi e}\ll {\mathrm{\Gamma }}_{\chi \nu }$, or $\Gamma_{\chi\gamma, \chi e} \gg \Gamma_{\chi\nu}$, the lower mass limit on the asymmetric dark matter is about $18$ MeV for $N_{\mathrm{e}\mathrm{f}\mathrm{f}}\simeq 3.0$.